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A Gallery of Ray Tracing for Geometers

Michael Grady (Southern Utah Univ.)

3. Projections

Projections are of fundamental importance in geometry, and ray tracers make them easily visible. Riemann, for example, used stereographic projection in his model of the complex plane. The POV-Ray generated video Mobius Transformations Revealed provides a beautiful exposition of Riemann's brilliant insight into the geometry of the Mobius functions where translations, dilations, rotations and inversions are all revealed to correspond to simple motions of the sphere [1].

Figure 4 depicts a sphere tangent to a plane at its south pole. There are three circles: the equator in yellow (a great circle) and two general circles passing through the north pole (in black). A point light located at the north pole projects the equator onto a circle concentric with the south pole. Circles passing through the poles are projected onto lines [5].

stereographic projection of circles onto the plane
Figure 4: Stereographic projection of circles onto the plane.

Figure 5 is a view of the same scene looking down on the plane from above.

sterieographic projection of circles onto the plane, view from above
Figure 5: Top view of the scene in figure 4.

Figure 6 depicts the central projection of a dodecahedron onto the sphere. This gives a regular partition of the sphere into 12 congruent spherical pentagons. The same process can be repeated for the other regular solids. Felix Klein used central projection in his study of the finite groups of rotations in three dimensions [6, p.3-30]. Using a ray tracer, projections like this come for free: just place a light at the center of the sphere and let the shadows of the edges of polyhedron fall on the surface of the sphere. Rotating the dodecahedron, as shown in the accompanying animation, reveals this projection more vividly.

central projection of dodecahedron onto sphere, with linked animation
Figure 6 with linked animation: Central projection.

Michael Grady (Southern Utah Univ.), "A Gallery of Ray Tracing for Geometers," Loci (September 2010), DOI:10.4169/loci003350