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D'Arcy Thompson transformations

Author:
hawe
Topic:
Matrices, Plane Figures or Shapes, Similarity Transformation or Similarity, Geometric Transformations, Translation

DArcyThompson Transformation

Argyropelecus olfersi - Sternotyx diaphana (70° Shear)

Argyropelecus olfersi - Sternotyx diaphana (70° Shear)
Fig={(-2.59,0.5),(-2,1),(-1,1.5),(-0.5,1.5),(0,1),(0.3,0.8),(2,0.5),(2.5,0.8),(2.8,0.9),(3,0.7),(2.7,0.1),(3,-0.5),(2.8,-0.7),(2.5,-0.6),(1.9,-0.2), (1.2,-0.3),(0.3,-0.7),(-0.2,-2.2),(-0.5,-2.1),(-0.7,-2.066),(-1,-2),(-1.2,-1.87),(-1.8,-1.7),(-2.12,-1.72),(-2.41,-1.29),(-2.74,-1),(-2.933,-0.6), (-2.98,-0.2),(-2.82,-0.3),(-2.58,-0.3),(-2.55,-0.2),(-2.77,-0.1),(-3,0),(-2.51,0.1),(-1,1.5),(-0.9,1.7),(-0.5,2.1),(-0.6,1.7),(-0.5,1.5), (0,1.9),(0.4,1.6),(0.8,1.2),(0.5,1),(0.3,0.8),(1.2,-0.3),(0.9,-0.7),(0.8,-1.13),(0.3,-0.7),(-0.86,-1.54),(-0.29,-0.94),(-0.5,-0.4),(-1.19,-1.27)} Grid={{1,2,3,4},{4,5,6,7},{7,8,9,10},{10,11,12},{12,13,14},{14,15,16,17,18},{18,19,20},{20,21,22},{22,23,24},{24,25,26},{26,27,28}, {28,29,30,31,32,33,28},{30,34,1},{35,36,37,38,39,40,41,42,43,44},{45,46,47,48}, {49,50,51,52}} ab_ij={0, 0.22, 0, 1, 0, -0.09, 0, 0, 0, 1} ab_ij={-1/10, 9/50, 17/50, 9/10, 9/50, 3/100, -3/100, 1/20, 3/50,1}

Scarus sp. - Pomacanthus

Scarus sp. - Pomacanthus
Fig={(-581/200,29/100),(-129/100,53/50),(9/5,43/100),(131/50,71/100),(147/50,-3/100),(67/25,-21/25),(46/25,-41/100), (63/100,-33/50),(-1/4,-43/50),(-43/20,-12/25),(-1377/500,-2/25),(-1323/500,1/50),(-1377/500,9/100),(-581/200,1/10), (-22/25,133/100),(1/20,63/50),(38/25,23/25),(7/4,3/4),(29/20,21/50),(147/100,-21/50),(167/100,-33/50),(119/100,-107/100), (31/100,-111/100),(3/100,-81/100),(-47/50,-3/4),(-79/100,-109/100),(-43/50,-1497/1000), (-149/100,-67/100)} A=(-2.25, 0.25) Grid={{1,2,3,4}, {4,5,6}, {6,7,8,9}, {9,10,11}, {11, 12,13,14,1}, {2,15,16,17,18,19}, {20,21,22,23,24}, {25,26,27,28}} ab_ij={0, 0, 31 / 50, 1, 0, 0, -43 / 100, 1 / 100, 1 / 25, 137 / 100}

Skulls

Skulls
Fig={(483/100,-6/25),(47/10,8/5),(53/25,349/100),(-101/100,381/100),(-359/100,147/50),(-423/100,197/100),(-483/100,1/25), (-497/100,-37/25),(-41/10,-263/100),(-54/25,-141/50),(-11/10,-82/25),(-9/50,-159/50),(133/100,-71/20),(9/5,-61/20),(7/4,-217/100), (59/20,-213/100),(387/100,-199/100),(451/100,-113/50),(479/100,-167/100),(114/25,-93/100),(26/5,-7/10),(493/100,-97/50),(21/4,-143/50), (93/20,-157/50),(4,-3),(191/50,-67/25),(31/10,-3),(13/5,-13/5),(11/5,-13/5),(27/10,-33/10),(59/20,-447/100),(419/100,-507/100),(529/100,-122/25), (26/5,-113/25),(21/4,-411/100),(41/10,-397/100),(267/50,-329/100),(127/25,-67/20),(5,-3),(467/100,-343/100),(107/25,-343/100), (419/100,-307/100),(197/50,-67/20),(261/50,-363/100),(491/100,-369/100),(97/20,-4),(227/50,-183/50),(89/20,-199/50),(207/50,-179/50), (391/100,-18/5),(22/5,-37/100),(211/50,-117/100),(357/100,-169/100),(211/100,-169/100),(159/100,-189/100)} Grid={{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,1},{1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,25},{23,37,38,39}, {38,40,24},{40,41,42}, {41,43,50,49,36},{49,47,48},{47,45,46},{45,44,35},{52,53,54,55}} ab_ij={3 / 100, -2 / 25, 9 / 100, 53 / 50, 0, 1 / 20, 0, -3 / 100, 7 / 100, 19 / 20}

Bedienung

Fig enthält die Punkte-Liste mit den figurbeschreibenden Punkten. Die Punktliste Fig (abgelegt in FigO) wird durch Abtasten eines Bildes erstellt. Mit dem Marker (der rote Punkt rechts) werden die (Splinekurven)-Punkte markiert und mit [++] nach Fig geschrieben. (POS=n einfügen an Pos n, POS=-n ersetze POS n) Grid enthält die Spline-Beschreibung, die Nummern der Punkte die mit einem Spline-Polynom verbunden werden sollen. Der Fisch Scarus Sp. wird mit 5 Splines gezeichnet durch die Punkte {{1, 2, 3, 4}, {4, 5, 6}, {6, 7, 8, 9}, {9, 10, 11}, {11, 12, 13, 14, 1}} Ich hab in diesem Beispiel ein Bild (aus mathshistory) des Scarus sp. einkopiert und mit dem Marker abgetastet und die Punkte mit Spline-Kurven verbunden -> FigOShape. FigDT (P(x,y),Q(x,y)), FigDT stellt die mit den Funktionen P,Q abgebildetet Punkte bzw. FigDTShape die damit generierten Splines-Kurven dar. ab_ij Einstellung für Slider a,b bei update, Input {} übertrage Slider a,b, nach ab_ij Die App zeigt Instabilitäten - vermutlich die Splines betreffend - für Entwurfsarbeiten ggf. Π für Polylines. Snoopy-Diddle morph Fig={(2.08, 1.74), (1.54, 2.61), (0.46, 2.82), (-0.47, 3.48), (-2.09, 2.76), (-2.18, 0.84), (-1.25, 0.72), (-1.16, 2.01), (-0.77, 0.78), (-0.77, -0.84), (-0.95, -1.29), (-0.68, -1.38), (-0.64, -1.64), (-0.44, -1.47), (-0.17, -1.5), (-0.08, -1.26), (-0.26, -0.51), (1.57, 0.96), (0.1, 0.48), (0.43, -0.33), (0.16, -0.93), (0.31, -1.65), (1.15, -1.02), (1.27, -2.07), (0.52, -2.34), (-0.71, -2.25), (-1.64, -2.19), (-1.88, -2.07), (-1.28, -1.86), (0.94, -0.59), (1.27, -0.3), (1.72, -0.96), (1.75, -2.07), (2.38, 1.67), (2.38, 1.27), (1.93, 1.29), (-0.92, -1.81)} Grid={{1, 2, 3, 4, 5, 6, 7, 8}, {7, 9, 10, 11, 12, 13, 14, 15, 16, 17}, {1, 18, 19, 20, 21, 22, 15}, {20, 23, 24, 25, 26, 27, 28, 29, 26}, {30, 31, 32, 33, 24}, {1, 34, 35, 36}, {11, 37, 26}} ab_ij={0, -0.1, -0.02, 0.78, 0.06, 0.02, -0.02, -0.14, -0.02, 1} Links https://mathshistory.st-andrews.ac.uk/Darcy/darcy/ https://observablehq.com/@stringertheory/stretchy-fish http://www.mcs.st-and.ac.uk/~dat/page4.html

Transformation Show

Transformation Show

Exzenter Switch [ )( ]

Exzenter Switch [ )( ]
Exzenter switch [ )( ] gegenläufige Biege a22 rechts/links, b11 oben/unten, + Exzenter-Point: Biege-Punkt ab_ij={-0.07, -0.04, 0.59, 1.1, 0.23, -0.02, -0.39, -0.04, 0.03, 1.6}

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