Vector Field ..... Courtesy Alvaro

5c41f3ad-7bb1-4bb8-8c4b-e8d336aeac89 131 4 5 decimal

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

∞ 1010^:

CycleColors 5 :

P V L 1 sf pVec t V norme eval : a 0.5 0.3 0.15 0.65 0.35 14 mat * t sf * eval : 12 mat x L 1 el a 1 el - 000 a 1 el 51 mat * eval : y L 2 el a 3 el a 4 el - a 3 el a 4 el + a 3 el a 4 el + (- a 3 el a 4 el + a 3 el a 4 el - 51 mat * eval : 12 mat θ t 0 ≡ 0 V 1 el sign V 2 el t / acos * - if eval : c θ cos : s θ sin : 13 mat k 15 range x k el y k el 21 mat P c x k el * s y k el * - s x k el * c y k el * + 21 mat + eval : for x y augment eval 51 line :

2 F B N L 1 sf pVField x1 x2 y1 y2 22 mat B : nx ny 21 mat N : n nx ny * : R 0 : t# Clear 15 mat M 5 n * 5 + 1 matrix : V n 1 matrix : P V : 13 mat Δx x2 x1 - nx / eval : Δy y2 y1 - ny / eval : 12 mat c 0 nx 1 - (range r 0 ny 1 - (range x y 21 mat c 0.5 + (Δx * x1 + r 0.5 + (Δy * y1 + 21 mat : ν r c ny * + 1 + : P ν el x y 21 mat : V ν el x y F eval : 13 mat 21 line for for V V Δy V norme vectorize max eval / * eval : k 1 n range M R R 1 + : el P k el V k el L t# sf pVec : η 1 CycleColors range M R R 1 + : el ∞ : for 21 line for M mat2sys.1 eval 71 line :

ic to tf N 2 D pRKFam S IC t u Clear k 1 ic rows range : 12 mat S k el ic k el to tf N t u D rkfixed : IC k el ic k el transpose o augment : mat2sys.1 S k el S k el 2 col S k el 3 col augment : mat2sys.1 21 sys 31 line :

Rkadapt RKF45 ... wrt 'β'  ►  D(t,X,β)) β 0.75 0.375 21 mat : t X β D X 2 el β 1 el X 2 el * X 1 el (2^1 - (* - β 2 el (2^X 1 el * - 21 mat : t.max 22 : N 1000 : ic 0 0.125 - 21 mat : t.0 ic 1 el : 14 mat r ic t.0 t.max N t X β D Rkadapt : vU r 1 col : vY r 2 col : vZ r 3 col : 13 mat IntegralPath vY vZ augment : IP IntegralPath : 71 sys

t0 1 time :

μ β 1 el : ω β 2 el : ε 1.01 : 13 mat t X d' X 2 el μ X 2 el * X 1 el (2^1 - (* - ω 2^X 1 el * - 21 mat : x v D 0 x v 12 mat d' : B 2.5 - 2.5 2 - 222 mat : N 202021 mat : L 1121 mat : t sf t ε + ln (0.4^: 14 mat Field x v D B N L t sf pVField : 51 sys

Gorgeous vector field Alvaro, arrows slightly refactored Jean.

IP Field 2 1 sys

1 time t0 - min 0.5

Geometric Polygon(s)

Y 0 :

N 0 :

U 0 :

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

add/remove '0' Y 1000000118 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

explore j 1 a rows range :

a b augment 0.7071 0 0.6036 0.25 0.3536 0.3536 0.1036 0.25 0 1.3 1014 -^* 0.1036 0.25 - 0.3536 0.3536 - 0.6036 0.25 - 0.7071 2.7 1014 -^* - 92 mat

Draw odd polygon

sX a j el * sY b j el * augment

x a : y b : 21 sys

As well, you can shift up/dn ... left/right and in combination with scaled directions.

Area Polygon wrt vertices coordinates x y 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 x y 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 x y Xo 16 x y 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 * : x y Yo 16 x y 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

Center of gravity x y Xo abs x y Yo abs 21 mat 0.3536 1.4095 1016 -^* 21 mat

XY sX a * sY b * augment :

Xo x y Xo abs : Yo x y Yo abs : 12 mat

Area
Perimeter x y A x y P 21 mat 0.3536 2.1648 21 mat

Draw odd polygon

XY Xo Yo + 200 red 1 5 mat 2 1 sys

Zoom grayscale image

grayscale Venturi

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17.8395 26.2346 33.5802 40.9259 48.2716 55.6173 64.0123 71.358 79.7531 87.0988 93.3951 101.7901 109.1358 116.4815 124.8765 132.2222 138.5185 146.9136 155.3086 163.7037 170 176.2963 184.6914 194.1358 201.4815 208.8272 215.1235 222.4691 229.8148 235.0617 239.2593 243.4568 242.4074 239.2593 237.1605 235.0617 232.963 224.5679 215.1235 208.8272 202.5309 126.9753 126.9753 126.9753 120.679 113.3333 104.9383 93.3951 85 76.6049 68.2099 58.7654 49.321 40.9259 32.5309 23.0864 13.642 5.2469 2.0988 5.2469 12.5926 22.037 30.4321 38.8272 48.2716 56.6667 66.1111 74.5062 83.9506 92.3457 101.7901 110.1852 119.6296 128.0247 136.4198 145.8642 155.3086 163.7037 172.0988 181.5432 189.9383 200.4321 207.7778 216.1728 225.6173 234.0123 239.2593 244.5062 246.6049 242.4074 240.3086 237.1605 227.716 217.2222 208.8272 201.4815 193.0864 181.5432 173.1481 165.8025 157.4074 145.8642 137.4691 130.1235 122.7778 126.9753 126.9753 125.9259 116.4815 107.037 95.4938 85 75.5556 67.1605 54.5679 44.0741 35.679 25.1852 13.642 4.1975 1.0494 7.3457 17.8395 27.284 36.7284 47.2222 58.7654 68.2099 77.6543 88.1481 99.6914 108.0864 119.6296 128.0247 139.5679 149.0123 159.5062 168.9506 180.4938 189.9383 200.4321 209.8765 220.3704 230.8642 239.2593 245.5556 248.7037 244.5062 241.358 232.963 221.4198 211.9753 202.5309 193.0864 181.5432 171.0494 162.6543 151.1111 140.6173 132.2222 121.7284 110.1852 100.7407 91.2963 82.9012 71.358 62.963 54.5679 45.1235 126.9753 126.9753 121.7284 112.284 100.7407 87.0988 77.6543 67.1605 53.5185 41.9753 32.5309 19.9383 8.3951 0 5.2469 17.8395 28.3333 39.8765 51.4198 62.963 73.4568 86.0494 97.5926 108.0864 119.6296 131.1728 142.716 153.2099 164.7531 177.3457 188.8889 199.3827 210.9259 222.4691 235.0617 242.4074 250.8025 247.6543 243.4568 232.963 221.4198 210.9259 199.3827 186.7901 175.2469 165.8025 153.2099 140.6173 130.1235 119.6296 107.037 96.5432 87.0988 73.4568 65.0617 54.5679 43.0247 34.6296 27.284 18.8889 13.642 14.6914 16.7901 17.8395 126.9753 126.9753 119.6296 108.0864 93.3951 80.8025 70.3086 55.6173 43.0247 31.4815 16.7901 5.2469 0 12.5926 25.1852 35.679 50.3704 62.963 74.5062 88.1481 101.7901 113.3333 125.9259 138.5185 151.1111 163.7037 176.2963 188.8889 201.4815 215.1235 227.716 240.3086 250.8025 249.7531 245.5556 232.963 220.3704 208.8272 195.1852 181.5432 170 157.4074 143.7654 132.2222 119.6296 105.9877 94.4444 82.9012 69.2593 58.7654 47.2222 37.7778 28.3333 18.8889 12.5926 14.6914 13.642 17.8395 29.3827 38.8272 49.321 59.8148 71.358 81.8519 126.9753 126.9753 116.4815 102.8395 87.0988 74.5062 60.8642 45.1235 32.5309 16.7901 4.1975 2.0988 15.7407 28.3333 43.0247 57.716 70.3086 85 99.6914 112.284 126.9753 140.6173 154.2593 167.9012 181.5432 197.284 210.9259 224.5679 238.2099 250.8025 249.7531 243.4568 229.8148 215.1235 201.4815 187.8395 174.1975 160.5556 145.8642 132.2222 118.5802 103.8889 91.2963 75.5556 65.0617 52.4691 40.9259 28.3333 18.8889 10.4938 12.5926 11.5432 20.9877 31.4815 44.0741 55.6173 67.1605 79.7531 89.1975 101.7901 113.3333 124.8765 137.4691 149.0123 126.9753 125.9259 112.284 97.5926 81.8519 68.2099 50.3704 36.7284 20.9877 5.2469 2.0988 16.7901 30.4321 46.1728 61.9136 76.6049 92.3457 107.037 122.7778 138.5185 153.2099 167.9012 182.5926 198.3333 215.1235 229.8148 244.5062 253.9506 248.7037 234.0123 219.321 203.5802 187.8395 173.1481 157.4074 141.6667 126.9753 112.284 96.5432 81.8519 68.2099 54.5679 41.9753 28.3333 17.8395 8.3951 9.4444 13.642 25.1852 37.7778 51.4198 64.0123 77.6543 88.1481 101.7901 113.3333 126.9753 139.5679 153.2099 164.7531 177.3457 189.9383 202.5309 215.1235 126.9753 124.8765 110.1852 92.3457 76.6049 59.8148 43.0247 27.284 9.4444 1.0494 15.7407 30.4321 48.2716 64.0123 80.8025 97.5926 113.3333 131.1728 145.8642 163.7037 179.4444 196.2346 213.0247 228.7654 245.5556 253.9506 245.5556 229.8148 213.0247 196.2346 179.4444 163.7037 146.9136 130.1235 113.3333 97.5926 81.8519 66.1111 52.4691 37.7778 23.0864 11.5432 8.3951 10.4938 23.0864 35.679 50.3704 64.0123 77.6543 90.2469 104.9383 117.5309 132.2222 146.9136 159.5062 173.1481 186.7901 200.4321 214.0741 228.7654 239.2593 246.6049 244.5062 240.3086 126.9753 122.7778 107.037 87.0988 71.358 51.4198 35.679 15.7407 1.0494 10.4938 27.284 46.1728 64.0123 81.8519 99.6914 116.4815 134.321 152.1605 171.0494 187.8395 205.679 223.5185 242.4074 255 247.6543 229.8148 211.9753 194.1358 176.2963 157.4074 140.6173 122.7778 104.9383 88.1481 70.3086 55.6173 40.9259 25.1852 9.4444 8.3951 10.4938 25.1852 39.8765 55.6173 70.3086 85 99.6914 112.284 129.0741 144.8148 158.4568 173.1481 186.7901 202.5309 217.2222 231.9136 244.5062 247.6543 243.4568 234.0123 218.2716 205.679 189.9383 176.2963 126.9753 119.6296 101.7901 81.8519 64.0123 44.0741 26.2346 6.2963 3.1481 23.0864 40.9259 60.8642 79.7531 99.6914 117.5309 136.4198 155.3086 175.2469 193.0864 214.0741 231.9136 251.8519 251.8519 235.0617 215.1235 196.2346 176.2963 158.4568 139.5679 120.679 101.7901 82.9012 66.1111 49.321 32.5309 15.7407 6.2963 9.4444 20.9877 36.7284 53.5185 70.3086 85 100.7407 115.4321 132.2222 149.0123 164.7531 179.4444 195.1852 211.9753 228.7654 242.4074 248.7037 245.5556 234.0123 217.2222 204.6296 186.7901 172.0988 156.358 139.5679 124.8765 108.0864 126.9753 118.5802 98.642 78.7037 57.716 37.7778 16.7901 1.0494 15.7407 34.6296 56.6667 75.5556 97.5926 115.4321 136.4198 156.358 178.3951 198.3333 217.2222 239.2593 255 245.5556 224.5679 204.6296 184.6914 163.7037 143.7654 122.7778 102.8395 82.9012 65.0617 47.2222 29.3827 12.5926 6.2963 12.5926 28.3333 46.1728 64.0123 80.8025 96.5432 112.284 130.1235 147.963 164.7531 180.4938 196.2346 214.0741 231.9136 247.6543 249.7531 243.4568 225.6173 208.8272 193.0864 176.2963 159.5062 141.6667 125.9259 109.1358 93.3951 75.5556 57.716 41.9753 3364 mat :

lo 20 :

hi 235 :

t0 1 time :

Smath 5346 takes "if/else"
linterp. 6179 does not x f x lo ≤ 0 x hi ≥ 255 lo hi 21 mat 025521 mat x linterp if if :

rgb min rgb max 21 mat 025521 mat

Typical RGB light intensifier

x f

i 1 rgb rows range j 1 rgb cols range V i j el rgb i j el f eval : for for

<= Demo ... enhance a grayscale image with some luminosity fnct.

N V mwround :

rgb cols rgb rows 12 mat 643312 mat

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rgb

1 time t0 - 3 s *

Pad(M,nc,nr), Scale(M,lo,hi)

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

M nc nr Pad A M mwfft2 : insert 'nc' cols in 'A' i 1 A rows range j 1 nc range vert i j el 0 : for for a A 1 A rows 1 A cols 2 / mwfloor submatrix : b A 1 A rows A cols 2 / mwfloor 1 + A cols submatrix : S a vert b augment transpose : insert 'nr' rows in the transpose 'S' i 1 S rows range j 1 nr range horiz i j el 0 : for for a S 1 S rows 1 S cols 2 / mwfloor submatrix : b S 1 S rows S cols 2 / mwfloor 1 + S cols submatrix : S a horiz b augment transpose : 111 line :

c 48 : r 24 : 12 mat

t0 1 time :

S rgb c r Pad : iVrgb S mwifft2 eval : RGB iVrgb mwabs eval : s RGB 0255 Scale : zoom s mwround : 51 sys

practical zoom limit 3 ≤ Insert/center ncols/nrows inverse Fourier absolute of the inverted matrix scale to rgb grayscale round OFF the zoomed image 61 line

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1 time t0 - min 2.9

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V Venturi

Venturi 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mat :

Round OFF Matrix/Vector [Smath prior 6179] Mat n Round i 1 Mat rows range j 1 Mat cols range m i j el Mat i j el n round : for for m 21 line :

Generate the symmetric square matrix x y f U V Sym r 1 : i 1 U rows range c 1 : j 1 V rows range Ω r c el U i el V j el f eval : c c 1 + : 21 line for r r 1 + : 31 line for Ω 31 line :

Mesh U, V  within the bounds L, H L H N xd N intervals U 0 : dx H L - N / : i 1 N (range U i el L dx i * dx - (+ : for U 51 line :

L 1 - :

H 1 :

N 12 :

xx yy f 255 xx 2^yy 2^+ (- 0.25 / exp * :

U L H N xd :

V U :

G xx yy f U V Sym :

V Venturi :

A B r c replace U A : i 1 B rows range j 1 B cols range U r i + c j + el B i j el : for for U 31 line :

N G 3 Round :

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 V iVBORw0KGgoAAAANSUhEUgAAAAwAAAAMCAIAAADZF8uwAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAACtSURBVChTjdC7DYRADIThBd0KiQAKINuAhBbogpicemiBdiiDPrhvxT3SdWAN/9iDIfyqqqq6rl/fopGP9xQUY2zbtuu6vu91GsH/E03T8IZhGMdxmiadRvA8J9OG55TSPM/LsqzrqtMIzs0xku2h27bt+34ch04jODe40QXybfPO87yuS6cRnJuHZLrDW2SYuO9bpxGcW5ZUdFPR1xX9p6coG5JdwNNpJE+E8AZ0rVKuWUoN3wAAAABJRU5ErkJggg== N

Rep V N 110 replace :

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 Rep

V Min V Max 21 mat 8616721 mat

Rotate image 4 quadrants

Davide Carpi: 10 April, 20017 V mirror V 1 V rows range V cols 1 range el :

Inverse 92 V - :

N V mirror :

V rotate_90 V mirror transpose :

V rotate90 V transpose :

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vertical V rotate_90 V rotate90 stack :

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255 V - 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 Inverse

opposed N M augment :

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 opposed iVBORw0KGgoAAAANSUhEUgAAAEAAAAAhCAIAAADmozp2AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAh5SURBVFhHndjnllTVFobhJgclg4rkLEkERTGQkyIg3v/NnKd5a0y2hcfjOPNHj6pda835pbVrV29cvXr1xo0b33777Y8//vjo0aMXL168ffv2zz//fPXq1bNnz+7fv//DDz988803ll24cOHkyZOfffbZgQMH9uzZs2XLlq1bt+7evdvbY8eO+ej8+fOWWfz999//8ssvT548+fXXX3X7448/3rx547Urrmt4+/ZtQy9fvnz27NkTJ07oefDgwU8++WTHjh16ql27du3bt+/w4cNffPHFqVOnjP7qq6++/vrrcD548AC23377TdsVgTt37ty7d68Pfv/993fv3hk8I+/evXvr1q0lB/Nw2LZt28bGxs6dO81u2JkzZy5dunT9+nUQAcU/GiahoaepRhgEx3fffafttWvXLl68aOOXX35JCJ337t0bE/1r7qKhFiCsvy1ksv2nn37aJGCe9+Yl28uXLzPh9evXDGHLzz//3DCLDSMJrIcOHfr0008NGCv2799/9OhRY9ZoaPv48WOtSIOG0tkUszAk3DCxy948oQgTtN2+fTsm/maLuT61JjKbBOy0n8wIgfv8+fNMUILk7cOHD320Jtjx48fBBZoVuqNhjNfR8Kk1VupvV2oR/unTpwxBIEMMikmewEBKibhy5Qq3T58+nS2JtSSTM9K7IlC8iEESMzRlOhOMGQ58mAHie+7cOXH6/PPPRyqto+F1CfapNVZabxdDkkm3piCAhjJOugziFceypVnSj0zOaHjkyBEaiRlDzNokoG7evOkYsBtKJnRE9MXB3zi43hG0EmG7OMjHJY1xQ6hGpAwRPKcckxILH8npBbHm2fKey4qMyOUMwu4KDbV3zInPigCimpKnpnZqCrd2+bD0WkcryYl2NHLDwYC1+0nyZMgwIZ6RxVe6SJstkaFO2iFjHBqRQcwVeGY6EfMHgBUBzPDLhMJqNQ3GB+dBo3EZySJrvI1CSNqlMLBCHBOedEvxQrqkGckhIxs20rUkw1DMjIgP4cw1PTIKvaG0ImAzNjlbkOIw5iIwVrQZjY/10ComYOVJR3BSO/fHDqKLgocP6yzGhwQlTaty4XCKkEEs6vBQFisBAWxFQFk9AjhDcYAyDawmwNCwkyqua0cnag2TPGGphEgXNJFxDLo5ggs06Ajgg0x8vMaQadjijBKL3IUcHpEjyrByWwuniR8IZMLysyHNiqEBOgKbt9j3X3bDxAJsLbZlyGiloZGaC8nwgQky+PizpCRgwypK3mLl3mCBTC6JEYVdHwgossWhg8U4VgyN3ChUa0y8KKCuI2MNZ5Z89CGKMOisvykosYioTvOwEiFGwQdlxLjhzMQNE3yGWNz+QkDhUJaEj35Dw5EVKlGJCXzIyOKQQQCNqW4gDoxPrbHSek7Km2Tqo5uejIqYWSaam12Q5NiSHskJH8OsY8g6AWV/VmiXGzEpV0OGtKDEBzKUCB8loQI9VgqZJSvXVdyKX/QyDUM9MdR/SBIRgHjm4VD9GwKVz2Ji3ZAh0vCJktbDSlQillGgABS9stfdA0MFPQ5TsYpw1RUfWRlV1V5Noq3nfyWwVsMnShViwy16w1ARbHgO1dhGeDgPbZWryC/55zAVEiItqn9L4F9WPP+hkmBqpcT7So4pulSpM5VMFbH+TwIrOO9rheUjZ9QKwgLEavL7Mn5p18eOVfmmsk7lnmLg/yYQxEE2gEKwFpUZ2Zi1hKzFYy0YRWIOSaHvAHROqs7G1N8TGNAhHqyDciAuwa3BGjRwLAd3Rt2RqrXzGm67Oqb6aKit5sh3C+4G1dfLOoGgL3ETGOgQp6gWCRlW4gXU+PCB1X1TDcTwhcwWG20PVpgCRB3jDKUXAKwGBiTY+uKbLwfPF+vfxNBTvYSkt3bhNiCNTU3dQRzcvpKDCysVByj9Qok802gRROos8fXN5ZnHwzlkyy/m+dryNOGZwvPSfDf/hUDo8dZ9Cd34wQ0ZfICGOI3RCHGO4xncFCVEWPWn0QAl4drjwxJi+Dw1eHZYPhepnmF7NPpAoORAn/CkWkIHjt7UBbqfB14XCTJbY2UxIHCIaZG0PWAP3J6xKRpWaAbo8hnOA5ynoHmAs6snU48VPZlKFDk+ECAPkdI+9IQUgFQf6CRPb9cHt/VSIXU6BHoePOdXQYiTFkRY/TAILgLLp+j5YSBL+nBMT3IQhTQE6hwGb0VgVizRJ7zAAD0/ypJcrLWgt/WDmySOl6l0kopAgwUciLDOT5lBbEH/8LEluJpoNVIaEdZlgKmpgFEfCNiDn9TiN+itSHgbUn2gkxxhqbPd4CVuQsqGDFAX6HAHmgl+TKYxfxBO4A5e6pZbMhFr4CpgJrHuKFZavyIgP8kPmc9Ys0TfTpS0ti3VORZ04S4n8w8PCQ43sQUGGZRSWjAkGGgTO28Mpwhd9A9xtzJ/vQYDGIIKBXFHNel3DKR0RQAamBCyTi9K27xErwtihhnJK/tTHXQ3EEmAElaI+58KvR3BwZ3YDUrpSanmRsw3Bp8noh2tf/Z5kwA0+iYGlJrqot2gZ5l2BEBSLzGlokajurQIyfxXy5VyYo2V1tu1FAhEdzAjwu31hDPQzGERo2xHngT9np5kdpa8WBEgqmCIR/LDDb3WmNBeX7MhqCkZJFhHP/kERqONjY2B7jZCIVMJpjMoAM25SppwJ7bry1imzoAm89o/FfPWaCYYtEnAGWITiICCq68B8kMYI3mS9tCLncRrDWWZobrS1FuqD/TpSXIGdqLCLZw0GtwlhC4CveYquNII94BGiXYwpA7dNwlABh+UzihJkt9fY1zhqU/NCD2IrNRu0773wjNBayfBeEHXd+QAXQIlG3R/O0ul0UlI7yXuRNGzG4DO7lpjaaeouyVUsD179uw/k3+FuQaRkEYAAAAASUVORK5CYII= 167 V - 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 V 168 - 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 166 V +

Mat 85191762281917811131101286315187131771920182014101048 mat :

Flip rows ▲ M2 Mat reverse : 71920182014101086315187131719178111311012851917622848 mat

M2 transpose 78198196175203819181511172018136147121013102101712884 mat

Bloc Extract col(s)/rows(s)

Bloc ... Col ... Row ... Collect ... Decimate

data u v BlocUser ct 1 : i v a 1 ct el data i col : ct ct 1 + : 21 line for V a 1 el : j 2 a cols range V V a j el augment : for ct 1 : i u b 1 ct el V i row : ct ct 1 + : 21 line for U b 1 el : j 2 b cols range U U b j el stack : for U transpose 91 line :

data list Collect collects index list ct 1 : i list a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 41 line :

data Lo n Hi Decimate every 'n' pairs range Lo Hi Lo 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 :

The col/row extractors utilities

add/remove/insert/shuffle
in col(,,,) argument ... at will in-situ data v Col ct 1 : i v a 1 ct el data i col : ct ct 1 + : 21 line for a V a 1 el : j 2 a cols range V V a j el augment : for V 61 line :

add/remove/insert/shuffle
in row(,,,) argument ... at will in-situ data v Row ct 1 : i v a 1 ct el data i row : ct ct 1 + : 21 line for a V a 1 el : j 2 a cols range V V a j el stack : for V 61 line :

lookup wrt value in col value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

Symmetric/asymmetric "pivot" pivot n m S U 0 : V 0 : 12 mat i 0 m range U i el i : for j 1 n range V j 1 + el j - : for V reverse U stack pivot + 41 line :

1. Matrix extractors

M 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat :

col 13513 mat :

row 43213 mat :

New M col row BlocUser :

New 45125245208018054512533 mat

M col Col 020805451252080180451252458018032012524540563 mat

M row Row 4580125180245320204580125180245520458012518036 mat

algo style ...
Σum of iterated Πroduct in a matrix
from user sub-range command line. m ΣofΠ m i j el j 1 m cols product i 1 m rows sum :

New ΣofΠ 1694250

M 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat :

r1 2 : r2 5 : c1 4 : c2 6 : 14 mat sub M r1 r2 range c1 c2 range el : 21 sys

sub 8012518012518024518024532024532040543 mat

sub ΣofΠ 53176500

M 52238432 mat :

M ΣofΠ 48

M 0 :

Grocery Manager

lookup wrt value in col value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

supply Store 1 apple 1.0 credit * stock 1 Store 1 banana 0.75 credit * stock 1 Store 1 orange 1.75 credit * stock 0 Store 1 peach 1.25 credit * stock 7 Store 1 grape 2.5 credit * stock 4 Store 2 apple 0.5 credit * stock 0 Store 2 banana 1.5 credit * stock 3 Store 2 orange 1.25 credit * stock 100 Store 2 peach 1.5 credit * stock 24 Store 2 grape 0.5 credit * stock 0105 mat :

supply 1 row Store 1 apple credit stock 115 mat

idx 1.25 credit * supply 3 lookup : 4821 mat

count 1 : i idx search count el supply i row : count count 1 + : 21 line for search 31 line

search Store 1 peach 1.25 credit * stock 715 mat Store 2 orange 1.25 credit * stock 10015 mat 21 mat

s 1 supply rows range : x y in x y ≡ vectorize max : 21 line

delrow 13531 mat :

x y INCLUDE j 1 : t x w j el t : j j 1 + : 21 line t y in 0 ≡ Exclude cases for w 31 line :

1. vector filtered data
2. filtered supply
3. length of 'filtered' vr s delrow INCLUDE : Grocery supply vr 1 supply cols range el : k 1 Grocery rows range : 31 sys

Grocery Store 1 banana 0.75 credit * stock 1 Store 1 peach 1.25 credit * stock 7 Store 2 apple 0.5 credit * stock 0 Store 2 banana 1.5 credit * stock 3 Store 2 orange 1.25 credit * stock 100 Store 2 peach 1.5 credit * stock 24 Store 2 grape 0.5 credit * stock 075 mat

ORDER Grocery select ... by default delete data list OrderCol M 0 : r 1 : i list c 1 : j 1 data cols range M r c el data i j el : c c 1 + : 21 line for r r 1 + : 31 line for M 41 line :

compose at will list 1354215 mat :

Grocery supply list OrderCol :

Grocery Store 1 apple credit stock 1 Store 1 orange 1.75 credit * stock 0 Store 1 grape 2.5 credit * stock 4 Store 1 peach 1.25 credit * stock 7 Store 1 banana 0.75 credit * stock 155 mat

Irregular Polygon ... demo

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 Clear 1

x1 1.5 - 2 - 21 mat :

y1 23 - 21 mat :

x2 1321 mat :

y2 28 - 21 mat :

Construction

I1 x1 y1 Intercept (: S1 x1 y1 Slope (: xy1 I1 x S1 * + : 31 sys

I2 x2 y2 Intercept (: S2 x2 y2 Slope (: xy2 I2 x S2 * + : 31 sys

NUMERIC x x1 x2 reverse stack : y y1 y2 reverse stack : xy x y augment : X x x 1 el stack : Y y y 1 el stack : 12 mat XY X Y augment : 51 sys

Area Polygon wrt vertices coordinates x y 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 x y 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 x y Xo 16 x y 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 * : x y Yo 16 x y 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

Center of gravity x y Xo x y Yo 21 mat 0.3596 2.5789 - 21 mat

Area
Perim x y A x y P 21 mat 23.75 22.294 21 mat

Xo x y Xo : Yo x y Yo : 12 mat 0.36 2.579 - 12 mat

XY x1 1 el y1 1 el . 14 black 1 5 mat x1 2 el y1 2 el . 14 black 1 5 mat x2 1 el y2 1 el . 14 black 1 5 mat x2 2 el y2 2 el . 14 black 1 5 mat Xo Yo + 20 red 1 5 mat 6 1 sys

close the display XY 1.5 - 22 - 3 - 38 - 12 1.5 - 252 mat

r c Matrix i 1 r range j 1 c range Ω i j el i r + j c + 12 mat : for for Ω 21 line :

SQUARE Matrix inventor

r 6 : c 6 : 12 mat

M r c Matrix : 7712 mat 7812 mat 7912 mat 71012 mat 71112 mat 71212 mat 8712 mat 8812 mat 8912 mat 81012 mat 81112 mat 81212 mat 9712 mat 9812 mat 9912 mat 91012 mat 91112 mat 91212 mat 10712 mat 10812 mat 10912 mat 101012 mat 101112 mat 101212 mat 11712 mat 11812 mat 11912 mat 111012 mat 111112 mat 111212 mat 12712 mat 12812 mat 12912 mat 121012 mat 121112 mat 121212 mat 66 mat

data UnestV v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

j r 1 range : k 1 c range : jj 1 r range : kk c 1 range : 14 mat

N1 M 1 row : N2 M 1 col : N3 M r row : N4 M c col : 41 sys U N1 j el UnestV : V N2 k el UnestV : W N3 jj el UnestV : Z N4 kk el UnestV : 41 sys 12 mat

Demo is square easy to center.
Dispose otherwise at will. s u 12 mat 9.5 - 9.5 - 12 mat :

127 + 2 / 9.5

rectangle U V W Z stack :

Rectangle rectangle 1 col s + rectangle 2 col u + augment :

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 :

pts Rectangle o 5 black plot :

Rectangle pts 2 1 sys

U V W Z 14 mat 71271171079787762 mat 77879710711712762 mat 12712812912101211121262 mat 12121112101291281271262 mat 14 mat

M r c Matrix i 1 M rows range j 1 M cols range Ω i j el i r + j c + 12 mat : for for Ω 21 line :

RECTANGULAR Matrix inventor

Truncated times table M 1234524681036912154812162045 mat :

data UnestV v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

U 0 :

V 0 :

W 0 :

Z 0 :

N1 0 :

N2 0 :

N3 0 :

N4 0 :

j 51 range : k 14 range : jj 15 range : kk 41 range : 14 mat

up/dn shifter n 3 - :

left/right λ 2.5 - :

N1 M λ n Matrix 1 row : N2 M λ n Matrix 1 col : N3 M λ n Matrix 4 row : N4 M λ n Matrix 5 col : 41 sys U N1 j el UnestV : V N2 k el UnestV : W N3 jj el UnestV : Z N4 kk el UnestV : 41 sys 12 mat

Rectangle U V W Z stack :

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 :

pts Rectangle o 5 black plot :

U V W Z pts 5 1 sys

U V W Z 14 mat 1.5 - 2 1.5 - 1 1.5 - 0 1.5 - 1 - 1.5 - 2 - 52 mat 1.5 - 2 - 0.5 - 2 - 0.5 2 - 1.5 2 - 42 mat 1.5 2 - 1.5 1 - 1.5 0 1.5 1 1.5 252 mat 1.5 2 0.5 2 0.5 - 2 1.5 - 242 mat 14 mat

data list ListCollect collects index list ct 1 : i list a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 41 line :

data U V W Z stack :

Strip duplicates but CLOSE the polygon ... [18] list 1234578911121314161718115 mat :

New data list ListCollect :

x New 1 col : y New 2 col : 12 mat

Area Polygon wrt vertices coordinates x y 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 x y 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 x y Xo 16 x y 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 * : x y Yo 16 x y 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

Center of gravity x y Xo x y Yo 21 mat 0021 mat

Area
Perimeter x y A x y P 21 mat 121421 mat

Explicit infinite algo codes

x y u v Clear 1

collapse in utilities x y u v code Algo x y u v code :

fabricate'code'
inifinitely wrt
u,v,w,x,y,z ... x y u v code x y sqrt * x sqrt y * x y * x y - x y + 51 mat :

32 / 112 / + ≡

inactive 'u, v'
units separate
the exponents. 2 m * 7 m *__code Algo 5.2915 m 32 /^* 9.8995 m 32 /^* 14 m 2^* 5 m * - 9 m * 51 mat

27 sqrt * 2 sqrt 7 * 27 * 27 - 27 + 51 mat m * 5.2915 m * 9.8995 m * 14 m * 5 m * - 9 m * 51 mat

27__code Algo 5.2915 9.8995 145 - 951 mat

27__code Algo m * 5.2915 m * 9.8995 m * 14 m * 5 m * - 9 m * 51 mat

Function explorer

User scaled slider click blue,Settings

User function from project x f x 3^:

Notes ... We explore in the plot range ± 2 Set min/max ± 20 scale 0.1 Each tick will be in 0.1 plot range. Click turquoise: read 'x'.

scale 0.1 : x scale x * : Y x f : XY x Y augment : 41 sys

x 1.2 -

track 3 - 1.5 0 1.5 x Y 32 mat :

V x 0 x Y 22 mat :

x f V track 3 1 sys

XY 1.2 - 1.728 - 12 mat

... Using eval(,)

NOT commutative t X D C X 1 el X 2 el 21 mat * :

C 3 - 113 - 22 mat :

t Clear 1

evaporate 'X' X Clear 1

t X D 3 X 1 el * - X 2 el + X 1 el 3 X 2 el * - 21 mat

N 100 : knot 1 : 21 line

Rotate the plot ... augment(col(U0,2),col(U0,3))

Knot U0 00013 mat : p 1 - 1 0.75 - range W p knot 21 mat : U W 02 N t X D rkfixed : U0 U0 U stack : 31 line for U0 3 col U0 2 col augment 31 line :

Knot Knot eval :

Knot Knot - 2 1 sys

Knot 101 row Knot 102 row Knot 103 row 31 mat 0.0004 0.0004 - 12 mat 0.0003 0.0003 - 12 mat 1 0.75 - 12 mat 31 mat

Piecewise collector
Help from plot 'N' idx 1 Knot rows range : N idx Knot augment : 21 sys

N1 N 2102 range 23 range el : N2 N 103203 range 23 range el : N3 N 204304 range 23 range el : N4 N 305405 range 23 range el : N5 N 406506 range 23 range el : N6 N 507607 range 23 range el : N7 N 608708 range 23 range el : N8 N 709809 range 23 range el : N9 N 810910 range 23 range el : 91 sys

plot N1 N2 N3 N4 N5 N6 N7 N8 N9 91 sys :

plot

Dual solver 'root'

evaporate from above use x Clear 1

Approximations ..... Prepare

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 :

Entropy saturation

Entropy 10 0.03596 2.1262 50 0.1681 1.9293 100 0.3121 1.7568 150 0.4398 1.6327 200 0.5567 1.5353 250 0.6673 1.4499 300 0.7774 1.3636 370 0.9820 1.1498 83 mat :

X Entropy 1 col : Y Entropy 2 col : 12 mat EntropyWaterWater X Y augment . 12 black plot : 21 line

X Entropy 1 col : Y Entropy 3 col : 12 mat EntropySteam X Y augment . 12 black plot : 21 line

se 1.87804 104 -^* - 3.8091 103 -^* 1.76923 105 -^* - 3.3417 107 -^* 4.7508 109 -^* - 3.9818 1011 -^* 1.9778 1013 -^* - 5.7415 1016 -^* 8.998 1019 -^* - 5.8805 1022 -^* 101 mat :

kcal_kg 1 :

sv 2.18785 6.6924 103 -^* - 4.34896 105 -^* 3.62069 107 -^* - 2.4730 109 -^* 1.0247 1011 -^* - 2.2014 1014 -^* 1.8911 1017 -^* - 81 mat :

Entropy water phase x se se j el x j 1 -^* j 110 sum 11 line :

Entropy vapor phase x sv sv j el x j 1 -^* j 18 sum 11 line :

0.01 sv 374 sv 21 mat 2.1878 1.1203 21 mat

s 0.01 x ≤ (x 374 ≤ (* :

x se s / x sv s / EntropyWaterWater EntropySteam 4 1 sys

0.01 sv 374 sv 21 mat 2.1878 1.1203 21 mat

x f x sv :

Dual Solving from undocumented 'root' ...

Reduced form

w sol t f w - t root : 4375720000 / t 11 -^* 167312500000 / - (t 21 -^* + 27181625000000 / t 31 -^* + 3620691000000000000 / - (t 41 -^* + 24731000000000000 / t 51 -^* + 102471000000000000000 / - (t 61 -^* + 11007500000000000000000 / t 71 -^* + 189111000000000000000000000 / - (t 81 -^* + w - t root

t q 4375720000 / (t 11 -^* 167312500000 / - (t 21 -^* + 27181625000000 / (t 31 -^* + 3620691000000000000 / - (t 41 -^* + 24731000000000000 / (t 51 -^* + 102471000000000000000 / - (t 61 -^* + 11007500000000000000000 / (t 71 -^* + 189111000000000000000000000 / - (t 81 -^* + w - :

Given 'w' ... solve for 't' w t Sol_1 t q 0 ≡ t 0374 solve :

Given 't' ... solve for 'w' w t Sol_2 t q 0 ≡ w 1.12 2.187 solve :

t 220 :

1.5004 WASP Katmar w w t Sol_2 (: 1.5016

sanity confirmed °C w t Sol_1 (: 220

Renew thw plot after you have Sol_1(w,t) from ... Sol_2(w,t)

x q s / t 0.5 t 0.5 - 2 2 mat 2 1 sys

x q s / x 0.01 374 int 35.2883

x q s / x 0.01 t int 58.0725

Difficult solve 'seeded inverser'

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 00102 0.143 3 0.727 4 0.889 5 0.875 6 0.714 7182 mat :

β 7 2.5 21 mat :

Transmute Sigmoid

x β f β 1 el 1 x / (β 2 el^* - exp :

plot XY o 5 black plot :

1000 β f 1.000000

ε 1012 -^:

t0 1 time :

Design a discrete function inverser  in the L, Δ increment , N points f L Δ N Inverse i 1 N range ω i 1 + el L Δ i * + : ω 1 el L : 21 line for ω n 1 ω rows range sol n el x f ω n el - 0 ≡ x 0100 solve : n n 1 + : 21 line for ω sol augment 41 line :

x f x β f eval :

'N' 100 reverts to singular 
end point ...  not solvable SOL f ε 0.01 99 Inverse :

SOL SOL rows row 0.99 13.714 12 mat

SOL

Thiele fit suggested

1 time t0 - 33 s *

5 min f ε 0.001 999 Inverse 0.999 34.5106 12 mat ≡

Rotavirus . . . Pathogen, Microorganism

Collected from Steven Finch [Mathsoft].

Solved otherwise [jmG], Transmute Sigmoid

]]>

Command line solve Y 0.825 :

X x f Y - 0 ≡ x 0100 solve :

track 0 Y X Y X 032 mat :

track x β f plot 3 1 sys

Y X 12 mat 0.825 4.211 12 mat

Thiele fit 'seeded inverser'

XY

XY 11012 -^* 0.5 0.01 1.1823 0.02 1.262 0.03 1.3185 0.04 1.3644 0.05 1.4042 0.06 1.4399 0.07 1.4728 0.08 1.5034 0.09 1.5324 0.1 1.5601 0.11 1.5867 0.12 1.6124 0.13 1.6375 0.14 1.6619 0.15 1.6858 0.16 1.7093 0.17 1.7324 0.18 1.7553 0.19 1.7779 0.2 1.8004 0.21 1.8227 0.22 1.8449 0.23 1.867 0.24 1.8891 0.25 1.9112 0.26 1.9332 0.27 1.9553 0.28 1.9775 0.29 1.9997 0.3 2.0221 0.31 2.0445 0.32 2.0671 0.33 2.0899 0.34 2.1128 0.35 2.136 0.36 2.1593 0.37 2.1829 0.38 2.2068 0.39 2.231 0.4 2.2554 0.41 2.2802 0.42 2.3053 0.43 2.3308 0.44 2.3567 0.45 2.383 0.46 2.4098 0.47 2.437 0.48 2.4647 0.49 2.493 0.5 2.5218 0.51 2.5512 0.52 2.5812 0.53 2.6119 0.54 2.6434 0.55 2.6755 0.56 2.7085 0.57 2.7423 0.58 2.7769 0.59 2.8126 0.6 2.8492 0.61 2.887 0.62 2.9259 0.63 2.966 0.64 3.0074 0.65 3.0503 0.66 3.0946 0.67 3.1406 0.68 3.1883 0.69 3.2379 0.7 3.2895 0.71 3.3433 0.72 3.3996 0.73 3.4584 0.74 3.5201 0.75 3.5849 0.76 3.6531 0.77 3.7251 0.78 3.8013 0.79 3.8822 0.8 3.9683 0.81 4.0603 0.82 4.1589 0.83 4.2651 0.84 4.3799 0.85 4.5048 0.86 4.6414 0.87 4.7919 0.88 4.9589 0.89 5.1459 0.9 5.3576 0.91 5.6003 0.92 5.8832 0.93 6.2192 0.94 6.6288 0.95 7.1452 0.96 7.8286 0.97 8.8014 0.98 10.3723 0.99 13.7141 1002 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 :

t0 1 time :

XY rows 100

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

list 1410203555759097100110 mat :

data XY list ListCollect :

support data . 10 black plot :

X data 1 col : Y data 2 col : 12 mat

Thiele system

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 :

X Y Δ Thiele ε 1018 -^: n X rows : i 1 n range M i 1 el Y i el : for j 2 n range i j n range dM M i j 1 - el M j 1 - j 1 - el - : dM dM abs ε < ε M i j el X i el X j 1 - el - dM / : if : 21 line for for i 1 n range c i 1 el X i el : c i 2 el M i i el : 21 line for Δ 0 ≡ c M if 61 line :

length Thiele j data rows : 10

Thiele on length of support points 'j' K X x Cfr N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (0 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 51 line :

recast the Cf on the system  [K,X,x] x Cf K X x Cfr :

Thiele K's continued fraction K X Y 0 Thiele 2 col :

Thiele v 0 : x 01 0.01 range : i 1 x rows range v i el x i el Cf eval : for x v augment 41 line :

support Thiele 2 1 sys

0.825 Cf 4.2123

1 time t0 - 0.9 s *

X Y K x Clear 1

j 5 :

Thiele on length of support points 'j' K X x Cfr N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (0 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 51 line :

Create a user Thiele continued fraction K X x Cfr K 1 el x X 1 el - K 2 el x X 2 el - K 3 el x X 3 el - K 4 el x X 4 el - K 5 el / + / + / + / +

X Y K Clear 1

Thiele fit 'common style'

This style may fail some projects wrt try/on error

XY ,,,

XY 0.0024 0.0294 0.1152 8.9573 0.2376 17.6451 0.36 26.2891 0.492 35.1373 0.6288 43.912 0.7824 52.5544 0.9504 61.1106 1.1544 69.4454 1.3728 76.5483 1.6008 82.3313 1.8456 86.9767 2.1048 90.5245 2.3784 93.006 2.6808 94.3653 152 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

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 :

k 0.0294 0.012635 170.669427 0.001563 214.319145 0.010379 9.660668 - 0.016125 - 867.718882 0.000104 - 1138.092233 - 0.02598 - 28.076798 - 0.025723 - 35.190142 151 mat

... NUMERIC ...

Recast the Project

isolate 'K' to get rid of "round"

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 :

K 0.0294 0.012635 170.669427 0.001563 214.319145 0.010379 9.660668 - 0.016125 - 867.718882 0.000104 - 1138.092233 - 0.02598 - 28.076798 - 0.025723 - 35.190142 151 mat :

x λ 10 x ≤ (x 3 ≤ (& cases :

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

NUMERIC

SYMBOLIC

vx XY 1 col : vy XY 2 col : 21 sys

x interp vx vy x ainterp :

1.23456789 Cf 1.23456789 interp 21 mat 72.25 72.24 21 mat

XY . 12 black plot x Cf x λ * 2 1 sys

Discontinuous 'min(x)'

min("matrix") - 
Returns the smallest element of vector or matrix.
If any value is complex, returns 
min(Re(...)) + i*min(Im(...)). x constrains 2110 x - (* 260 x - 2 / 5128 x - (* 4 / 31 mat min :

x bound 1080 x - (* 7 / :

x constrains x bound 41 200 41 64 - 2 2 mat 80 200 80 64 - 2 2 mat 4 1 sys

D x 01001 range : i 1 x rows range C i el x i el constrains : for x C augment 31 line :

x constrains x bound 402004064 - 22 mat 802008064 - 22 mat fill 51 sys

x y1 x constrains : x y2 x bound : 21 line

'N' random in critical region N 1000 :

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 :

size/define critical region.
Plot first before filling xmin 32 - : xmax 200 : ymin 200 - : ymax 200 : 22 mat space_x xmax xmin - (: space_y ymax ymin - (: 21 line 21 sys

A function that tests an arbitrary point 
for its membership in the critical region. x y Test 1 y x y1 ≤ (* y x y2 ≥ (* :

A routine that finds an array of N random 
pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

a b c spec . 6 #808080 13 mat : fill Pts a b c spec pts : 21 line

Filling a range of a critial domain

x constrains x bound 40 200 40 64 - 2 2 mat 80 200 80 64 - 2 2 mat fill 5 1 sys

More geometrics 'from user vertices'

isolate for convenience X 3.08 2.73 531 mat : Y 20.77 17.27 1531 mat : 12 mat

Centroid X 1 el X 2 el + X 3 el + 3 / Y 1 el Y 2 el + Y 3 el + 3 / augment : 3.6033 17.68 12 mat

Area of Polygons wrt vertices coordinates A 1 - 2 / X i el Y i 1 + el * X i 1 + el Y i el * - (i 1 X rows 1 - sum * : 24.45525

a X 1 el X 3 el - (2^Y 1 el Y 3 el - (2^+ sqrt : 6.0811

Geom X Y augment 3.08 20.77 12 mat stack :

Centoid 3.6033 17.68 12 mat :

b X 1 el X 2 el - (2^Y 1 el Y 2 el - (2^+ sqrt : 3.5175

Geom 3.08 20.77 2.73 17.27 515 3.08 20.77 42 mat :

Geom Centroid 2 1 sys

c X 2 el X 3 el - (2^Y 2 el Y 3 el - (2^+ sqrt : 3.2103

Perimetre Perim a b + c + : 12.8088

clear(Y,U,,) before next

t Clear 1

Y Clear 1

U Clear 1

A 11 - 21 - 22 mat :

t U D A U 1 el U 2 el 21 mat * :

022 - 022 mat

t U D U 1 el U 2 el - 2 U 1 el * U 2 el - 21 mat

N 100 :

center U0 00013 mat : p 1 - 11 - 0.25 + (range center p 221 mat : U center 02 π * N t U# D rkfixed : U0 U0 U stack : 31 line for U0 3 col U0 2 col augment 31 line :

center center eval :

center

ellipse U0 00013 mat : p 1 - 11 - 0.25 + (range ell p 1.3333 21 mat : U ell 02 π * N t U# D rkfixed : Ω U0 U stack : 31 line for Ω 3 col Ω 2 col augment 31 line :

ellipse ellipse eval :

Ellipse ellipse 2 N 2 + (range 12 range el :

1.5 1.25 1.5 0.5 - 2 2 mat Ellipse 2 1 sys

Center U0 00013 mat : p 1 - 11 - 0.25 + (range Ctr p 121 mat : U Ctr 02 π * N t U# D rkfixed : U0 U0 U stack : 31 line for U0 2 col U0 1 col augment U0 1 col U0 2 col augment 21 mat 31 line :

center center eval :

Center 1 el Center 2 el 2 1 sys

circles from observatory

r 1 :

A 11 - 21 - 22 mat :

A 0 r r - 022 mat :

circles
ellipses/ p 010 0.5 + (range (≡ p 1 - 11 - 0.25 + (range (≡ 21 sys

t U D A U 1 el U 2 el 21 mat * :

a set of straight lines 11 - 11 - 22 mat

t U D U 2 el U 1 el - 21 mat

N 100 :

experiment
◄  observatory p 0.25 21 mat

circles U0 00013 mat : p 010 0.5 + (range cir p 0.25 21 mat : U cir 02 π * N t U# D rkfixed : U0 U0 U stack : 31 line for U0 3 col U0 2 col augment 31 line :

circles circles eval :

circles

The svg circle ...

r 1 : svg 1.25 : N 360 : 13 mat

A 0 r r - 022 mat :

t U D A U 1 el U 2 el 21 mat * :

t U D U 2 el U 1 el - 21 mat

svgCir U0 00013 mat : p 010 0.5 + (range cir p svg 21 mat : U cir 02 π * N t U# D rkfixed : U0 U0 U stack : 31 line for U0 3 col U0 2 col augment 31 line :

circles svgCir eval :

circles rows 1084

C circles 2272 range 12 range el :

XY C C rows row (: 0 1.25 - 12 mat

C svg 1 svg 1 - 2 2 mat 2 1 sys

circles 2362 range 12 range el circles 363723 range 12 range el circles 7241084 range 12 range el 31 sys

XY 0 1.25 - 12 mat

Pie chart quick X_Y

ODE produces exact [0 ... 360°]

expand to see cleared system

Radial Stem [q:=0,v:=0,S:=0]

--------
radius circle
-------- Y0 01031 mat :

t0 0 : t1 2 π * : N 360 : 31 line

u t X Y Clear 1

Rename: Y => .. else variable name t u D u 2 el u 3 el 1 u 1 el (2^- (* u 2 el * (u 1 el - 031 mat :

Either 'rkfixed' or 'Rkadapt' solution Y0 t0 t1 N t u D dn_GearsBDF :

X solution 2 col : Y solution 3 col : 12 mat circ X Y augment : 21 mat

Core component of the stem style P v replicate k 1 : i 1 v length range P i el 1 ≡ continue j 1 P i el range T k j + el v i el : for k k P i el + : 21 line if for T 31 line :

Stem construction vy stem q 0 : v 0 : S 0 : 13 mat N vy length : i 1 N range q i el 3 : q N el 2 : 21 line for 11 line i 1 N range v i el i : for i 1 N range S 3 i * el vy i el : for u q v replicate : V u S augment : V V 3 V rows 12 submatrix : 81 line :

i j pie circ i j range 12 range el :

i j p i j pie 1 col stem 2 col i j pie 2 col stem 2 col augment :

x y axes x y / - :

p(i,j) ...
conformal Transform pie 172 p 72122 p 122230 p 230284 p 284327 p 327360 p 61 sys :

Get the 'j' from your Impressario ... compose 'pie'

pie axes 2 1 sys

DeCasteljau svg primer

DeCasteljau

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 :

Understanding SVG

Model from jmG Mathcad 8 Pro. made work in Smath by Dave Carpi

ndp =  number of discretization points (> 1) in the prescribed range  0..1
p = control points ... each segment is populated 'ndp'. ndp p DeCasteljauNDP 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 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 : ---------------------- 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 81 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 :

XY 100020201015 - 05050015 - 26 mat transpose :

a b c spec . 14 black 13 mat :

xy XY a b c spec pts :

svg 1 :

p svg 1010015 - 15 - 023 mat 000005023 mat 0102050605023 mat 202020500023 mat 201010015 - 15 - 23 mat 15 mat transpose * :

curved region...adjust '60'

NDP 100 :

C 0200020000505025 mat transpose :

level 510151022 mat :

bin NDP p 1 el DeCasteljauNDP :

NDP 100 : bin NDP p 1 el DeCasteljauNDP : i 2 p length range bin bin NDP p i el DeCasteljauNDP stack : for 31 line

Thanks & credits to Davide Carpi 3/4 2015 tutoring MCD 8 Pro. So gorgeous Smath !!!

C bin level 3 1 sys

p 1015 - 1015 - 0032 mat 000005032 mat 0501060205032 mat 205020020032 mat 2001015 - 1015 - 32 mat 51 mat

bin 2 col Max 55.00

Solve for a given aperture ... plot/svg

solving washer aperture Ω 1.25 : OD 2 : 21 sys

clear imaginary i Clear 1

once rougly solved ...
manage the bracket  range sol OD w i π * w /^eval * Re Ω - 0 ≡ w 12 solve :

sol 1.56019262501239

svg 1 :

'n' population
'w' aperture  
radius OD svg n 2000 : w sol : OD 2 : 31 sys *

a svg OD w i π * w /^eval * Re * : 1.25000000

x circle a 2^x 2^- sqrt ± :

Disk j 1 n range F j el OD w i 2 * π * j w / *^eval * : for F Re F Im augment 21 line :

a 0 + 200 blue 1 5 mat x circle svg Disk * 4 1 sys

CWD C:\SmathFile CurrentDirectory :

Disk CWD DiskAperture wfile 1

Disk CWD DiskAperture rfile :

disk Disk 18 range 12 range el :

disk

Analog clock ...

Settings

Starting Time

hour hand HH 16 hr * :

minute hand MM 26 min * :

second hand SS 50 s * :

Duration

duration D 0 hr * 2 min * + 35 s * + :

Accuracy

frame skip accuracy 1 s * :

Clock geometry

radius of hour hand r.hh 1.25 :

radius of minute hand r.mm 2 :

radius of second hand r.ss 2 :

dial radius r.dial r.hh r.mm r.ss 31 mat max 1.1 * eval :

rim radius r.rim r.dial 1.1 * :

Algorithms

dial k 112 range dial k 1 el r.dial 2 π * 12 / k * sin * eval : dial k 2 el r.dial 2 π * 12 / k * cos * eval : dial k 3 el . : dial k 4 el 15 : dial k 5 el #cc000000 : 51 line for

rim k 1120 range rim k 1 el r.rim 2 π * 120 / k * sin * eval : rim k 2 el r.rim 2 π * 120 / k * cos * eval : rim k 3 el . : rim k 4 el 8 : rim k 5 el #cc000000 : 51 line for

fastest way to plot the rim rim r.rim (2^x 2^- sqrt ± :

tRange 0 D s / accuracy s / range eval :

s clock ss SS s / s + eval : mm MM min / ss 60 / trunc + eval : hh HH hr / mm 60 / 2 round + eval : r.mm 002 π * 60 / mm * sin 2 π * 60 / mm * cos 22 mat * eval r.ss 002 π * 60 / ss * sin 2 π * 60 / ss * cos 22 mat * eval r.hh 002 π * 12 / hh * sin 2 π * 12 / hh * cos 22 mat * eval dial rim 00.20 #101010 15 mat 61 sys 41 line :

use this to have movements of hour hand every 60 minutes hh HH hr / ss 3600 / trunc + :

This version ... adapted Friedel.

t clock

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