Hermite interpolation Utilities

2b29d242-2db2-42cb-b348-fe2a3808e791 123 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

t0 1 time :

tWS 1 time :

tWS 8 min ≡

Example 1 . . . Hermite cubic tensors

]]>

data 0 0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 1 0.6 0.167 0.7 0.048 0.8 0.022 0.9 0.012 1 0.00794 112 mat :

X data 1 col : Y data 2 col : 12 mat support data . 14 black plot : 4 appVersion 0.98.6179.21440 ≡ 31 mat

n data rows : 11

Initial end points tensors f0 0.00875 : fn 0.00875 - : 12 mat

i 1 n range deriv i el f0 i 1 ≡ fn i n ≡ 0.5 Y i 1 + el Y i el - X i 1 + el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* cases : for

Local derivative tensors deriv 5 el 2.5 : deriv 7 el 2.5 - : 12 mat deriv 3 el 0.075 : deriv 9 el 0.075 - : 12 mat deriv 10 el 0.1 - : 31 sys

store/isolate deriv deriv eval :

i 1 n 1 - (range b i el 3 Y i 1 + el * X i 1 + el X i el - (deriv i 1 + el * - 2 deriv i el * X i 1 + el X i el - (* - 3 Y i el * - X i 1 + el X i el - (2^/ : for

i 1 n 1 - (range a i el deriv i 1 + el X i 1 + el * deriv i 1 + el X i el * - 2 Y i 1 + el * - deriv i el X i 1 + el * + deriv i el X i el * - 2 Y i el * + X i 1 + el X i el - (3^/ : for

Hermite cubic scalar
piecewise segments j x H a j el x X j el - (3^* b j el x X j el - (2^* + deriv j el x X j el - (* + Y j el + :

j 5 : jj 6 : 12 mat

Scalar algo style interpolation 
over all 'J' segment(s) x Xdata J search index n n 1 - : x Xdata 2 el ≤ n 1 : x Xdata n el ≥ n n 1 - : resume next if if i 1 n 1 + (range x Xdata i el ≤ break close if for J i 1 - (: 51 line :

j x H jj x H 0 0.00794 . 14 black 0.1 0.012 . 14 black 0.2 0.022 . 14 black 0.3 0.048 . 14 black 0.4 0.167 . 14 black 0.5 1 . 14 black 0.6 0.167 . 14 black 0.7 0.048 . 14 black 0.8 0.022 . 14 black 0.9 0.012 . 14 black 1 0.00794 . 14 black 11 5 mat 3 1 sys

J JJ 5647382952 mat

local segment j 6 :

x λ 10 x ≤ (x 1 ≤ (& cases :

u Interp u X J u H : x f X Y x cinterp : 21 sys

solve wisely from piecewise segments U 0.5 :

j x H U - 0 ≡ x 0.375 0.5 solve 0.458

Wise Hermite Cubic rescues freak cinterp ...

j x H x Interp x λ * x f x λ * 0 0.00794 . 14 black 0.1 0.012 . 14 black 0.2 0.022 . 14 black 0.3 0.048 . 14 black 0.4 0.167 . 14 black 0.5 1 . 14 black 0.6 0.167 . 14 black 0.7 0.048 . 14 black 0.8 0.022 . 14 black 0.9 0.012 . 14 black 1 0.00794 . 14 black 11 5 mat 4 1 sys

U 0.375 :

U Interp 1 U - Interp 21 mat 0.1166484375 0.1166484375 21 mat

deriv 0.00875 0.0703 0.075 0.725 2.5 0 2.5 - 0.725 - 0.075 - 0.1 - 0.00875 - 111 mat

a 0.215 - 5.47 - 28 84.5 1416 - 1416 84.5 - 28 - 2.5 2.755 - 101 mat

b 0.34 0.844 0.95 - 3.8 - 199.9 224.9 - 21.55 7.45 0.5 - 0.8695 101 mat

in-situ evaluation 0.25 Interp 0.026875

1 time t0 - 5 s *

CONCLUSION ... 1/1 Mathcad 11 [jmG] Smath rejects FD derivative ... accepts infinitesimal derivative. One may be able to optimize the tensors up to a pleasant 2nd derivative. Not so bad as offered from gusstimated tensors ... The alternate option is the Akima 'ainterp' ...

x f x Interp :

h 0.1 :

x h D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / :

x h D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / :

x h D1 x λ * 0.5 10 0.5 10 - 2 2 mat 2 1 sys x h D2 x λ * 0.5 200 0.5 250 - 2 2 mat 2 1 sys

Example 2 . . . Hermite cubic interpolation

]]>

t0 1 time :

XY 366.59 4990 375.88 4882 380.525 4441.6 381.419 4036.8 42 mat :

user original polygon Xo 366.59 375.88 380.525 381.419 14 mat transpose : Yo 49904882 4441.6 4036.8 14 mat transpose : 21 sys

recast the data for easy handling plot Xo Xo Xo 1 el - : Yo Yo Yo 4 el - : 12 mat

populate reasonably for export t 01 0.125 range :

Hermite interpolation Utilities

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

data Strip nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

r 2 : X Xo r RepEnd : Y Yo r RepEnd : 31 sys

x c 10 x ≤ (x 1 ≤ (& cases :

Hermite interpolate t P 16 / t 3^- 3 t 2^* + 2 t * - 3 t 3^* 6 t 2^* - 3 t * - 6 + 3 t 3^* - 3 t 2^* + 6 t * + t 3^t - 41 mat * :

Hermite basis cubics 0...1

x P 1 el x c * x P 2 el x c * x P 3 el x c * x P 4 el x c * 4 1 sys

t j XP t P 1 el X j el * t P 2 el X j 1 + el * + t P 3 el X j 2 + el * + t P 4 el X j 3 + el * + : t j YP t P 1 el Y j el * t P 2 el Y j 1 + el * + t P 3 el Y j 2 + el * + t P 4 el Y j 3 + el * + : 21 sys

U j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hx i ct el t i el j XP eval : for ct ct 1 + : 21 line for Hx 41 line :

V j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hy i ct el t i el j YP eval : for ct ct 1 + : 21 line for Hy 41 line :

collect the project xy U Unwrap V Unwrap augment : contour xy Strip : contour contour t rows contour rows t rows 2 - (- (range 12 range el : 31 sys

contour 11 mat 0 953.2 0.939 950.2082 1.9959 945.0906 3.1435 937.409 4.3547 926.725 5.6021 912.6004 6.8586 894.5969 8.097 872.2762 9.29 845.2 10.1063 800.7812 10.8518 751.8875 11.5282 699.2375 12.1372 643.55 12.6807 585.5438 13.1603 525.9375 13.5778 465.45 13.935 404.8 14.1933 344.6816 14.3986 285.8406 14.5564 228.998 14.6723 174.875 14.752 124.1926 14.8009 77.6719 14.8247 36.034 14.829 0252 mat 11 mat

a b c spec . 12 black 13 mat : data contour a b c spec pts : XoYo Xo Yo augment + 20 black plot : 31 line

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

vx contour 1 col : vy contour 2 col : 12 mat x f vx vy x cinterp : x λ 1 Xo 1 el x ≤ (x Xo 4 el ≤ (& cases : x spline Xo Yo x cinterp : x Akima Xo Yo x ainterp : 51 mat

contour contour rows row 14.829 012 mat

x f x λ * x spline x λ * x Akima x λ * XoYo data 5 1 sys

1 time t0 - 5 s *

XYdata Xo Yo augment :

Nice plot ▼ digitize

XYdata XYdata 2 1 sys

blue ► HermiteCubic red ► freak cinterp

Example 3 . . . Short form Quadratic Holistic . . .

]]> Tridiagonal System: [A, S]

Matrix whose inverse will solve for the quadratic coeff's X A n X rows : i 13 n 1 - (* (range j 13 n 1 - (* (range M i j el 0 : for for i 1 n 1 - (range j 12 range k 13 range M 2 i * 1 - j + (3 i * 3 - k + (el X i 1 - j + el (k 1 -^: for for for i 2 n 1 - (range j 12 range k 12 range M 2 n 1 - (* i + 3 i * 8 - k + j 3 * + (el 1 - (j^- (2 X i el * (k 1 -^* : for for for M 13 el 1 : M 61 line :

X Y S n X rows : i 13 n 1 - (* (range M i el 0 : for i 1 n 1 - (range j 01 range M 2 i 1 + (* 2 - j + el Y i j + el : for for M 41 line :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Quadratic/Holistic adapted Mathcad 11 jmG: 2021-05-16

data 1015 -^0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 162 mat :

X data 1 col : Y data 2 col : 12 mat support data . 14 black plot : 21 mat

4 appVersion 0.98.6179.21440 ≡

A X A :

S X Y S :

x f C 1 el x C 2 el C 3 el x * + (* + C 4 el x C 5 el C 6 el x * + (* + C 7 el x C 8 el C 9 el x * + (* + C 10 el x C 11 el C 12 el x * + (* + C 13 el x C 14 el C 15 el x * + (* + 51 mat :

C A 1 -^S * :

A 1 -^S * sometimes needed ≡ 11 line

isolate C C eval :

C rows 15

X data 1 col : Y data 2 col : 12 mat x Csp X Y x cinterp : x Asp X Y x ainterp : 31 sys

x y axes x y / - :

Collect piecewise plot for solving ...

x f 1 el x f 2 el x f 3 el x f 4 el x f 5 el data 0.4584 0.5 1 2 mat axes 8 1 sys

x h 10 x ≤ (x 0.5 ≤ (& cases :

Scalar companion interpolator ω 0 : t Q n X rows : i 1 n 1 - (range t X i 1 + el ≤ (t X i el ≥ (& ω 0 : j 02 range ω ω C 3 i * 2 - j + el t j^* + : for 21 line out range if for t Q ω : 31 line : 21 line

x f 5 el 0.5 - 0 ≡ x 0.375 0.5 solve 0.4584

x Q x h * x Csp x h * 0 0.0079 . 14 black 0.1 0.012 . 14 black 0.2 0.022 . 14 black 0.3 0.048 . 14 black 0.4 0.167 . 14 black 0.5 1 . 14 black 6 5 mat 3 1 sys

PROJECT NOTE ... 1. In this plot at right, plot Asp(x) the Akima spline specific to Smath. 2. Another method exist: Hermite Cubic Draftsman spline ... Top document.

Source code to segments  f(x)... above α u f C α el u C α 1 + el C α 2 + el u * + (* + :

preset longer Holistic C 1 el x C 2 el C 3 el x * + (* + C 4 el x C 5 el C 6 el x * + (* + C 7 el x C 8 el C 9 el x * + (* + C 10 el x C 11 el C 12 el x * + (* + C 13 el x C 14 el C 15 el x * + (* + C 16 el x C 17 el C 18 el x * + (* + C 19 el x C 20 el C 21 el x * + (* + # 81 mat

Export the reduced segments ... Optimiz NONE 1 u f 4 u f 7 u f 10 u f 13 u f 51 mat 39750000 / u 2035000 / 0 u * + (* + 6940000000003500000000000000 / u 195500000001125000000000000 / - 5940000000008191000000000000000 / u * + (* + 15179999999991500000000000000 / u 2429999999990911000000000000000 / - 2514999999999325000000000000 / u * + (* + 85785000000003125000000000000 / u 288487499999976250000000000 / - 8294000000000310000000000000 / u * + (* + 23640499999999725000000000000 / u 48465399999999310000000000000 / - 63105999999999310000000000000 / u * + (* + 51 mat

Example 3 . . . Quadratic interpolation

]]>

t0 1 time :

X 1 1.5 2 2.5 41 mat :

Y 1.3 3.1 5.2 0.8 41 mat :

NOTE ... This module is generous in three counts: 1. It complies with the symbolic expansion. 2. The user front end shapes the all quadratic. 3. It solves from the quad(x)... if/otherwise.

user predict front end shape user 1 :

D 1 el user : i 2 X rows range D i el 2 Y i el Y i 1 - el - X i el X i 1 - el - / (* D i 1 - el - : for 21 line

D 1 6.2 2.2 19.8 - 41 mat

t Clear 1

Quadratic interpolation components for symbolic end use. i 2 X rows range f i 1 - el D i el 2 / t X i 1 - el - (2^X i el X i 1 - el - (/ * D i 1 - el 2 / t X i el - (2^X i el X i 1 - el - (/ * - Y i el + D i el 2 / X i el X i 1 - el - (* - : for t f f : 21 line

Quadratic-Derivative components for symbolic end use. i 2 X rows range d i 1 - el D i el 2 / t X i 1 - el - (2^X i el X i 1 - el - (/ * D i 1 - el 2 / t X i el - (2^X i el X i 1 - el - (/ * - Y i el + D i el 2 / X i el X i 1 - el - (* - (t diff : for t D d : 21 line

Piecewise Quadratic t f 62121 - t + (2^* + (* - 53 - 2 t * + (2^* + 31 + 20 / - 426312 - t + (2^* - (* 113 - 2 t * + (2^* + 11 - 20 / 1611362 - t + (2^* 5 - 2 t * + (2^+ (* - 99 + 20 / 31 mat

Piecewise Quadratic-Derivative t D 621 - t + (* - 53 - 2 t * + (* + 5 / - 622 - t + (* - 113 - 2 t * + (* + 5 / 115 - 2 t 92 - t + (* + (* + (* 5 / - 31 mat

t quad t f 1 el t X 2 el ≤ t f 2 el t X 3 el ≤ t f 3 el cases :

Sy . :

size 14 :

clr black :

plotSample Sy size clr Sy size clr Sy size clr Sy size clr 43 mat :

t quadDeriv t D 1 el t X 2 el ≤ t D 2 el t X 3 el ≤ t D 3 el cases :

Data X Y plotSample augment :

Data 1 1.3 . 14 black 1.5 3.1 . 14 black 2 5.2 . 14 black 2.5 0.8 . 14 black 45 mat

U 0.5 :

Ω 3 :

Interpolate V U quad : 2.1

sol x quad Ω - 0 ≡ x 01 solve : 0.324046

Plot Collectors

Piecewise joined Quadratic Spline

x quad x quadDeriv Data sol Ω + 40 red 1 5 mat 4 1 sys

Deriv x f 1 el x diff x f 2 el x diff x f 3 el x diff Data 41 sys :

Fnct x f 1 el x f 2 el x f 3 el Data 41 sys :

cinterp & ainterp ► strictly freak

1 time t0 - 1 s *

Piecewise Quadratic segments

x f 1 el x f 2 el x f 3 el Data 4 1 sys

Example 4 . . . Thiele . . . interpolation . . . clean data

]]>

Trpo/Mesosphere
km[alt]/hPa[P] XY 0 0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 1 0.6 0.167 0.7 0.048 0.8 0.022 0.9 0.012 1 0.00794 112 mat :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

X XY 1 col : Y XY 2 col : 12 mat support XY . 14 black plot : 21 sys

Clear K 0 : Table 0 : 12 mat

Thiele table of inverse differences ... Carlos i 1 Y rows range Table i 1 el Y i el : for j 2 Y rows range i j Y rows range Table i j el X j 1 - el X i el - Table j 1 - j 1 - el Table i j 1 - el - / : cont j : 21 line break try for for i 1 cont range K i el Table i i el : for 31 line

K K 6 round vectorize :

... NUMERIC ...

K 0.00794 24.630542 0.00961 - 48.605756 - 0.050922 - 1.150783 0.052401 1067.849624 0.000243 427.64605 - 0.002065 111 mat

isolate K K eval :

Thiele on length of support points 'j' K X x Cfr j K rows 1 - : N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (1 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 61 line :

isolated 'K'

Recast inclusive of
parameters K's , X's x Cf K X x Cfr :

NONE

x λ 10 x ≤ (x 1 ≤ (& cases :

x Cf x λ * 0.5 1.25 0.5 0.25 - 2 2 mat support 3 1 sys

t0 1 time :

Canvas 2nd derivative may create freak glitches ...

x Cf x 2 diff x λ * 2 1 sys x Cf x diff x λ * 1 1 sys

1 time t0 - min 1.3

405 x 152 pixels

Example 5 . . . subspace equally both directions

]]>

Xo 0 :

Yo 0 :

Unk sub subspace j 1 Unk rows 1 - (range Vsub j el i 1 sub range vxSub i el Unk j el i sub / Unk 1 j + el Unk j el - (* + : for vxSub 21 line eval : 11 line for v Vsub 1 el : j 2 Vsub rows range v v Vsub j el stack : for v 31 line 21 line :

user original polygon Xo 0.5 - 123456718 mat transpose : Yo 6 7.5 7.5 1.25 1.25 64718 mat transpose : 21 sys

sub 3 :

select subspaces

New Xo sub subspace Yo sub subspace augment : New Xo 1 el Yo 1 el 12 mat New stack : 21 line

isolate New New :

New 0.5 - 60 6.5 0.5 71 7.5 1.333333 7.5 1.666667 7.5 2 7.5 2.333333 5.416667 2.666667 3.333333 3 1.25 3.333333 1.25 3.666667 1.25 4 1.25 4.333333 2.833333 4.666667 4.416667 56 5.333333 5.333333 5.666667 4.666667 64 6.333333 5 6.666667 677222 mat

Origin Xo Yo augment o 8 black plot : sub New . 10 red plot : 21 sys

subspace core component polyline

Origin sub 2 1 sys

Example 6 . . . Quadratic Lagrange knots

]]>

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

k x B1 k 4 el x - (2^k 4 el k 2 el - (k 4 el k 3 el - (* / :

Lagrange Quadratic Generatrix x b k 1 el x ≤ (x k 2 el ≤ (& k x B1 k x B2 + k x B3 + k 2 el x ≤ (x k 3 el ≤ (& k x B1 k x B2 + k 3 el x ≤ (x k 4 el ≤ (& k x B1 0 if if if 11 line :

k x B2 k 4 el k 1 el - (k 3 el x - (2^* - k 4 el k 3 el - (k 3 el k 2 el - (* k 3 el k 1 el - (* / :

k x B3 k 4 el k 1 el - (k 2 el x - (2^* k 4 el k 2 el - (k 3 el k 2 el - (* k 2 el k 1 el - (* / :

Experimental knots.
Play knots, watch graphs. k 3 3.5 4.25 5.75 14 mat transpose :

Piecewise Quadratic segments x Q k 1 el x ≤ (x k 2 el ≤ (& k x B1 k x B2 + k x B3 + junction if k 2 el x ≤ (x k 3 el ≤ (& k x B1 k x B2 + junction if k 3 el x ≤ (x k 4 el ≤ (& k x B1 N/A if 31 mat :

knots k 1 el 0.12 black k 2 el 0.12 black k 3 el 0.12 black k 4 el 0.12 black 45 mat :

There are "Lagrange cubic knots". Not offered in Smath style. Quad knots: adapted from Robert A. Mathsoft Collaboratory.

Lagrange basis interpolating funct's

k x B1 k x B2 k x B3 knots 4 1 sys

Lagrange Quadratic "Generatrix"

x b k 1 el 0 . 12 black k 2 el 0 . 12 black k 3 el 0 . 12 black k 4 el 0 . 12 black 4 5 mat 2 1 sys

Piecewise Quadratic segments

x Q 1 el x Q 2 el x Q 3 el k 1 el 0 . 12 black k 2 el 0 . 12 black k 3 el 0 . 12 black k 4 el 0 . 12 black 4 5 mat 4 1 sys

b(x)+knots

Q(x)+knots

User applied example

X 0 0.5 1 1.5 2 2.5 3481 mat :

Y 12 3.1 2.9 1.7 2.8 6281 mat :

u 1 - 0.75 1.25 1.75 2.75 3 3.25 71 mat :

Add two auto knots on the LHS
there are 'n' intervals. i 1 t0 i el u i el 2 u i 1 + el u i el - (* - : t1 i el u i el u i 1 + el u i el - (- : 21 line for u t0 t1 u stack : 21 line

u transpose 4.5 - 2.75 - 1 - 0.75 1.25 1.75 2.75 3 3.25 19 mat

u 9 el 3.5 :

u 10 el 4 :

u 11 el 5.5 :

u 1 el 1.5 - :

u 2 el 10 - :

User adapt first & last point

t u : isolate  'u' for 't' ≡

t transpose 1.5 - 10 - 1 - 0.75 1.25 1.75 2.75 3 3.5 4 5.5 111 mat

k x b k 1 el x ≤ (x k 2 el ≤ (& k x B1 k x B2 + k x B3 + k 2 el x ≤ (x k 3 el ≤ (& k x B1 k x B2 + k 3 el x ≤ (x k 4 el ≤ (& k x B1 0 if if if 11 line :

Lagrange poly from Quad knots. n x Bp t n el t n 1 + el t n 2 + el t n 3 + el 41 mat x b :

1 x Bp 2 x Bp 3 x Bp 4 x Bp 5 x Bp 6 x Bp 7 x Bp 8 x Bp 9 x Bp 10 x Bp 11 x Bp 111 sys

For the 11 knots, here are the 11 Lagrange polynomials

1 x Bp 2 x Bp 3 x Bp 4 x Bp 5 x Bp 6 x Bp 7 x Bp 8 x Bp 9 x Bp 10 x Bp 11 x Bp 11 1 sys

Quad tridiagonal matrix

r 1 X rows range :

Tridiag of QuadCoeff i r j r M i j el j X i el Bp : for for 21 line

s M 1 -^Y * :

The Quadratic tridiagonal matrix M 0.03 0.716 0.254 00000 0.003 0.425 0.571 000000 0.056 0.819 0.125 000000 0.125 0.792 0.083 000000 0.375 0.575 0.05 00000 0.042 0.508 0.45 0000000 0.667 0.333 00000000 0.75 88 mat

s 1.512 - 0.292 3.291 3.094 0.468 5.407 7.187 2.667 81 mat

s 0.295 - 0.292 3.291 3.094 0.468 5.407 7.187 2.571 81 mat :

<= user

x L s j el j x Bp * (j 1 s rows sum :

0 L 1.0361

compute 1 time :

vx 2 - 7 1.95 - range :

Original knots

u 1 - 0.75 1.25 1.75 2.75 3 3.25 71 mat :

v 000000071 mat :

Spline i 1 vx rows range vy i el vx i el L eval : for vx vy augment 21 line :

XY X Y . 10 black plotG :

knots u v . 10 black plotG :

Spline Spline :

t transpose 1.5 - 10 - 1 - 0.75 1.25 1.75 2.75 3 3.5 4 5.5 111 mat

i 1 t rows range to i el 0 t i el * : for

The augmented/adapted set of user knots MyKnots t to o 10 blue plotG :

Complete plot of Lagrange Quadratic Spline

XY Spline knots MyKnots 4 1 sys

1 time compute - 4 s *

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - range deriv i el 0.5 Y i 1 - el Y i el - X i 1 - el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X deriv augment 101 line :

Deriv(Spline) is same as the infinitesimal, but discrete.

x f x L :

h 0.001 :

x h f D1 x 2 h * - f x 2 h * + f - 4 h * / - :

x Deriv x h f D1 2 - x ≤ (x 6 ≤ (& cases :

D x 2 - 6 1.99 - range : i 1 x rows range D i el x i el Deriv eval : for x D augment 31 line :

Derivative of the Lagrange Knots Quadratic. Spline is quadratic, derivative is piecewise linear. Gradient is 2nd deriv of spline at support points 'X'

D X s augment Spline XY 4 1 sys

D 1rst derivative ≡ X s augment Gradient @ points ≡ Spline Lagrange Quadratic spline ≡ XY support points ≡ 41 sys

Stem fill plot for QuickPlot unicolor data STEM stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - (range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - (range V V stem j el stack : for V V data data rows row stack : 71 line :

Spline Spline STEM XY 6 1 sys

Example 7 . . . Advanced Hermite smoothlet

]]>

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

The Scientific/Engineering Smoothlet

This "Carbon α 12" appeared in Matlab for demonstrating their built-in Spline smoothing procedure. It works in some way and not in other way. Some Matlab user attempted Legendre orthopoly, not to avail. In data smoothing, everyting is permitted to arrive at a final result. Points X2, Y9 were corrected, I have no recollection how I did. The K0(u) was designed for this particular application. It works like the simpler "Gaussian Ksmooth", but "Gaussian Ksmooth", fails for this application. Otherwise, either "Ksmooth" work as well in applications done in Mathcad 11 and Smath.

α 0.12 :

γ 2 :

bw 0.15 :

t0 1 time :

u K0 1 γ / 1 α - (u * abs - γ sqrt / exp * 1 α + (u * γ sqrt / abs π 2 γ * / + sin * :

Carbon12 0.175 317.303 0.35 195.232 0.525 124.387 0.7 43.733 0.875 40.463 1.05 55.722 1.225 90.599 1.4 115.668 1.575 184.332 1.75 140.736 1.925 100.409 2.1 57.902 2.275 42.643 2.45 58.992 2.625 98.229 2.8 206.131 2.975 316.212 3.15 345.64 3.325 284.605 3.5 235.559 3.675 90.599 3.85 35.014 4.025 36.104 4.2 11.035 242 mat :

Adjustable Ksmooth Gaussian like kernel

10 - x ≤ ( x 10 ≤ ( & x K0 NO plot if 0 2 1 sys

Xo Carbon12 1 col : Yo Carbon12 2 col : 21 line

X Carbon12 1 col : Y Carbon12 2 col : 21 line

6179 did not recover "data" from 5346 data Xo Yo + 7 red plotG :

n X rows : 24

X 2 el 0.3325 :

Y 9 el 216.5901 :

The crappy Carbon12 data

Xo Yo augment data 2 1 sys

"Ksmooth" is a distance weighted average smoothing. It rectifies the Y's . This kernel has two supplementary parameters that create/adjust the negative trace. It's this feature that makes K0(u) rectify Carbon α12 wrt the X's spacing. The more traditional "Gausian Ksmooth" has only "bw" [bandwidth] parameter. Either kernel consume computing power !

XW 0.175 317.303 0.3325 217.0355 0.525 130.1479 0.7 61.3415 0.875 32.4856 1.05 47.5163 1.225 90.6788 1.4 143.2053 1.575 173.5314 1.75 155.6779 1.925 105.1301 2.1 56.8338 2.275 34.1471 2.45 51.1141 2.625 114.8755 2.8 214.0454 2.975 306.0658 3.15 341.7215 3.325 304.952 3.5 214.1093 3.675 111.3089 3.85 42.8756 4.025 15.8375 4.2 7.5659 242 mat :

j X Y S0 μ j 11 line : X i el X μ el - bw / K0 Y i el * (i 1 n sum X i el X μ el - bw / K0 (i 1 n sum / 21 line :

W j 1 n range s j el j X Y S0 : constrain 1rst node s 1 el Y 1 el : 31 line for s 21 line :

XW X W augment :

Isolate XW =>

XW data 2 1 sys Hermite Utilities

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

data Strip nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

Hermite PolyLine t P 16 / t 3^- 3 t 2^* + 3 t * - 1 + (3 t 3^* 6 t 2^* - 4 + (3 t 3^* - 3 t 2^* + 3 t * + 1 + (t 3^41 mat * :

Hermite PolyLine 0...1

x P 1 el x P 2 el x P 3 el x P 4 el 4 1 sys

Setup Hermite

X XW 1 col : Y XW 2 col : 21 line

r 2 : X X r RepEnd : Y Y r RepEnd : 31 line

r 2 : X X r RepEnd : Y Y r RepEnd : 31 line

X XW 1 col : Y XW 2 col : 21 line

Setup Hermite

t j XP μ j 11 line : t P 1 el X μ el * t P 2 el X μ 1 + el * + t P 3 el X μ 2 + el * + t P 4 el X μ 3 + el * + 21 line :

t j YP μ j 11 line : t P 1 el Y μ el * t P 2 el Y μ 1 + el * + t P 3 el Y μ 2 + el * + t P 4 el Y μ 3 + el * + 21 line :

<=  populate at will in 't' t 01 0.1 range :

'j' runs 1..rows(X)-(r+1)

U j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hx i ct el t i el j XP eval : for ct ct 1 + : 21 line for Hx 41 line :

V j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hy i ct el t i el j YP eval : for ct ct 1 + : 21 line for Hy 41 line :

xy U Unwrap V Unwrap augment : xy xy Strip : xw XW 1 col XW 2 col o 7 black plotG : 31 sys

1 time t0 - 17 s *

Hermite Smoothlet to "Carbon α 12"

xw xy 2 1 sys

Each "j" segment populated 't' Each "j" segment is collected in matrix of 't' rows, 'j' cols. Each U,V is Unwrapped in column, augmented in 'xy', then stripped of repeated pairs for export. Once clean/smooth 'xy' in hand, proceed scalar C12(x) for + maths. Hermite exports the smoothed data populated at will .

x C12 xy 1 col xy 2 col x ainterp :

wanted 128 :

x C12 wanted - 0 ≡ x 05 solve 0.534 1.354 1.844 2.64 3.653 4.705 61 mat

Hermite Smoothlet "Carbon α12"...prefered conclusion

wanted x C12 xw 3 1 sys

Akima w/o ▼ Hermite

In there, we Akima directly from the K0(u) smmothe 'XW'

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Raw experimental Lab α12 0.175 317.303 0.35 195.232 0.525 124.387 0.7 43.733 0.875 40.463 1.05 55.722 1.225 90.599 1.4 115.668 1.575 184.332 1.75 140.736 1.925 100.409 2.1 57.902 2.275 42.643 2.45 58.992 2.625 98.229 2.8 206.131 2.975 316.212 3.15 345.64 3.325 284.605 3.5 235.559 3.675 90.599 3.85 35.014 4.025 36.104 4.2 11.035 242 mat :

K0(u) kernel smooth XW 0.175 317.303 0.3325 217.0355 0.525 130.1479 0.7 61.3415 0.875 32.4856 1.05 47.5163 1.225 90.6788 1.4 143.2053 1.575 173.5314 1.75 155.6779 1.925 105.1301 2.1 56.8338 2.275 34.1471 2.45 51.1141 2.625 114.8755 2.8 214.0454 2.975 306.0658 3.15 341.7215 3.325 304.952 3.5 214.1093 3.675 111.3089 3.85 42.8756 4.025 15.8375 4.2 7.5659 242 mat :

X XW 1 col : Y XW 2 col : 12 mat data α12 + 7 red plot : x α12 X Y x ainterp : x λ 1 X 1 el x ≤ (x X 24 el ≤ (& cases : 41 sys

x α12 x λ * data 2 1 sys

Example 8 . . . Esplina circumscribed Hermite

]]> plotG, RepEnd, Unwrap, Strip

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

data Strip nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

Hermite Interpolation Project

t0 1 time :

XW 1.5 2.6 0 4.8 3.5 - 3.4 4.1 - 0 1.5 - 1.5 - 0 4.1 - 3.2 3.1 - 4.7 0 1.5 2.6 92 mat :

r 2 :

X XW 1 col r RepEnd : Y XW 2 col r RepEnd : XY X Y augment : 31 line

X XY 1 col : Y XY 2 col : 12 mat Cntrl X Y . 14 black plotG : 21 sys

α 16 / :

at α = 1/6 first & last points join, other α we have a zoom ± effect

Hermite PolyLine t P α t 3^- 3 t 2^* + 2 t * - 3 t 3^* 6 t 2^* - 3 t * - 6 + 3 t 3^* - 3 t 2^* + 6 t * + t 3^t - 41 mat * :

Each "j" segment is collected in matrix of 't' rows x 'j' cols. Each U,V is Unwrapped in column, augmented in 'xy', then stripped of repeated pairs for export.

<=  populate at will t 01 0.05 range :

Hermite Procedure

t j XP μ j 11 line : t P 1 el X μ el * t P 2 el X μ 1 + el * + t P 3 el X μ 2 + el * + t P 4 el X μ 3 + el * + 21 line :

t j YP μ j 11 line : t P 1 el Y μ el * t P 2 el Y μ 1 + el * + t P 3 el Y μ 2 + el * + t P 4 el Y μ 3 + el * + 21 line :

j runs 1..rows(X)-(r+1)

U j 1 X rows r 1 + (- range : ct 1 : j j i 1 t rows range Hx i ct el t i el j XP eval : for ct ct 1 + : 21 line for Hx 41 line :

V j 1 X rows r 1 + (- range : ct 1 : j j i 1 t rows range Hy i ct el t i el j YP eval : for ct ct 1 + : 21 line for Hy 41 line :

xy U Unwrap V Unwrap augment :

xy xy Strip :

polygon XW 1 col XW 2 col . 10 black plotG :

strip both ends
for clean junction span xy rows t rows - 2 + : xy xy t rows span range 12 range el : 21 sys

Hermite Polyline to Esplina

xy XW polygon 3 1 sys

Hermite PolyLine 0...1

x P 1 el x P 2 el x P 3 el x P 4 el 4 1 sys

1 time t0 - 15 s *

Example 9 . . . Esplina inscribed DeBoor

]]>

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

http://www.chemicalaid.com/tools/reactionstoichiometry.php?equation=S+%2B+O2+%3D+SO3

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

X 1.5 0 3.5 - 4.1 - 1.5 - 0 3.2 4.7 1.5 91 mat :

Y 2.6 4.8 3.4 0 1.5 - 4.1 - 3.1 - 0 2.6 91 mat :

Cntrl X Y . 14 black plotG :

r 1 :

n 2 :

t0 1 time :

X X r RepEnd : Y Y r RepEnd : 21 line

XY 1.5 2.6 1.5 2.6 0 4.8 3.5 - 3.4 4.1 - 0 1.5 - 1.5 - 0 4.1 - 3.2 3.1 - 4.7 0 1.5 2.6 1.5 2.6 112 mat :

<= isolate the repeated polygon

X XY 1 col : Y XY 2 col : 21 line

DeBoor 'u' vector

u 1 X rows n 2 + (+ range :

v j N0 μ j 11 line : u μ el v ≤ (v u μ 1 + el < (& 10 if 21 line :

v j N1 μ j 11 line : v u μ el - u μ 1 + el u μ el - / v μ N0 * u μ 2 + el v - u μ 2 + el u μ 1 + el - / v μ 1 + N0 * + 21 line :

v j N2 μ j 11 line : v u μ el - u μ 2 + el u μ el - / v μ N1 * u μ 3 + el v - u μ 3 + el u μ 1 + el - / v μ 1 + N1 * + 21 line :

v j N μ j 11 line : v μ N1 21 line μ j 11 line : v μ N2 21 line 21 mat :

Collect the respective De boor

<= Spline order, set from above n 2

Defined as local definition v Sx v k N n el X k el * (k 1 X rows sum 11 line :

Defined as local definition v Sy v k N n el Y k el * (k 1 X rows sum 11 line :

<= ppulate at will vx 312 3.1 range :

i 1 vx rows range SX i el vx i el Sx : for

i 1 vx rows range SY i el vx i el Sy : for

DeBoor SX SY augment :

1 time t0 - 30 s *

U DeBoor 1 col : V DeBoor 2 col : 12 mat

De Boor parametric order 2 ... kind of Convex Hull

XY DeBoor Cntrl 3 1 sys

De Boor components

vx U augment vx V augment 2 1 sys

1 time tWS - min 4.1

Example 10 . . . interpolate digitized image

]]> Gray Image

Gray 255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858585858585858585858585858585858585858585858585858585858585858585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585858585858585858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858585858585858585858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858585858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858585858585858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858585858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585858585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858585858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255858525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255852552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552558525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255852552552552552558525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525525585255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255255107307 mat :

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 Gray

Gray rows Gray cols 21 mat 10730721 mat

Gray Min Gray Max 21 mat 8525521 mat

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

λ 85 :

Convert existing gray pixels to black N i 1 Gray rows range j 1 Gray cols range N i j el 0 Gray i j el λ ≤ 255 cases : for for N 21 line :

isolate N N :

Extract black pixels r N rows : c N cols : 12 mat k 1 : i 1 c range j 1 r range N j i el 1 ≤ X k el i : Y k el r j - : k k 1 + : 31 line No pixel if for for 31 line

XY X Y augment :

iVBORw0KGgoAAAANSUhEUgAAATMAAABrCAIAAACUkPInAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAANGSURBVHhe7dVbcqMwEEDR7H/TGQw9U5nEjjHWoyWd8wd+AFLf4uMTyEeZkJEyISNlQkbKvOKjibgYS7L990UcD8SXKouL/Sq+ynRs7f0A4rP04nZ/iI8Z1qJbGPO7i1NziWf7Ij5gEAttWEzoLk6tJJ78i/iAlCbfnpjBXZzir1iXL+IDEphwM2LKdnGKc2LVdnGKTibZgJimXZziPbGauzhFQ8MvutFp4FjkQ5yisoEX2qB0cSz7IU5RwZCLayySODbiEKcoZKQFjREwBCnF3uziFG8YYxHt91iO/drEMa/LvnY2eGjH9m3imNPyLpkdncmxm5s45pmMK2ULJ3Zs7iaOeSDXAtmzdRx7vYlj/pdoXWzSmo4+N3HMLsVy2Bg2xxhs4nhtnVfBTvDTMRWbOF5Sz4dffOl5auU++zz2yivOq9aclg4PrEkuWK3Ppo+62uJS3DFCmzieV6MnXGQ1aWb6iWrxbJqkkon7rP5UsqS2Kfus+zyypJnJ+qz4JLKkvWn6rPUMsqSjCfqscveyJIOh+yx/37IklUEHsvBNy5KEbq/O0Saz5O3KkszG6rPYjcqSIYwyqGXuUpYM5PbqTD+xBe5Plowo+dy+e3OyZFy3V2fWAX7rtmTJBHL2ef2GZMlMss2zMiHcXp1ppvrifciSWSWZbWXCd7dXZ+8Jv3J5WbKCvnOuTHjo9ursNO0vX1WWrKbLzCsTnms/9q9dT5Ysaxv+lvOvTHhBswReuIwsYXN7ddZvQZlwRe0czv67LOGbqlEoE66r18Wp/5UlPFKpDmXCu2oE8vwfZQlPFc9EmVBG2VKe/Jcs4byCvSgTSiqVjDKhsCLVKBPKez+c334vS7jszXyUCbW8U5AyoaLLET38mSyhiGspKRPqUiYkdaGm+z+QJZT1alPKhEZeykqZ0IgyIanzZd35niyhnpN9KRNaO5OYMqG1K2XKEhp4GpoyoY/fW1Mm9KFMSOqX3JQJ3SgTknpUnDKhJ2VCRqfKlCW0d7c7ZUJnyoSkfqanTOhPmZDUt/qUCSkoEzJSJmSkTMhImZDU1wCVCVkoEzK6X6YsoS9lQkbKhIyUCUn9y1CZkIgyIaM7ZQJ5KBMyUibk8/n5B3abQo2Fsx4KAAAAAElFTkSuQmCC N

Raw extraction to be cleaned

data StripX strip/nest repeated X u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - (range u i 1 + el u i el - 0 ≠ R k k 1 + : el u i el v i el 12 mat : 11 line StripX done if 11 line for Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 91 line :

stripped of repeated X xy XY StripX :

XY rows xy rows 21 mat 36630121 mat

Digitized ... cleaned

143.4 45.1 + 50 blue 1 5 mat xy 2 1 sys

143.4 45.1 12 mat

list 123510255075100125150175200225250270285295299300301121 mat :

data ListCollect collects index list select list : ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 51 line :

support xy ListCollect : ax support 1 col : ay support 2 col (: 12 mat x spline ax ay x ainterp : S support o 5 black plot : x λ 1 xy 11 el x ≤ (x xy 3011 el ≤ (& cases : 51 sys

Digitized ... cleaned/listed for interpolation

S x spline x λ * 2 1 sys

view list transpose support augment :

view list X Y 13 mat view stack :

view list X Y 143259362158321013482528715053887578971001031011251281021501531001751789620020389225228802502536727027354285288402952982629930217300303123013046223 mat

x spline x xy 11 el xy 3011 el int 24188

jmGiraud: 2016/09/12

Int(f,a,b,n) ... Simpson/Romberg Kernel

Basic Romberg-Simpson ... integration acc from options f a b Int w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 31 line eval :

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from  l_p_c_spline 
Integration accuracy from user command line 'n' f a b n Int w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : α 0 : β 1 : h β α - n / : 14 mat Simpson 13 / h 2 / α G β G + (* h α k h * + G k 1 n 1 - sum * + (* 23 / h * α h 2 / - k h * + G k 1 n sum * + eval : 32 / b a - (* Simpson * 41 line :

x f x spline :

f xy 11 el xy 3011 el 10000 Int 24187

1 time tWS - min 6.2

Example 11 . . . colibri-Aliens in both directions

]]>

Xo 0 :

Yo 0 :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Unk sub subspace j 1 Unk rows 1 - (range Vsub j el i 1 sub range vxSub i el Unk j el i sub / Unk 1 j + el Unk j el - (* + : for vxSub 21 line eval : 11 line for v Vsub 1 el : j 2 Vsub rows range v v Vsub j el stack : for v 31 line 21 line :

user original polygon Xo 1.5 35745 0.5 - 17 mat transpose : Yo 010424617 mat transpose : 21 sys

sub 15 :

select subspaces

New Xo sub subspace Yo sub subspace augment : New Xo 1 el Yo 1 el 12 mat New stack : 21 line

isolate New New :

New 1.5 0 1.6 0.066667 1.7 0.133333 1.8 0.2 1.9 0.266667 2 0.333333 2.1 0.4 2.2 0.466667 2.3 0.533333 2.4 0.6 2.5 0.666667 2.6 0.733333 2.7 0.8 132 mat

Origin Xo Yo augment o 8 black plot : sub New . 9 red plot : 21 sys

subspace core component polyline

Origin sub 2 1 sys

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

X 1.5 35745 0.5 - 71 mat :

Y 010424671 mat :

t0 1 time :

Cntrl X Y . 14 black plotG :

r 1 :

spline order for below invoke n 2 :

X X r RepEnd : Y Y r RepEnd : 21 line

<= isolate the repeated polygon XY X Y augment eval :

X XY 1 col : Y XY 2 col : 21 line

DeBoor 'u' vector

u 1 X rows n 2 + (+ range :

v j N0 μ j 11 line : u μ el v ≤ (v u μ 1 + el < (& 10 if 21 line :

4 appVersion 0.98.6179.21440 ≡

1. correct
2. wrong u μ el v ≤ (v u μ 1 + el < (& u μ el v ≤ (v u μ 1 + el ≤ (& 21 sys

v j N1 μ j 11 line : v u μ el - u μ 1 + el u μ el - / v μ N0 * u μ 2 + el v - u μ 2 + el u μ 1 + el - / v μ 1 + N0 * + 21 line :

v j N2 μ j 11 line : v u μ el - u μ 2 + el u μ el - / v μ N1 * u μ 3 + el v - u μ 3 + el u μ 1 + el - / v μ 1 + N1 * + 21 line :

v j N μ j 11 line : v μ N1 21 line μ j 11 line : v μ N2 21 line 21 mat :

Collect the respective De boor

n 2

<= Spline order, set from above

Defined as local definition v Sx v k N n el X k el * (k 1 X rows sum 11 line :

Defined as local definition v Sy v k N n el Y k el * (k 1 X rows sum 11 line :

vx 310 3.05 range :

<= n=2 populate for more points at will

i 1 vx rows range SX i el vx i el Sx eval : for

i 1 vx rows range SY i el vx i el Sy eval : for

DeBoor SX SY augment :

1 time t0 - 38.8 s *

De Boor parametric order 2 wrt Smath ... kind of Convex Hull

XY DeBoor Cntrl 3 1 sys

De Boor components

vx SX augment vx SY augment 2 1 sys

DeBoor rows 141

x Sx x Sy 2 1 sys