# Stereographic Projection of Some Double Groups

Stereographic Projection of Some Double Groups

Stereographic projection provides geometric insight into the double cover . Each rotation of the sphere corresponds to exactly two linear transformations of homogeneous coordinates , . The projection remains a bijection because the Möbius transformation of a complex plane coordinate retains only the relative sign of , [1]. Taking a linear perspective, you can view points in the plane as the cosets of inversion by a rotation. Plotting complex vectors , off each plane coordinate reveals the hidden coset structure, which sometimes gets overlooked. The double groups, also called binary groups, contribute a foundational element in the analysis of quintic equations [2] and for quantum mechanics [3]. Furthermore, there is an aesthetic value in these dynamic images, in which variation of parameters appears to create a whirling dance of the points across the plane.

SU(2)~SO(3)

μ

1

μ

2

z=/

μ

1

μ

2

μ

1

μ

2

2π

μ

1

μ

2

z