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.
Bloody Tai-Chi. Varing concentric rotation applied to a hexagonal grid. Function[{x,y},({{Cos[#1 n],-Sin[#1 n]},{Sin[#1 n],Cos[#1 n]}}&)[Norm[{x,y}]].{x,y}]
Varying concentric rotation applied to half of a polar grid.
transformation
Starwave. The Sin[Norm[v]] * 0.4 * Normalize[v] + v applied to a wallpaper design.
transformation
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.
transformation
Polar mutate. Function[{#2, Cos[#1*#2]}] applied to a polar grid.

Notation Used

The notation used on this page is from Mathematica.

These graphics are generated by my Mathematica packages Transform2DPlot.m and PlaneTiling.m. You can get them at Mathematica Package: Geometric Transformation and Complex Function Plot and Geometry: Plane Tiling Mathematica Package.