Appendix: Spherical (and other) geometry with vectors¶
In the geometric dome world, geometry is usually synthetic rather than analytic. That is, geometric constructions are usually expressed in terms of step-by-step geometric constructions rather than in equations and numeric values. See, for instance, [Kenner]. Implementing a synthetic geometric construction on a computer can be a pain; analytic specifications are more amenable to programming idioms.
There are a number of ways to numerically specify points on a sphere. By far the most common is by latitude and longitude, which appear on every modern map of the Earth and probably some other planets. Latitude and longitude are familiar and convenient, but performing extended geometry calculations using latitude and longitude is a complicated task.
Doing spherical geometry using a 3-component unit vector is more convenient in a number of ways: the equations are often simpler, and there are no singularities at the poles. This appendix isn’t a full course in spherical geometry with vectors, but is here to explain the notation and methods that are used in this text.
A 3d unit sphere can be defined as the set of all unit vectors in 3-space; i.e., vectors \(\mathbf v = [v_x, v_y, v_z]\) such that the vector norm \(\|\mathbf v \|=1\). Unit vectors are often denoted using a hat: \(\hat{\mathbf v}\). We’ll often find ourselves normalizing vectors, so if the numerator and denominator are the same, we’ll supress the denominator with an ellipsis like so:
Basics¶
The shortest distance between two points in Euclidean space is a straight line. On the sphere, the shortest distance is an arc of the great circle between those points. The great circle is the intersection of the sphere and a plane passing through the origin. A plane through the origin can be specified as \(\hat{\mathbf n} \cdot \mathbf v = 0\), where \(\hat{\mathbf n}\) is a unit vector normal to the plane; this vector \(\hat{\mathbf n}\) can be used to specify a great circle. Given two points \(\mathbf{\hat{v}_1, \hat{v}_2}\) on the sphere, the \(\hat{\mathbf n}\) of the great circle between those two points is (up to normalization) their cross product: \(\mathbf{\hat{n}} = \frac{\mathbf{\hat{v}}_1 \times \mathbf{\hat{v}}_2}{\|\dots\|}\)
Small circles are the intersection of the sphere with a plane not through the origin. All planes may be specified in Hessian normal form as \(\mathbf{\hat{n}} \cdot \mathbf v = r\), where r is the minimum distance between the plane and the origin.
| r | Intersection with unit sphere |
|---|---|
| 0 | Great circle |
| > 0 and < 1 | Small circle |
| 1 | Point |
| > 1 | None |
Measurements¶
| Measurement | Euclidean | Spherical |
|---|---|---|
| Length | \(\eta = \|\mathbf v_1-\mathbf v_2\|\) | Central angle: \(\theta = \arctan\left( \frac{\|\mathbf{\hat{v}}_1 \times \mathbf{\hat{v}}_2\|} {\mathbf{\hat{v}}_1 \cdot \mathbf{\hat{v}}_2}\right)\) |
| Angle | \(\cos \phi_1 = \frac{(\mathbf v_1 - \mathbf v_2) \cdot (\mathbf v_1 - \mathbf v_3)} {\|\mathbf v_1 - \mathbf v_2\|\|\mathbf v_1 - \mathbf v_3\|}\) | Dihedral angle: \(\mathbf{\hat{c}}_{12} = \frac{\mathbf{\hat{v}}_1 \times \mathbf{\hat{v}}_2}{\|\dots\|}\), \(\mathbf{\hat{c}}_{13} = \frac{\mathbf{\hat{v}}_1 \times \mathbf{\hat{v}}_3}{\|\dots\|}\), \(\cos\phi_1 = \mathbf{\hat{c}}_{12} \cdot \mathbf{\hat{c}}_{13}\) |
| Triangle area | \(A = \frac{\|(\mathbf v_1-\mathbf v_3)\times (\mathbf v_2-\mathbf v_3)\|}{2}\) | Solid angle: \(\tan(\Omega/2) = \frac{|\mathbf{\hat{v}_1} \cdot \mathbf{\hat{v}}_2 \times \mathbf{\hat{v}}_3|} {1+\mathbf{\hat{v}}_1\cdot \mathbf{\hat{v}}_2+\mathbf{\hat{v}}_2 \cdot \mathbf{\hat{v}}_3+\mathbf{\hat{v}}_3\cdot \mathbf{\hat{v}}_1}\) [Oosterom] |
In general, the spherical measures approach the Euclidean measures when the measures are small. (A sphere is locally Euclidean.)
Constructions¶
| Construction | Euclidean | Spherical |
|---|---|---|
| Mean (midpoint when n=2, centroid when n=3) | \(\mathbf v_\mu = \frac{\sum\mathbf v_i}{n}\) | \(\mathbf{\hat{v}}_\mu = \frac{\sum\mathbf{\hat{v}}_i}{\|\dots\|}\) |
| Interpolation | \(\mathrm{Lerp}(\mathbf{v_1}, \mathbf{v_2}; t) = (1-t) \mathbf{v_1} + t \mathbf{v_2}\) | \(\mathrm{Slerp}(\mathbf{\hat{v}_1}, \mathbf{\hat{v}_2}; t) = \frac{\sin {((1-t)w)}}{\sin (w)} \mathbf{\hat{v}_1} + \frac{\sin (tw)}{\sin (w)} \mathbf{\hat{v}_2}\) where \(\cos(w) = \mathbf{\hat{v}_1} \cdot \mathbf{\hat{v}_2}\) |
Face normals¶
This program uses this definition for the normal to a Euclidean polygon:
i should be treated as if it’s \(\mod n\): it loops around. This definition allows for a somewhat sensible extension to skew polygons: the normal points in the general direction that’s expected.
The normal will be outward-facing if the points are ordered counterclockwise, and inward-facing if the points are ordered clockwise.
Face bentness¶
There’s no standard measure of face bentness, so this program uses an ad-hoc measure that seems to work well. This program measures the bentness of a face with 4 or more vertices by these steps:
- Let \(\mathbf x_i = \mathbf{v}_i - \bar{\mathbf{v}}\), where \(\bar{\mathbf{v}}\) is the (Euclidean) average of the points
- Calculate the SVD decomposition of the matrix that has \(\mathbf x_i\) as rows (or columns). We only need the singular values: since we’re in 3d space, there will be 3 singular values.
- The “bentness” is the smallest singular value divided by the sum of the other two singular values.