Geometry: Transformation of the Plane II
The following are images of transformations in the plane.
{{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.
Clear[n]; n = N[Pi/3]; Transform2DGraphicsPlot[TriangularGrid[Pi, 49], Function[{x, y}, ({{Cos[#1*n], -Sin[#1*n]}, {Sin[#1*n], Cos[#1*n]}} & )[ Norm[{x, y}]] . {x, y}], ResolutionLength -> 0.2, AspectRatio -> Automatic]
Sin[Norm[v]] * 0.4 * Normalize[v] + v applied to a wallpaper design.
Function[{#1,#2}/(Sqrt[#1^2 +#2^2] + 5)] applied to a wallpaper design of stars. This function is often called fish-eye lens.
Function[{#2, Cos[#1*#2]}] applied to a polar grid.
Notation Used
The notation used on this page is from Wolfram Language.
- A 2D vector is written as
{x,y}. - A square function is written as
Function[#1^2]. The#1is the first argument and#2indicate second argument, and so on. For example,Function[#1+#2]is the same asFunction[{x,y},x+y], andFunction[{#2, Cos[#1*#2]}]is the same asFunction[{x,y}, {y, Cos[x*y]}]orf[x_,y_]:={y, Cos[x*y]}. {a,b}*cmeans{a*c,b*c}. It is automatically distributive.{a,b}/cmeans{a/c, b/c}.- A 2 by 2 square matrix is written as
{{a,b},{c,d}}, with{a,b}being the top row. {{a,b},{c,d}} . {x,y}means matrix multiplication with the vector{x,y}, resulting:{a x + b y, c x + d y}.Norm[{x,y}]is the length of a vector{x,y}.Normalize[{x,y}]is the unit vector of{x,y}.
These graphics are generated by my Mathematica packages Transform2DPlot.m and PlaneTiling.m. You can get them at WolframLang: Transform2DPlot Mathematica Package and WolframLang: Plane Tiling Mathematica Package .