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 my Mathematica Packages page.
above: 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.
above: 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}]
above: Varying concentric rotation applied to half of a polar grid.
above: 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.
above: 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.
above: 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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Page created: 1998-05-22. © 1998-2006 by Xah Lee.