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

bdd52abc-79bb-4f3a-aa5a-e602810e4264 15 15 12 decimal

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

Revisit Hermite interpolation . . . RHMSD

]]> 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 :

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

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 / - :

t0 1 time :

populate @ will t 01 0.125 range :

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

XY contour axes 3 1 sys

data contour :

data rows 65

ρ 1.25 : n data rows : ε 1012 -^: 13 mat X ρ data 1 col (* : Y ρ data 2 col (* : 31 line

x ρ uscale range x Max x Min - : middle x Max x Min + : 2 x * middle - 1 ρ / range * / 31 line :

data X ρ uscale Y ρ uscale augment :

mesh data rows : border 1.375 : 21 line

index a b scale 2 index 1 - (* a / 1 - (b * :

i 1 mesh 1 + (range : k i : z k mesh border scale : 31 line

Root Harmonic Mean Square Distance [RHMSD] calculated @ grid points rhmsd i 1 n range j 1 n range rhsmd i j el 1 n / data k 1 el z i el - (2^data k 2 el z j el - (2^+ ε + (1 -^k 1 n sum * (0.5 -^eval : for for rhsmd 21 line :

rhmsd rhmsd :

M Low High Scale High Low - (M M Min - M Max M Min - 1012 -^+ / (* Low + :

W rhmsd 0255 Scale :

rhmsd Min rhmsd Max 21 mat 0.0185 1.8144 21 mat

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 W -

1 time t0 - min 8.1

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 48.6 27.5 21 mat

Center of gravity Xo Yo 21 mat 0.084 0.298 21 mat

Region-wise interpolation

range wrt 't' population sub contour 4057 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 sub axes xy 5 1 sys

Ellipse perimeter

a 1 :

b 0.5 :

eccentricity e 1 b 2^a 2^/ - : 0.75

p 4 a * 1 e t sin (2^* - sqrt t 0 π 2 / int * :

p 4.844