Xah Lee, 1998-03
The following are images of transformations in the plane. These graphics are generated by my Mathematica packages Transform2DPlot.m and PlaneTiling.m. You can get them at Geometric Transformation on the Plane.
Saturn. The preimage and image of a linear transformation on a polar grid by the matrix {{3,-2},{1,0}}. The matrix has two independent eigenvectors {1,1} and {2,1}, indicated by blue lines. Their significance is that points on those lines will remain on those lines.
Bloody Tai-Chi. Varingconc entric rotation applied to a hexagonal grid. In Mathematica notation, the function is: Function[{x,y},({{Cos[#1 n],-Sin[#1 n]},{Sin[#1 n],Cos[#1 n]}}&)[Sqrt[x^2+y^2]].{x,y}]
Varying concentric rotation applied to half of a polar grid.
Starwave. The function Function[With[{l = Sqrt[#1^2 + #2^2]},Sin[l]*0.4* ({#1,#2}/l)+{#1,#2}]] applied to a wallpaper design.
Stareye. The function Function[{#1,#2}/(Sqrt[#1^2 +#2^2] + 5)] applied to a wallpaper design of stars. This function is often called fish-eye lens.
Polar mutate. The function Function[{#2, Cos[#1*#2]}] applied to a polar grid.
A note about the notation. The notation used on this page is from Mathematica. For example: Function[{#2, Cos[#1*#2]}] means f(x,y):=(y,Cos[x*y]). In general, “#1” means the first argument, and “#2” is the second argument and so on. A pair (a,b) is written as {a,b}. And, {a,b}*c means (a*c, b*c). And, {a,b}/c means (a/c, b/c). A 2 by 2 square matrix is written as {{a,b},{c,d}}, with {a,b} being the top row. And, “{{a,b},{c,d}} . {x,y}” means matrix multiplication on the vector {x,y}, resulting: {a x + b y, c x + d y}.
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