Join the segments ... myData spline type [l_p

82930fbf-7acb-44c4-b396-d6a9d167a108 180 15 12 decimal

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

Part 1. The 3 types of cubic splines l_p_c

Essentially, the coefficients of the cubic splines consists in solving the basis tridiagonal matrix wrt the 'br' vector of derivatives . Independently of the type of cubic spline [type: 1, 2, 3 = l, p, c] two steps of preparation are needed: 1. Install the constant vector 'br' 2. Install the blank tridiagonal matrix.

XY 3.52 - 2.358 - 3 - 2.213 - 2 - 1.256 - 1 - 1.739 - 0 1.482 - 1 1.277 - 62 mat :

X XY 1 col :

Y XY 2 col :

N Y rows 1 - :

N 5

r 2 N range :

Prep.0 Install the constant vector 'br'

i r br i el 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 - / (- : br N 1 + el 0 : 21 line for

br 0 0.6782 1.44 - 0.74 0.052 - 061 mat

Prep.1 Install the tridiagonal matrix

Initialise the blank
diagonal matrix M 11 el 1 : M N 1 + N 1 + el 1 : 21 sys

M 10000000000000000000000000000000000166 mat

Observe the population of the rows upper left row ---------------------- ----------------------lower right row respectively to linear/parabolic/cubic. The basis tridiagonal is the "linear", further modified to suit user l_p_c .

0. Construct the tridiagonal linear spline matrix [lspline]

i r M i i 1 - el X i el X i 1 - el - 6 / : for

i r M i i el X i 1 + el X i 1 - el - 3 / : for

i r M i i 1 + el X i 1 + el X i el - 6 / : for

lspline M 1 -^br * :

M 100000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00000166 mat

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

pspline M 1 -^br * :

M 11 - 0000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00001 - 166 mat

2. Construct the tridiagonal cubic spline matrix [cspline]

M 12 el X 1 el X 3 el - (X 3 el X 2 el - / (:

M N 1 + N el X N el X N 2 - el - (X N 2 - el X N 1 - el - / :

M 13 el X 2 el X 1 el - (X 3 el X 2 el - (/ :

M N 1 + N 1 - el X N 1 - el X N el - (X N 1 - el X N el - (/ :

cspline M 1 -^br * :

M 1 1.52 - 0.52 000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00012 - 166 mat

data n Mround i 1 data rows range j 1 data cols range m i j el data i j el n round : for for m 21 line :

s lspline pspline cspline augment :

s s 3 Mround :

s 0 2.034 7.166 2.419 2.034 1.062 3.285 - 3.174 - 2.887 - 2.081 2.02 1.845 0.598 - 0.466 - 0.052 - 0 0.466 - 1.949 - 63 mat

spline type type 3 :

1 linear end points ≡ 2 parabolic end points ≡ 3 cubic end points ≡ 31 mat

s s type col :

Part 2. The l_p_c segments & junction

Collect  cubic spline components for C(j,u) abcd i 1 X rows 1 - range hs i el s i 1 + el s i el - : hy i el Y i 1 + el Y i el - : hx i el X i 1 + el X i el - : 31 line for i 1 X rows 1 - range a i el hs i el 6 hx i el * / : b i el s i el 2 / : c i el hy i el hx i el / hx i el 6 / s i 1 + el 2 s i el * + (* - (: d i el Y i el : 41 line for a b c d augment 31 line :

abcd 1.9564 - 3.583 1.0553 - 2.358 - 0.6582 - 0.531 1.0842 2.213 - 0.7887 1.4435 - 0.1718 1.256 - 0.3162 - 0.9225 0.3493 - 1.739 - 0.3162 - 0.026 - 0.5472 1.482 - 54 mat

Recover segments coeff's [a,b,c,d]

a abcd 1 col : b abcd 2 col : 21 sys

c abcd 3 col : d abcd 4 col : 21 sys

view/plot selected triplets segments j u S i j : a i el u X i el - (3^* b i el u X i el - (2^* + c i el u X i el - (* + d i el + 21 line :

Join the segments ... myData spline type [l_p_c]

The while loop form of the spline interpolation function s X Y t interpspline Find the knots that bracket 't' i0 1 : j X rows : j i0 - 1 > i i0 j + 2 / Floor : X i el t > j i : i0 i : if 21 line while Evaluate the spline based on those knots h X j el X i0 el - : a X j el t - h / : b t X i0 el - h / : a Y i0 el * b Y j el * + h 2^6 / a a 2^1 - (* s i0 el * b b 2^1 - (* s j el * + (* + 91 line :

X_Y rejects the scalar plot x mySpline s X Y x interpspline :

myData x 3.75 - 1.5 3.7 - range : j 1 x rows range vy j el x j el mySpline eval : for x vy augment 31 line :

myData myData :

myData

plotXY 1 x S 2 x S 3 x S 4 x S 5 x S XY myData 71 sys :

plotXY

type 3

Observe . . .

type 2 LHS tracks segment 2

type 3 LHS tracks segment 1

]]> j j x S

j x S 11831500 / x 2 - (- (3^* 28872000 / - (x 2 - (- (2^* + 10316000 / x 2 - (- (* + 157125 / - +

Demo: collect the 'j' segments [there are few]. u Spline j local assignment n X rows : u X 1 el ≤ n 1 : u X n el ≥ n n 1 - : i 1 n range u X i el ≤ break return 'j' < (i-1) if for if if i 1 - (41 line : j u S 21 line :

x segments x X 1 el ≥ (x X X rows el ≤ (& x Spline Not interpolated if :

Sy . :

size 14 :

clr black :

ptsSpec k 1 : i k X rows range data i el Sy size clr 13 mat : k k 1 + : 21 line for v data 1 el : j 2 data rows range v v data j el stack : for v 51 line :

About Hermite cubic . . .

is of different construction, segments junctions are discontinuous.

Will fit better some Gaussian. Also called "Draftsman spline".

The Mathcad Odesolve wrapper is essentially built around lspline

[linear spline] and can't be extrapolated. In other types of works,

be careful if you extrapolate.

Both Hermite, l_p_c_splines can be constrained at end points,

just manipulate the 1rst and/or last 's' coefficients.

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support X Y ptsSpec augment :

"Proofing" the cubic spline project

x segments j x S 3.52 - 2.358 - . 14 black 3 - 2.213 - . 14 black 2 - 1.256 - . 14 black 1 - 1.739 - . 14 black 0 1.482 - . 14 black 1 1.277 - . 14 black 6 5 mat 3 1 sys

Sy . :

size 14 :

clr black :

Global pts plot

ptsSpec very fast plot points k 1 : i k X rows range data i el Sy size clr 13 mat : k k 1 + : 21 line for Unest Points v data 1 el : j 2 data rows range v v data j el stack : for v 71 line :

On large number of points, plotG(vx,vy,a,b,c) is prohibitive of computation time. "ptsSpec" is about 12 times faster.

Data X Y ptsSpec augment :

plot 1 x S 2 x S 3 x S 4 x S 5 x S Data 61 sys :

Cubic spline segments through data points

plot

2 2.5 - S 1.620438 -

piecewise integration compatible Σ X XY 1 col : i 2 X rows range vΣ i el i 1 - x S x X i 1 - el X i el int : for vΣ sum 31 line :

Σ 7.3305 -

Part 3. The complimentary scalar dspline

s X Y t dspline t X 2 el ≤ k 1 : t X X rows 1 - el ≥ k X rows 1 - : k 1 : t X k 1 + el > k k 1 + : while 21 line if if c Y k 1 + el Y k el - X k 1 + el X k el - / s k 1 + el 2 s k el * + (X k 1 + el X k el - 6 / (* - : s k 1 + el s k el - X k 1 + el X k el - / (t X k el - (2^2 / * s k el t X k el - (* + c + 31 line eval :

x D s X Y x dspline : x c 1 3.5 - x ≤ (x 1.5 ≤ (& cases : 21 sys

x D x c *

collect/simplify plot x cubic X 1 el x ≤ (x X X rows el ≤ (& (s X Y x interpspline OUTSIDE domain if :

4. The complimentary Simpson-Romberg

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from this linear spline demo 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 cubic :

set reasonable 'n'
decimals ≤ "s" f X 1 el X X rows el 100 Int 7.3305 -

5. The complimentary arc length

Discretise for export L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

arc vx 3.52 - 1100 xd : i 1 vx rows range vy i el 1 s X Y vx i el dspline (2^+ sqrt eval : for vx vy augment 31 line :

data Σ Finite differences Cumul Integration 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 61 line :

Discrete cumulative arc length

arc Σ 2 1 sys

arc Σ arc Σ rows row 1 5.3 12 mat

6. The complimentary fintegrate

Prepare the pseudo-integrand ... 'N' from initial  spline setup s X Y fintegrate i 1 N range h i el X i 1 + el X i el - : f i el h i el 2 / Y i el Y i 1 + el + (* h i el (3^24 / s i el s i 1 + el + (* - : 21 line for f 21 line :

s X Y fintegrate sum 7.3305 -

Sources and credits . . .

1. Mathsoft: for the l_p_c types.

2. Mathsoft: dspline(s,X,Y,t),

3. Robert Adair: interpspline(s,X,Y,t).

4. Robert Adair: fintegrate(s,X,Y)

5. Smath document: Jean Giraud

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Hermite cubic . . . interpolation

. . . ½ split improves cinterp . . .

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EXPLAIN: you may need to smooth the given data set. Whether a data set is "true" or just data, it would not be correct to spline interpolate directly in sparse region(s). A good way to resolve for greater accuracy is to populate Hermite Interpolation, then spline over the Hermite populated, eventually pre-smoothed.

plot(data,char,size,color) ... Hermite Polyline modules

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 :

data sub subspace j 1 data rows 1 - (range Vsub j el i 1 sub range vx i el data j el i sub / data 1 j + el data j el - (* + : for vx 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 :

Round OFF Matrix/Vector M n Round i 1 M rows range m i el M i el n round : for m 21 line :

user original polygon Xo 3.52 - 3 - 2 - 1 - 0116 mat transpose : Yo 2.358 - 2.213 - 1.256 - 1.739 - 1.482 - 1.277 - 16 mat transpose : 21 sys

X Xo : Y Yo : 21 line

populate reasonably t 01 0.5 range :

Polyline construct

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

x c 10 x ≤ (x 1 ≤ (& cases :

interpolate 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 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

polyline 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 *

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 : polygon xy Strip : polygon polygon t rows polygon rows 1 - (range 12 range el : 31 sys

XoYo Xo Yo augment + 20 black plot : a b c spec . 14 black 13 mat : Hermite polygon a b c spec pts : vx polygon 1 col : vy polygon 2 col : 12 mat x h vx vy x cinterp : x λ 1 3.52 - x ≤ (x 1 ≤ (& cases : 61 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 :

x c Xo Yo x cinterp :

HermiteCubic polygon : XoYo Xo Yo augment : 21 sys

a Yo 2 el : b Yo 4 el Yo 3 el - : Xcept 0 2.213 - 12 mat : 31 mat 2.213 - 0.483 - 0 2.213 - 12 mat 31 mat

Amongs CAS, Smath renders Hermite cubic as a tool.

XY x c x h a b x * + XoYo HermiteCubic Xcept 61 sys :

polygon 3.52 - 2.358 - 3.322 - 2.345 - 3 - 2.213 - 2.53 - 1.695 - 2 - 1.256 - 1.5 - 1.454 - 1 - 1.739 - 0.5 - 1.654 - 0 1.482 - 0.562 1.363 - 1 1.277 - 112 mat

Algo style Memo formula of incremental analysis of this demo example h 0.1 : (increment ≡ x h f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x h f D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / : (1rst Derivative ≡ x h f D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / : (2nd Derivative ≡ 41 line

x D x f x h : x h f D1 x f Xo Yo x cinterp : x h f D1 41 sys :

x D 2 el x λ * x D 4 el x λ * 2 1 sys

The 3 's' splines companions

. . . fintegrate(s,X,Y)

. . . dspline(s,X,Y,t) 1rst derivative

. . . interpspline(s,X,s,t) 2nd derivative

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t0 1 time :

plot(data,char,size,clr)

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 collects every 'n' points 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 31 line 31 line :

data

data 0 0.0385 0.2 0.0416 0.4 0.0451 0.6 0.0491 0.8 0.0536 1 0.0588 1.2 0.0648 1.4 0.0716 1.6 0.0796 1.8 0.089 2 0.1 2.2 0.1131 2.4 0.1289 2.6 0.1479 2.8 0.1712 3 0.2 3.2 0.2358 3.4 0.2809 3.6 0.3378 3.8 0.4098 4 0.5 4.2 0.6098 4.4 0.7353 4.6 0.8621 4.8 0.9615 51 5.2 0.9615 5.4 0.8621 5.6 0.7353 5.8 0.6098 6 0.5 6.2 0.4098 6.4 0.3378 6.6 0.2809 6.8 0.2358 7 0.2 7.2 0.1712 7.4 0.1479 7.6 0.1289 7.8 0.1131 8 0.1 8.2 0.089 8.4 0.0796 8.6 0.0716 8.8 0.0648 9 0.0588 9.2 0.0536 9.4 0.0491 9.6 0.0451 9.8 0.0416 10 0.0385 512 mat :

data 0 0.0385 0.2 0.0416 0.4 0.0451 0.6 0.0491 0.8 0.0536 1 0.0588 1.2 0.0648 1.4 0.0716 1.6 0.0796 1.8 0.089 2 0.1 2.2 0.1131 2.4 0.1289 2.6 0.1479 2.8 0.1712 3 0.2 3.2 0.2358 3.4 0.2809 3.6 0.3378 3.8 0.4098 4 0.5 4.2 0.6098 4.4 0.7353 4.6 0.8621 4.8 0.9615 5.2 0.9615 5.4 0.8621 5.6 0.7353 5.8 0.6098 6 0.5 6.2 0.4098 6.4 0.3378 6.6 0.2809 6.8 0.2358 7 0.2 7.2 0.1712 7.4 0.1479 7.6 0.1289 7.8 0.1131 8 0.1 8.2 0.089 8.4 0.0796 8.6 0.0716 8.8 0.0648 9 0.0588 9.2 0.0536 9.4 0.0491 9.6 0.0451 9.8 0.0416 10 0.0385 502 mat :

X data 1 col : Y data 2 col : 12 mat L X Min : H X Max : 12 mat x c 1 L x ≤ (x H ≤ (& cases : pts data . 9 red plot : 41 sys

pts

data rows 51

l_p_c tridiagonal

n data rows : N n 1 - : r 2 N range : 31 line

Install the constant vector 'br' i r br i el 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 - / (- eval : br N 1 + el 0 : 21 line for

Prepare the cubic splines

100000000000000000000000155 mat

Initialise the blank
diagonal matrix M 11 el 1 : M n n el 1 : 21 sys

The respective l_p_c splines tridiagonal matrices

0. Construct the tridiagonal linear spline matrix [lspline]

i r M i i 1 - el X i el X i 1 - el - 6 / : for

i r M i i el X i 1 + el X i 1 - el - 3 / : for

i r M i i 1 + el X i 1 + el X i el - 6 / : for

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

2. Construct the tridiagonal cubic spline matrix [cspline]

M 12 el X 1 el X 3 el - (X 3 el X 2 el - / (:

M N 1 + N el X N el X N 2 - el - (X N 2 - el X N 1 - el - / :

M 13 el X 2 el X 1 el - (X 3 el X 2 el - (/ :

M N 1 + N 1 - el X N 1 - el X N el - (X N 1 - el X N el - (/ :

M M :

The respective spline coefficients.
Watch possible 'matrix NON-invertibe'. s M 1 -^br * eval :

Disable the 'cspline' tridiagonal to collect 'pspline' ... disable 'pspline & cspline' tridiagonal to collect 'lspline.

s s 6 round vectorize :

fintegrate(s,X,Y)

Prepare the pseudo-integrand s X Y fintegrate i 1 N range h i el X i 1 + el X i el - : f i el h i el 2 / Y i el Y i 1 + el + (* h i el (3^24 / s i el s i 1 + el + (* - : 21 line for f 21 line eval :

Σ s X Y fintegrate :

Σ Σ :

Export the cumulative integral Σcumul n X rows 1 - : Sum 1 el 0 : Sum n 1 + el 0 : i 1 n range Sum i 1 + el Sum i el Σ i el + eval : for X Sum augment 51 line :

l_p_c spline cumulative integrate

Σcumul . 9 blue plot 0 2 1 sys dspline(s,X,Y,t)

s X Y t dspline t X 2 el ≤ k 1 : t X X rows 1 - el ≥ k X rows 1 - : k 1 : t X k 1 + el > k k 1 + : while 21 line if if c Y k 1 + el Y k el - X k 1 + el X k el - / s k 1 + el 2 s k el * + (X k 1 + el X k el - 6 / (* - : s k 1 + el s k el - X k 1 + el X k el - / (t X k el - (2^2 / * s k el t X k el - (* + c + 31 line eval :

x D s X Y x dspline :

3 D 0.15988

x D1 L x ≤ (x H ≤ (& (x D out spline if :

The scalar dspline 1rst deriv.

x D1 0 2 1 sys

2nd derivative of [lpc] spline

X s augment 0 2 1 sys interpspline(s,X,Y,t)

The while loop form of the spline interpolation function s X Y t interpspline Find the knots that bracket 't' i0 1 : j X rows : j i0 - 1 > i i0 j + 2 / Floor : X i el t > j i : i0 i : if 21 line while Evaluate the spline based on those knots h X j el X i0 el - : a X j el t - h / : b t X i0 el - h / : a Y i0 el * b Y j el * + h 2^6 / a a 2^1 - (* s i0 el * b b 2^1 - (* s j el * + (* + 91 line :

x Spline s X Y x interpspline : x D1 s X Y x dspline : vx X : vy Σcumul 2 col : 12 mat x Σspline s vx vy x interpspline : x D2 s X s x interpspline : x u 1 L x ≤ (x H ≤ (& cases : 61 sys

from dichotomy below var 5.987910238278 :

2nd derivative "scalar"

x D2 x u * 0 2 1 sys

x Spline x u * x D1 x u * 0.5 x Σspline * x u * var var Spline + 25 black 1 5 mat x D2 x u * 5 1 sys $x f x Spline x D2 - :$ $a 5.75 : b 6.25 : xn b a + (2 / : 31 line$ $ε 1012 -^:$ $Dichotomy n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line$ $sanity check xn n 21 mat 5.98791 3721 mat$ $var Spline var D1 var Σspline var Σspline π Σspline - var D2 51 mat 0.5061 0.5066 - 2.1515 1.8554 0.5061 51 mat$ $1 time t0 - 33 s *$ $L H N xd U 0 : dx H L - N / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :$ $Discretize for X_Y
not scalar wrt 'x' L X Min : H X Max : N 100 : 13 mat U L H N xd : 21 sys$ $cumul i 1 U rows range Σ i el 1 U i el Σspline * eval : for U Σ augment 21 line :$ $S i 1 U rows range spline i el U i el Spline eval : for U spline augment 21 line :$ $D1 i 1 U rows range d1 i el U i el D1 eval : for U d1 augment 21 line :$ $D2 i 1 U rows range d2 i el U i el D2 eval : for U d2 augment 21 line :$ S D1 D2 cumul 0 5 1 sys $The 2nd derivative sub X s augment :$ $X s 5 cinterp 2.0884 -$ X s x cinterp sub 2 1 sys $user original polygon Xo 12 4.5 5.55 7.5 911 8.5 18 mat transpose : Yo 7.5 7765 3.5 5718 mat transpose : 21 sys$ $x f Xo Yo x ainterp :$ $XY Xo Yo augment :$ $x c 11 x ≤ (x 11 ≤ (& cases :$

Not a case for cubic splines

x f x c * XY 2 1 sys

X Xo : Y Yo : 21 line

Hermite smooth travel

XoYo Xo Yo augment :

plotG(vx,vy,char,size,color) ... Hermite Polyline modules

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 :

data sub subspace j 1 data rows 1 - (range Vsub j el i 1 sub range vx i el data j el i sub / data 1 j + el data j el - (* + : for vx 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 :

Round OFF Matrix/Vector M n Round i 1 M rows range m i el M i el n round : for m 21 line :

t 01 0.05 range :

<= populate for more points at will

Polyline construct

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

x c 10 x ≤ (x 1 ≤ (& cases :

interpolate 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 PolyLine 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

polyline 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 *

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 : 21 sys

XoYo contour 2 1 sys

The Akima cubic spline . . .ainterp(X,Y,x)

has this almost unique application.

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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 :

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 :

X XY 1 col : Y XY 2 col : x f X Y x ainterp : 31 sys

L H N # f range Algo U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for j 1 range range u j el U j el : y j el U j el f eval : 21 line for u y augment 51 line :

Discrete 01400 x f 400 Algo : support XY + 10 red plot : 21 sys

Discrete support 2 1 sys

data Σ Finite differences Cumul Integration 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 61 line :

Discrete Σ

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 1 deriv * augment 101 line :

Discrete Deriv

The first derivative suggests some smoothing be applied to "Discrete" ... so to speak smoothe the discontinuous junctions, not offered.

Hermite-D . . . Draftsman cubic spline

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The Hermite spline [Draftsman spline] is not more obsolete than "mathematicians". This example is given to demonstrate its "fabricability". Surely, you will not encounter such task in real life. The other point is about scalar and pseudo-scalar. Each Hermite segment is scalar. The jointed segments are "pseudo-scalar". The global cubic does not support the derivative operator, it discretises piece-wise This project originated from Robert [Mathcad 11]. l_p_c splines failed miserably. Hermite 'D' Draftsman style was done by Jean in final collaboration with Robert ... re-adapted for Smath.

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 :

plot(data,char,size,clr)

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 :

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 :

X XY 1 col : Y XY 2 col : data XY + 10 black plot : 31 line

Revisit Hermite 'D' cubic

α 2.75 : β 0.5 : γ 0.1303 : 31 line

D 1 el 0.025 : D 2 el 0.070 : D 3 el γ : D 4 el β : D 5 el α : 51 line

D 6 el 0 : D 7 el α - : D 8 el β - : D 9 el γ - : D 10 el 0.070 - : D 11 el 0 : 61 line

2.75 0.5 0.1303 31 line

D transpose 0.025 0.07 0.1303 0.5 2.75 0 2.75 - 0.5 - 0.1303 - 0.07 - 0111 mat

isolate D D :

Hermite

b n D rows : i 1 n 1 - range b i el 3 Y i 1 + el * X i 1 + el X i el - (D i 1 + el * - 2 D i el * X i 1 + el X i el - (* - 3 Y i el * - X i 1 + el X i el - (2^/ : for b 31 line :

a n D rows : i 1 n 1 - range a i el D i 1 + el X i 1 + el * D i 1 + el X i el * - 2 Y i 1 + el * - D i el X i 1 + el * + D i el X i el * - 2 Y i el * + X i 1 + el X i el - (3^/ : for a 31 line :

j u H i j : a i el u X i el - (3^* b i el u X i el - (2^* + D i el u X i el - (* + Y i el + 21 line :

Pseudo-interpolate wrt segments u Her j n X rows : u X 1 el ≤ n 1 : u X n el ≥ n n 1 - : i 1 n range u X i el ≤ break if for if if i 1 - (31 line : j u H 21 line :

Nest/check/interpolate

NOT compatible plot points x f x Her :

x c 10 x ≤ (x 1 ≤ (& cases :

Global segments

Global segments & pts plot

Data XY . 13 black plot :

plot 1 x H 2 x H 3 x H 4 x H 5 x H 6 x H 7 x H 8 x H 9 x H Data 101 sys :

Hermite cubic segments

plot 0 0.00794 . 13 black 0.1 0.012 . 13 black 0.2 0.022 . 13 black 0.3 0.048 . 13 black 0.4 0.167 . 13 black 0.5 1 . 13 black 0.6 0.167 . 13 black 0.7 0.048 . 13 black 0.8 0.022 . 13 black 0.9 0.012 . 13 black 1 0.00794 . 13 black 11 5 mat x f x c * 2 1 sys

U1 0 0.1 0.01 range : U2 0.1 0.2 0.11 range : U3 0.2 0.3 0.21 range : U4 0.3 0.4 0.31 range : U5 0.4 0.5 0.401 range : 51 sys

S1 U1 1 U1 H vectorize augment : S2 U2 2 U2 H vectorize augment : S3 U3 3 U3 H vectorize augment : S4 U4 4 U4 H vectorize augment : S5 U5 5 U5 H vectorize augment : 51 sys

The Hermite α circle X 4 el Y 4 el 21 mat 0.3 0.048 21 mat

Export .. strip repeated pairs H S1 S2 S3 S4 S5 stack :

data StripX strip repeated Xdata 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 Unest StripX if 11 line for v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

H 4 x H 5 x H Data X 4 el Y 4 el o 140 black 1 5 mat 5 1 sys

The circle is the impossible l_p_c splines. Smath Akima 'ainterp' is much better but too discontinuous. Hermite-D is slightly discontinuous wrt 1rst derivative.

... both rejected by X_Y plot ...

'J' => index of segment(s) u Xdata J search index n Xdata rows : u Xdata 1 el ≤ n 1 : u Xdata n el ≥ n n 1 - : resume next if if i 1 n range u Xdata i el ≤ break Done if for J i 1 - (: J 61 line :

u InterpHer u X J u H :

Pseudo-interpolate wrt segments u Her j n X rows : u X 1 el ≤ n 1 : u X n el ≥ n n 1 - : i 1 n range u X i el ≤ break End if for if if i 1 - (31 line : j u H 21 line :

u 0.35 :

u InterpHer u Her 21 mat 0.079375 0.079375 21 mat

Xmin 0.34794 : Ymin 0.0044 : 12 mat

x Ω X Y x cinterp :

0.34794 Her 0.34794 Ω 21 mat 0.077 0.004 21 mat

Hermite-D[red]... cinterp [blue]

X Y x cinterp x c * x f x c * 0.348 0.25 0.348 0.125 - 2 2 mat 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

There is an exact model to this Gaussian data set. This Tool spline session is our concern. Hermite-D is OK for publishing, Smath renders exceptional.

Golden ratio

x f X Y x cinterp : TOL 106 -^: 21 line

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

min f 0.3 0.4 y1 y2 ≤ TOL Extrema 1 el :

min X Y min cinterp 21 mat 0.34794 0.0044 21 mat

Revisit Hermite interpolation . . . brute force

stand alone as offered with associated utilities

]]> plotG(vx,vy,char,size,color) ... Hermite Polyline modules

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 :

data sub subspace j 1 data rows 1 - (range Vsub j el i 1 sub range vx i el data j el i sub / data 1 j + el data j el - (* + : for vx 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 :

Round OFF Matrix/Vector M n Round i 1 M rows range m i el M i el n round : for m 21 line :

X 1.5 0 3.5 - 4.1 - 1.5 - 0 3.2 4.7 1.5 19 mat transpose : Y 2 4.8 3.4 0 1.5 - 4.1 - 3.1 - 0219 mat transpose : XY X Y augment : 31 sys

x y axes x y / - :

t 01 0.05 range :

t rows 21

Interpolate construct

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

x c 10 x ≤ (x 1 ≤ (& cases :

interpolate 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 PolyLine 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

polyline 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 *

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 Y 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 : 21 sys

contour contour t rows contour rows range 12 range el : contour contour 1 contour rows t rows - 2 + (range 12 range el : 21 line

Region-wise interpolation

sub contour 100140 range 12 range el :

vx sub 1 col : vy sub 2 col : x Line vx vy x cinterp : 31 sys

u 2 :

u Line 3.81 -

xy u u Line augment :

XY contour axes xy 4 1 sys

NOTES ... This convex hull polygon has the counterpart via cinterp . Here, Hermite cubic interpolation looks more formal mathematically. More work for more exploitation.

x contour 1 col : y contour 2 col : 21 sys

Area Polygon wrt vertices coordinates A 12 / x i el y i 1 + el * x i 1 + el y i el * - (i 1 x rows 1 - sum * abs :

Perimeter  Polygon wrt vertices coordinates P x i 1 + el x i el - (2^y i 1 + el y i el - (2^+ sqrt i 1 x rows 1 - sum :

Center of gravity Xo 16 A * / x i el x i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : Yo 16 A * / y i el y i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : 21 mat

Area
Perimeter A P 21 mat 48.7 27.5 21 mat

Center of gravity Xo Yo 21 mat 0.084 0.298 21 mat

Hermite inscribed/circumscribed hull

stand alone as offered with associated utilities

Hermite Polyline

]]> 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 :

XW 1.5 20 4.8 3.5 - 3.4 4.1 - 0 1.5 - 1.5 - 0 4.1 - 3.2 3.1 - 4.7 0 1.5 292 mat :

r 2 :

Hermite Polyline hull

t0 1 time :

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 : Cntrl X Y . 14 black plotG : 31 line

α 16 / < (inscribed hull ≡ α 16 / > (circumscribed hull ≡ 21 sys

α 17 / :

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^* + 3 t * - 1 + (3 t 3^* 6 t 2^* - 4 + (3 t 3^* - 3 t 2^* + 3 t * + 1 + (t 3^41 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.

t 01 0.05 range :

<= populate at will

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 :

hull xy Strip :

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

Hermite Polyline to Esplina

hull 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

x hull 1 col : y hull 2 col : 21 sys

1 time t0 - 15 s *

Area Polygon wrt vertices coordinates A 12 / x i el y i 1 + el * x i 1 + el y i el * - (i 1 x rows 1 - sum * abs :

Perimeter  Polygon wrt vertices coordinates P x i 1 + el x i el - (2^y i 1 + el y i el - (2^+ sqrt i 1 x rows 1 - sum :

Center of gravity Xo 16 A * / x i el x i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : Yo 16 A * / y i el y i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : 21 mat

Area
Perimeter A P 21 mat 28.5 2021 mat

Center of gravity Xo Yo 21 mat 0.041 0.197 21 mat

Bezier Quadratic 3 points Arc

]]> plotG(vx,vy,char,size,color) ... Bezier utilities modules

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 :

data sub subspace subdivide the polyline j 1 data rows 1 - (range Vsub j el i 1 sub range vx i el data j el i sub / data 1 j + el data j el - (* + : for vx 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 31 line :

Round OFF Matrix/Vector M n Round i 1 M rows range m i el M i el n round : for m 21 line :

Xo 30413 mat transpose : Yo 211 - 13 mat transpose : 21 sys

polygon Xo Yo augment :

Xm Xo 1 el Xo 2 el + 2 / Xo 2 el Xo 3 el + 2 / 12 mat : Ym Yo 1 el Yo 2 el + 2 / Yo 2 el Yo 3 el + 2 / 12 mat : 21 mat 1.5 212 mat 1.5 012 mat 21 mat

X Xo : Y Yo : 21 sys

t 01 0.025 range :

Interpolate construct

r 1 : X X r RepEnd : Y Y r RepEnd : 31 sys

x c 10 x ≤ (x 1 ≤ (& cases :

Bézier Quadratic t P 1 t - (2^2 t * 1 t - (* t 2^31 mat :

Bézier basis quadratics 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 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 * + : 21 sys

U j 1 X rows r 2 + (- (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 Y rows r 2 + (- (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 : 21 sys

sub contour t rows 1 + (contour rows range 12 range el : end Xo 3 el Yo 3 el 12 mat : sub sub end stack : 31 sys

arc explorer j 13 :

U sub j row : 1.83 1.31 12 mat

polygon sub Xm 1 el Ym 1 el Xm 2 el Ym 2 el 2 2 mat Xm 1 el Ym 1 el . 14 black 1 5 mat Xm 2 el Ym 2 el . 14 black 1 5 mat U 1 1 el U 1 2 el + 20 black 1 5 mat 6 1 sys

polygon sub Xm 1 el Ym 1 el Xm 2 el Ym 2 el 22 mat 31 sys

Bezier Quadratic 3 points elliptic Arc

]]> plotG(vx,vy,char,size,color) ... Bezier utilities modules

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 :

data sub subspace subdivide the polyline j 1 data rows 1 - (range Vsub j el i 1 sub range vx i el data j el i sub / data 1 j + el data j el - (* + : for vx 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 31 line :

Round OFF Matrix/Vector M n Round i 1 M rows range m i el M i el n round : for m 21 line :

Xo 01113 mat transpose : Yo 11013 mat transpose : 21 sys

polygon Xo Yo augment :

Xm Xo 1 el Xo 2 el + 2 / Xo 2 el Xo 3 el + 2 / 12 mat : Ym Yo 1 el Yo 2 el + 2 / Yo 2 el Yo 3 el + 2 / 12 mat : 21 mat 0.5 112 mat 1 0.5 12 mat 21 mat

X Xo : Y Yo : 21 sys

t 01 0.0125 range :

Interpolate construct

r 1 : X X r RepEnd : Y Y r RepEnd : 31 sys

x c 10 x ≤ (x 1 ≤ (& cases :

Bézier Quadratic t P 1 t - (2^2 t * 1 t - (* t 2^31 mat :

Bézier basis quadratics 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 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 * + : 21 sys

U j 1 X rows r 2 + (- (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 Y rows r 2 + (- (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 : 21 sys

circle x C 1 x 2^- sqrt ± :

end Xo 3 el Yo 3 el 12 mat : sub contour end stack : 21 sys

arc explorer j 41 :

U sub j row : 0.75 0.75 12 mat

polygon sub Xm 1 el Ym 1 el Xm 2 el Ym 2 el 2 2 mat U 1 1 el U 1 2 el + 20 black 1 5 mat 4 1 sys

0.75 C 0.661437827766148 0.661437827766148 - 21 sys

vx sub 1 col : vy sub 2 col : x ω vx vy x linterp : 31 sys

sanity interpolate 0.75 ω 0.75