# A Projective Tetrahedral Surface

A Projective Tetrahedral Surface

This Demonstration shows a continuous morphing from a sphere to Steiner's Roman surface (a self-intersecting mapping of the real projective plane into 3D space) to a tetrahedron.

At , the surface is a sphere, which can be identified with the points of the special orthogonal group , which generates 3D rotations. As , the rotation matrices are squared to produce the Roman surface. As , the surface transforms to a tetrahedron, associated with the tetrahedral subgroup of .

(p,q)=(0,0)

SO(3)

q1

p1

SO(3)

The resulting tetrahedron is equivalent to one approximated by a Bezier surface.