Geometry: Transformation of the Plane II

By Xah Lee. Date: . Last updated: .

The following are images of transformations in the plane.

linear transformation
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.
bloodyTaiChi.png BM3rC
Bloody Tai-Chi Varing concentric rotation applied to a hexagonal grid.
Clear[n]; n = N[Pi/3];
Transform2DGraphicsPlot[TriangularGrid[Pi, 49],
   y}, ({{Cos[#1*n], -Sin[#1*n]}, {Sin[#1*n], Cos[#1*n]}} & )[
    Norm[{x, y}]] . {x, y}], ResolutionLength -> 0.2,
   AspectRatio -> Automatic]
Varying concentric rotation applied to half of a polar grid.
Starwave. The Sin[Norm[v]] * 0.4 * Normalize[v] + v applied to a wallpaper design.
Stareye. 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. Function[{#2, Cos[#1*#2]}] applied to a polar grid.

Notation Used

The notation used on this page is from Wolfram Language.

These graphics are generated by my Mathematica packages Transform2DPlot.m and PlaneTiling.m. You can get them at WolframLang: Transform2DPlot Package 📦 and WolframLang: Plane Tiling Package 📦 .