bSpline modules

abe7eaff-c8ba-4d28-8eb4-49a4b2cfabe1 6 4 5 decimal

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Fourier "ifft" from padded created vector [Numeric] v ifft N v length : s 1 N sqrt / : 12 mat η 0 N range I η 1 + el s (v k 1 + el e i 2 π * η * N / (* k * -^* (k 0 N 1 - sum eval * : for I 31 line :

add/remove '0' Y 100000117 mat transpose :

shape/scale directions sX 1 : sY 1 : 12 mat

U Y ifft mat2sys sys2mat : a U Re mat2sys sys2mat : b U Im mat2sys sys2mat : 31 line

Draw odd polygon

sX a * sY b * augment

a transpose 0.756 0.614 0.294 0.037 0.037 0.294 0.614 0.756 18 mat

b transpose 0 0.296 0.368 0.164 0.164 - 0.368 - 0.296 - 2.3 1014 -^* - 18 mat

. . . Close the open regular heptagon . . .

Delete the last RHS coordinates

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xx 0.756 0.614 0.294 0.037 0.037 0.294 0.614 17 mat transpose : yy 0 0.296 0.368 0.164 0.164 - 0.368 - 0.296 - 17 mat transpose : 21 sys

z Close n z rows : w 1 el z n el : i 2 n 1 + (range w i el z i 1 - el : for w w n 2 + el z 1 el : w n 3 + el z 2 el : w 71 line :

XY xx yy augment :

can be automated

CLOSE xx Close yy Close augment :

XY CLOSE 2 1 sys

1. closing a shape is borrowed from Mathcad "Create Amazing Images". 2. The system runs anti-clockwise

CLOSE transpose 1 row CLOSE transpose 2 row 21 mat 0.614 0.756 0.614 0.294 0.037 0.037 0.294 0.614 0.756 0.614 110 mat 0.296 - 0 0.296 0.368 0.164 0.164 - 0.368 - 0.296 - 0 0.296 110 mat 21 mat

bSpline modules

u b 0 u 2 - ≤ 16 / 2 u + (3^* 2 - u ≤ (u 1 - ≤ (& 16 / 2 u + (3^* 23 / 1 u + (3^* - 1 - u ≤ (u 0 ≤ (& 23 / 1 u - (3^* - 16 / 2 u - (3^* + 0 u ≤ (u 1 ≤ (& 16 / 2 u - (3^* 1 u ≤ (u 2 ≤ (& 0 cases :

The Gaussian b-spline

Gaussian basis 'b-spline'

2 - x ≤ x 2 ≤ & x b Undefined if

0 b 0.6667

u v Bsp v m el u m - b * (m 1 v rows sum :

bSpline t 2 X length 1 - (2.1 range : i 1 t rows range vx i el t i el ox eval : vy i el t i el oy eval : 21 line for vx vy augment 31 line :

3 appVersion 0.98.6179 ≡ timing 55 s ≡ 21 mat

once the bSpline is plotted the module is free again for next data set ox(u), oy(u)

t0 1 time :

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 :

special construct wrt  
closing square-circle Xo 0.614 0.756 0.614 0.294 0.037 0.037 0.294 0.614 0.756 0.614 110 mat transpose : Yo 0.296 - 0 0.296 0.368 0.164 0.164 - 0.368 - 0.296 - 0 0.296 110 mat transpose : 21 sys

sub 1 :

subspaces is 1 for interpolation

New Xo sub subspace Yo sub subspace augment : New Xo 1 el Yo 1 el 12 mat New stack : ctrl Xo Yo augment : 31 line

isolate New New :

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

<=  populate Hermite at will t 01 0.01 range :

Clean the system

Hermite Polyline construct

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

x c 10 x ≤ (x 1 ≤ (& cases :

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 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 : Polyline xy : 21 sys

X Xo 2 RepEnd : Y Yo 2 RepEnd : 21 line

s 2 t rows * :

Polyline Polyline s Polyline rows s - (range 12 range el :

u ox u X Bsp : u oy u Y Bsp : 21 line

u 21 :

bSpline bSpline u bSpline rows u - (range 12 range el :

Bspline bSpline o 5 black plot : XoYo Xo Yo augment + 20 red plot : 21 line

rows bSpline
rows Hermite Bspline rows Polyline rows 21 mat 7070821 mat

ctrl Polyline Bspline XoYo 0.378188 0 + 40 green 1 5 mat Bspline 1 1 el 0 . 15 black 1 5 mat Bspline 36 1 el 0 . 15 black 1 5 mat 7 1 sys

1 time t0 - 47 s *

Bspline 11 el Bspline 361 el 21 mat 0.708667 0.047708 21 mat

0.708667 0.047708 + 2 / 0.378188

This is the PID CPV discrete iterative control algorithm. CPV = Composite Past Value The single line diagram helps visualising what's going on. The field input is converted from Analog/Digital, then passed into the PID CPV, then converted back to Analog. The D/A to field may be optional, some field controlling devices are fit to discrete signal.