Planar Isometries

Helmut Knaust, Department of Mathematical Sciences, UTEP, El Paso TX 79968
hknaust@utep.edu
2/22/2012

Isometries in the Complex Plane

Type 1: Rotation and Translation

The definition of the isometry:

f[z_]:=a+Exp[I θ] z

The translation “vector” a:

a=2+3I

The rotation angle θ:

θ=-π/6

CLine[z_]:=Line[Transpose[{Re[z],Im[z]}]]
w:=Exp[I θ];

The blue rectangle gets rotated into the dashed triangle, then translated intored triangle:

Graphics[{{Blue,Thick,CLine[{-I,I,1+I,-I}]},{Blue,Thick,Dashed,CLine[w{-I,I,1+I,-I}]},{Red,Thick,CLine[{f[-I],f[I],f[1+I],f[-I]}]}},Axes->True,PlotRange->All]

Note that this isometry preserves the orientation of the triangle.

Type 2: Conjugation, Rotation and Translation

The translation “vector” a:

a=1+2I

The rotation angle θ:

θ=π/3

The definition of the isometry:

f[z_]:=a+Exp[I θ] z\[Conjugate]

(SuperStar[z]) denotes the complex-conjugate of z.

The blue rectangle is reflected about the real axis into the dotted triangle, then rotated into the dashed triangle, finally translated into the red triangle:

Graphics[{{Blue,Thick,CLine[{-1-I,I,1+I,-1-I}]},{Blue,Thick,Dotted,CLine[{-1+I,-I,1-I,-1+I}]},
{Blue,Thick,Dashed,CLine[Exp[I θ]{-1+I,-I,1-I,-1+I}]},{Red,Thick,CLine[{f[-1-I],f[I],f[1+I],f[-1-I]}]}},Axes->True,PlotRange->All]

Note that this isometry reverses the orientation of the triangle.

Isometries in the Real Plane

We will only consider isometries that have the origin as a fixed point (no translations).
Using the representation of a complex number a+b i as a matrix (({{a, -b}, {b, a}})), we can define a rotation matrix  and a reflection matrix:

Rot[θ]:={{Cos[θ],-Sin[θ]},{Sin[θ],Cos[θ]}}
Refl[θ]:={{Cos[θ],Sin[θ]},{Sin[θ],-Cos[θ]}}

Multiplication by Rot[θ] is a rotation about the origin by angle θ:

Manipulate[Graphics[{{Dotted,Circle[{0,0},Sqrt[5],{ArcTan[2],ArcTan[2]+θ}]},{Dotted,Circle[{0,0},Sqrt[13],{π+ArcTan[-3/2],π+ArcTan[-3/2]+θ}]},{Blue,Thick,Line[{{1,2},{-2,3}}]},{Red, Thick,Line[{{{Cos[θ],-Sin[θ]},{Sin[θ],Cos[θ]}}.{1,2},{{Cos[θ],-Sin[θ]},{Sin[θ],Cos[θ]}}.{-2,3}}]}},Axes->True,PlotRange->{{-4,4},{-4,4}}],{θ,0,2π,.1}]

Multiplication by Refl[θ] is a reflection about the line through the origin with slope tan(θ/2)

Manipulate[Graphics[{{Dotted,Circle[{0,0},2,{0,θ/2}]},{Text["(θ/2)",1.5{Cos[θ/4],Sin[θ/4]}]},{Dotted,Line[{{-Sqrt[13]Cos[θ/2],-Sqrt[13]Sin[θ/2]},{Sqrt[13]Cos[θ/2],Sqrt[13]Sin[θ/2]}}]},{Blue,Thick,Line[{{1,2},{-2,3}}]},{Red, Thick,Line[{{{Cos[θ],Sin[θ]},{Sin[θ],-Cos[θ]}}.{1,2},{{Cos[θ],Sin[θ]},{Sin[θ],-Cos[θ]}}.{-2,3}}]}},Axes->True,PlotRange->{{-4,4},{-4,4}}],{θ,0,2π,.1}]