A Projective Tetrahedral Surface

parametric variables

p

0

0

q

0

1

opacity

0.25

0.5

1

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

(p,q)=(0,0)

, the surface is a sphere, which can be identified with the points of the special orthogonal group

SO(3)

, which generates 3D rotations. As

q1

, the rotation matrices are squared to produce the Roman surface. As

p1

, the surface transforms to a tetrahedron, associated with the tetrahedral subgroup of

SO(3)

.

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