File:SO(3) 60-90-60.svg

English: w:Conjugacy in w:SO(3) acting on a 2-sphere shown in w:stereographic projection. The rotation f is a 90° rotation, the rotation g is a 60° rotation. In words, if a point is moved along the black arrow (g⁻¹), then blue arrow (f), and then yellow arrow (g), then the action of g f g⁻¹ (green arrows) is obtained. If one considers f as a one-parametric subgroup, then the picture shows how g (yellow) maps orbits of f (blue) to orbits of g f g⁻¹ (green).

There are two spherical triangles in the lower part of the image that show g(f(g⁻¹(x))) where x is a g’s fixed point and f(g⁻¹(x)) is a g’s fixed point.

A gradient-filled shape in the upper part shows the situation where all three actions are not trivial. It is not a “true” quadrilateral: yellow and black sides are bent arcs, not line (i.e. great circle) segments. “60°” inscriptions mean that g is a 60° rotation, not that arcs have π/3 length. Only arrows along the central vertical line have π/3 lengths on the sphere.

At the upper left corner there is a depiction of the same sphere in orthographic projection and a schema of the stereographic projection is present. Near side of the sphere corresponds to the lower side of the image and is seen upside-down (the main image shows the sphere from inside, whereas the orthographic projection shows its exterior view).