SVG(ndp,p) Companion

ac365594-12e5-4247-9bbe-3492f42abc7b 65 4 5 decimal

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DeCasteljau work shop ...

ndp p SVG t-> interpolation points [0,1] ... p-> control points t p DeCasteljau M 1 t - (p 1 row * t p 2 row * + : p rows 2 > i 2 p rows 1 - range M M 1 t - (p i row * t p i 1 + row * + stack : for t M DeCasteljau 21 line M if 21 line : t-> range of interpolation points [0,1] ... p-> control points t p DeCasteljauRange M t 1 el p DeCasteljau : i 2 t length range M M t i el p DeCasteljau stack : for M 31 line : ndp-> number of discretised points [>1] ... p-> control points t 011 ndp 1 - / range : M t 1 el p DeCasteljau : i 2 t length range M M t i el p DeCasteljau stack : for M 91 line :

svg 1.125 :

data set

data svg 22560 239.84 59.85 258.16 59.18 274.82 57.98 289.82 56.25 303.14 53.99 314.81 51.19 324.8 47.86 333.13 43.99 339.8 39.59 344.79 34.66 348.13 29.2 349.79 23.2 349.96 18.77 348.96 13.77 346.63 8.64 342.96 3.38 337.96 2.02 - 331.63 7.55 - 323.97 13.22 - 314.97 19.01 - 304.64 24.94 - 292.98 31.01 - 279.99 37.21 - 265.66 43.54 - 25050 - 262 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 :

data n Decimate every 'n' pairs range 1 data rows n range : k 1 : i range u k el data i 1 el : v k el data i 2 el : k k 1 + : 31 line for u v augment 51 line :

data data 4 Decimate :

support data . 10 black plot :

support

populate NDP 50 :

s 225 :

last 250 :

p svg 00 s 60606023 mat s 35035060602023 mat 350350 last 2010 - 50 - 23 mat 13 mat transpose * :

B svg s 60350603502035010 - last 50 - 52 mat * :

bezier user companion: NDP, p bezier NDP p 1 el SVG : i 2 p length range bezier bezier NDP p i el SVG stack : for 31 line

bezier rows 150

u 130 :

XY bezier u row : 354.34 21.39 - 12 mat

track 02019220 XY 11 el XY 12 el 32 mat :

plot B bezier track support 41 sys :

plot

The data set represents a reversible process ... not commonly found. We need lot more points. Spline(s) won't fit wrt the ascending order of the spline variate 'x'. DeCasteljau will fit/populate/zoom. Populate 'NDP' ... zoon 'svg'. This demo example done for your visit. Other data set need PP [Pain+Prestige] NO composite model was attempted.

XY 354.34 21.39 - 12 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 :

Hermite work shop ...

Hermite t H0 2 t 3^* 3 t 2^* - 1 + : t H1 2 t 3^* - 3 t 2^* + : t H2 t 3^2 t 2^* - t + : t H3 t 3^t 2^- : 41 sys

X 35810 - 14 mat transpose : Y 494414 mat transpose : 21 line

XY X Y augment : Cntrl XY + 14 black plot : t 1 - 1.375 0.99 - range : v 01 0.01 range : 41 line

t HX 1 t H0 X 1 el * t H1 X 2 el * + t H2 X 3 el * + t H3 X 4 el * + (* : t HY 1 t H0 Y 1 el * t H1 Y 2 el * + t H2 Y 3 el * + t H3 Y 4 el * + (* : 21 sys

Project 34 3.794 4.428 4.552 4.904 5.238 5.416 5.816 5.952 6.25 6.5 6.504 7.048 6.542 7.584 6.328 8.096 5.826 8.572 59112 mat :

P Project . 10 black plot :

Ht i 1 t rows range vx i el t i el HX : vy i el t i el HY : 21 line for vx vy augment 21 line :

Hv i 1 v rows range vx i el v i el HX : vy i el v i el HY : 21 line for vx 4 round vectorize vy 4 round vectorize augment 21 line :

plot Hv Ht XY Cntrl 41 sys :

scan 10 :

XY Hv scan row (: 3.716 4.383 12 mat

track 0747 XY 11 el XY 12 el 32 mat :

Hv rows 101

plot 6.5554 7.4252 + 20 red 1 5 mat track P 4 1 sys

Find index of key in reverse direction X M key FindIndex r M rows : idx 1 : M idx el key ≠ idx r < idx idx 1 + : idx if while idx 41 line eval :

XY 3.716 4.383 12 mat

idx Hv 1 col Hv 1 col Max FindIndex (: 68

data_1 Hv 16812 submatrix : data_2 Hv 69 Hv rows 12 submatrix : 21 sys

Hv idx row 6.5554 7.4252 12 mat

data_1 data_1 eval : data_2 data_2 eval : 21 sys

data Σ Finite differences Cumul Integr'n Y 0 : vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 71 line :

Σ1 data_1 Σ : Σ2 data_2 Σ : 21 sys

Σ1 Max 18.4619

Σ1 Σ2 2 1 sys

The serrated red Σ2 indicates reverse collection ... shifted in both directions.

N Hv 69 Hv rows 12 submatrix :

shiftY X 1 el sqrt : 1.732

data N eval :

shiftX Hv 1 col Max Y 1 el / : 1.639

Σ data Σ :

Σ2 Σ 1 col shiftX + reverse Σ 2 col shiftY Σ1 Max * + reverse augment :

Σ2 Σ2 rows row 8.1922 31.9769 12 mat

Cumul Σ1 Σ2 stack : vx Cumul 1 col : vy Cumul 2 col : 12 mat x Σ vx vy x cinterp : x λ 1 X 1 el x ≤ (x 8.192 ≤ (& cases : 41 mat

Closely true as given ...

Σ1 Σ2 6.5554 18.462 + 30 black 1 5 mat 3 1 sys

u 5 :

track 024424 u u Σ 32 mat :

8.1922 Σ 31.9765

... continuous cumulative

x Σ x λ * track 2 1 sys

u u Σ 12 mat 5 9.142 12 mat

Project explain . . . conclude

Once the sparse data sets have been fitted,

they are both populated for scanning demand.

Hermite in cumulative integration.

Observe flat DeBoor entrance similar to

flat DeCasteljau entrance .

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DeBoor work shop ... sparse data set

data 6.5 2.5 6.82 0.9 5.38 0.1 2.56 0 0.64 00062 mat :

control polygon X 258014 mat transpose : Y 050014 mat transpose : 21 sys

data data . 14 black plot :

DeBoor construction

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 :

r 1 :

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

Cntrl X Y + 14 black plotG :

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 :

respective DeBoor components v j N μ j 11 line : v μ N1 21 line μ j 11 line : v μ N2 21 line 21 mat :

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 :

<=  populate/range conveniently vx 57 5.01 range :

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 :

scan 75 :

XY DeBoor scan row (: 5.708 0.169 12 mat

track 10 - 141 XY 11 el XY 12 el 32 mat :

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

6.909 1.3323 + 40 blue 1 5 mat DeBoor Cntrl data track 6 1 sys

De Boor components

vx SX augment vx SY augment 2 1 sys

scan XY 12 mat 75 5.71 0.17 12 mat 12 mat

DeBoor rows 201

M key FindIndex r M rows : idx 1 : M idx el key ≠ idx r < idx idx 1 + : idx if while idx 41 line :

idx SX SX Max FindIndex : 28

SX idx row SY idx row 12 mat 6.909 11 mat 1.3323 11 mat 12 mat

N1 DeBoor 128 range 12 range el : N2 DeBoor 29 DeBoor rows range 12 range el : 21 sys

N1 N2 2 1 sys

Piecewise spline ... once populated/parted

◄  explorer  ►  ... try 6.5 up to 6.9 u 6.7 :

x s1 N1 1 col N1 2 col x ainterp : x s2 N2 1 col N2 2 col x ainterp : u λ1 1 6.5 u ≤ (u 6.909 ≤ (& cases : u λ2 1 u 6.909 ≤ (0 u ≤ (& cases : 41 sys

The cross hair visualizes existence of 1/2 Y's

x s1 x λ1 * x s2 x λ2 * u 1.33 + 200 black 1 5 mat 6.91 1.33 o 7 black 1 5 mat 4 1 sys

u s1 u λ1 * 2.13

u s2 u λ2 * 0.71

OBSERVE ... DeBoor data set is 2 decimals. Wise to spline 2 decimals. Spline wisely the region(s).

certified
vs adaptive s2 x s2 x 0 6.909 int : 0.641

uncertified s1 x s1 x 6.5 6.91 int : 0.8506

uncertified s3 x s2 x 6.5 6.909 int : 0.3084

s2 s1 s3 - (+ 1.1833

Data sets that exhibit reverse direction X are difficult to manage. If noisy, you can attempt Ksmoothing providing they are well populated. If too sparse, solution is via control point work shop. This document offers 3 work shops: DeCasteljau, Hermite, DeBoor. The three work shops have been fitted to comfort your lecture. Once fitted/populated/parted, you can spline piecewise, as well Thiele. For Thiele piecewise, you will have to de-populate the fit. Inst_Fit Deasteljau Thiele