Inspired by Mednykh and Pashkevich’s “Elementary Formulas for a Hyperbolic Tetrahedron”, I have compiled most of my disparate notes about hyperbolic hedronometry into one document: “Hedronometric Formulas for a Hyperbolic Tetrahedron”. I consider this a “living document” that I will update as I learn more about the subject matter.
It’s primarily a formula look-up list for myself, so I’ve left out a lot of exposition and proof. There may be a few typos, math-os, or outright errors. (There are at least two places where my results disagree with Mednykh and Pashkevich. One is a sign discrepancy, and the other an errant square root.) So, caveat reador.
As for content, “Hedronometric Formulas” covers familiar territory, from the foundational Laws of Cosines to the roadblock that is volume. There are also a few never-before-noted results, primarily involving “pseudo-altitudes” (each of which is a segment perpendicular to a pair of opposite edges); perhaps the coolest of these is the “Law of Side-Angle-Side-Altitude Sines”, which asserts that the product
$$\sinh{(\text{edge})} \;\cdot\; \sinh{(\text{edge})} \;\cdot\; \sin{(\text{angle})} \;\cdot\; \sinh{(\text{altitude})}$$
is a metric invariant of a hyperbolic tetrahedron, having a constant value (related to the Gram determinant) for any choice of two distinct “edge”s and the “angle” and “altitude” they determine. (For adjacent edges, “angle” is the angle between them and “altitude” is the altitude to the face they bound; for opposite edges, “altitude” is the corresponding pseudo-altitude between them, and “angle” is the “twist” of those edges about the pseudo-altitude.) This is analogous to the Euclidean result in which “edge \(\cdot\) edge \(\cdot\) sin(angle) \(\cdot\) altitude” always gives “\(6 \cdot \text{volume}\)”, which is also valid across standard and pseudo elements.
I also provide an appendix describing a tetrahedron in coordinatized hyperbolic space. “Coordination” happens to have provided the route to proving key results about pseudo-altitudes and twists, but generally it seems even messier than Euclidean analytic geometry can often be.
Some Open Questions raised (or repeated) here:
- What is the Pythagorean Theorem for Right-Corner Simplices?
- Is there a “monolithic” integral (in the spirit of the Derevnin-Mednykh formula) for the volume of a tetrahedron parameterized by its face and pseudo-face areas rather than its dihedral angles?
- What is the geometric interpretation of a hyperbolic pseudo-face?
Regarding the last: A Euclidean pseudo-face is the quadrilateral shadow of a tetrahedron projected into a plane parallel to a pair of opposite edges. Parallelism and projection are tricky concepts in hyperbolic space, but interestingly there is a shadow-like construction —valid only in certain cases— of a quadrilateral with a given pseudo-face area (so that it’s reasonable to recognize the quadrilateral as the pseudo-face). A construction that works in general eludes me.