Appendix: Goldberg-Coxeter Operation on improper spherical polyhedra

There are some improper spherical polyhedra that cannot be tesselated with the traditional geodesic dome methodology but can with some methods developed earlier. The Naive Slerp methods are able to project a triangle or square onto a full hemisphere, including the boundary.

Transformations to the disk

When the base vertices all lie on the same great circle (i.e. in a plane through 0), Naive Slerp before projection or normalization transforms the triangle or square to the disk. Unlike Triangular and Quadrilateral #1 Naive Slerp, Quadrilateral #2 simplifies nicely on the disk. If the vertices of the quadrilateral are \(\left<\pm 1, 0, 0\right>\) and \(\left<0, \pm 1, 0\right>\), and \(x,y\) are the xy coordinates, then:

\[\hat{\mathbf v}^* = \left< \cos \left(\frac{\pi}{2}(x+y)\right), \sin \left(\frac{\pi}{2}(x-y)\right), 0 \right>\]

The radial coordinate r of this vector in polar form has a nice form. (The angle coordinate \(\theta = \operatorname {arctan2} (v_y,v_x)\) doesn’t simplify usefully.)

\[r = \sqrt{1 - \sin (\pi x) \sin (\pi y)}\]

There are also some disk-specific transformations, accessible with gcopoly -p=disk. These amount to converting the vertex to polar form and scaling the r variable. Unlike Naive Slerp, there are points where the transformation is not smooth.

Triangular:

\[r = 1 - 3 \min(\beta_i)\]

\[\theta = \operatorname {arctan2} (\sum \beta_i v_{yi}, \sum \beta_i v_{xi})\]

Quadrilateral:

\[r = \max(|2x - 1|, |2y - 1|)\]

\[\theta = \operatorname {arctan2} (2y-1, 2x-1)\]

Dihedra

Fig. 1 The 3-dihedron

Fig. 2 The 4-dihedron

With the transformations defined above, performing the GC operation on a dihedron is fairly straightforward. These disk operations can be projected with a k factor the same as naive slerp, although here we require k to be >0: otherwise, all points would be projected onto the great circle.

The only caveat has to do with the presence of vertices with valence 2. On the quadrilateral, the valence-2 vertex and its adjacent edges can be smoothed into a single edge. On the triangle, this produces dangling faces, which can be removed. In both cases, one can also add or switch edges so that the vertex is no longer degenerate.

The process of doing the GC operation on a dihedron resembles stitching together two flat sheets and inflating them, like the children’s novelty the whoopi cushion. The dual of that could be called a Whoopi Goldberg polyhedron. (Thank you, I’ll be here all night.)

Monohedron

Fig. 3 The monohedron.

The spherical monohedron is the “polyhedron” with one face, one vertex, and no edges: the entire sphere is one big face. (Hey, it satisfies the Euler equations.) Topologically, we can tile this surface by taking the disk defined earlier and collapsing the boundary to a single point. This amounts to projecting the disk in the manner of a map projection of the entire Earth, which is a well-studied problem. Given \(\theta = \arctan2(v_x, v_y)\) and \(\mathbf{\hat{v}} = (\sin(\phi) \cos(\theta), \sin(\phi) \sin(\theta), \cos(\phi))\), two useful projections used in cartograph can be described as such:

Lambert azimuthal equal-area:

\[\phi = 2 \arcsin(\|\mathbf v\|)\]

Azimuthal equal-distance:

\[\phi = \pi \|\mathbf v\|\]

Really any continuous, monotonic function of \(\|\mathbf v\|\) that maps 0 to 0 and 1 to \(\pi\) could be used to calculate \(\phi\).

We’ll call these “balloon polyhedra” by analogy with blowing up a balloon. balloon.py can be used to calculate these. Because of the radical change to the boundary, many small-parameter GC operations produce the same polyhedron. Quad faces may be reduced to triangle faces, and triangle faces may be reduced to degenerate faces. Many produced polyhedra are not convex. Class III polyhedra tend to have dangling faces.