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#1
is the first argument and#2
indicate 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}*c
means{a*c,b*c}
. It is automatically distributive.{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. {{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 Package 📦 and WolframLang: Plane Tiling Package 📦 .