The Laws of Cosines for Non-Euclidean Tetrahedra

Some time ago, I derived a (the?) hedronometric (“area-based”) Pythagorean Theorem for tetrahedra in Non-Euclidean 3-space.

A Right-Corner Tetrahedron

$$\cos\frac{W}{2} = \cos\frac{X}{2} \cos\frac{Y}{2} \cos\frac{Z}{2} \pm \sin\frac{X}{2} \sin\frac{Y}{2} \sin\frac{Z}{2}$$

where, throughout, “\(\pm\)” is “\(+\)” in hyperbolic space, and “\(–\)” in spherical space. Naturally, this leads to a Law of Cosines:

$$\begin{align}\cos\frac{W}{2} = \cos\frac{X}{2} \cos\frac{Y}{2} \cos\frac{Z}{2} &\pm \sin\frac{X}{2} \sin\frac{Y}{2} \sin\frac{Z}{2}S \\[0.5em] &+ \cos\frac{X}{2}\sin\frac{Y}{2}\sin\frac{Z}{2} \cos DA \\[0.5em] &+ \sin\frac{X}{2} \cos\frac{Y}{2} \cos\frac{Z}{2} \cos DB \\[0.5em] &+ \sin\frac{X}{2} \sin\frac{Y}{2} \cos\frac{Z}{2} \cos DC \end{align}$$


$$S^2 := 1 – 2 \cos DA \cos DB \cos DC – \cos^2 DA – \cos^2 DB – \cos^2 DC$$

And, as in the Euclidean case, this Law —which I call “First”— gives rise to a version —“Second”— that involves opposing dihedral angles and invites introduction of “pseudoface” elements.

$$\cos\frac{W}{2} \cos\frac{X}{2} \pm \sin\frac{W}{2} \sin\frac{X}{2} \cos BC = \cos \frac{H}{2} = \cos\frac{Y}{2} \cos\frac{Z}{2} \pm \sin\frac{Y}{2} \sin\frac{Z}{2} \cos DA$$

(Actually, I began calling the opposing dihedral version without the pseudoface element the “Second Law”, and the version with the pseudoface element the “Second-and-a-Halfth Law”; this phrasing persists in a couple of my notes. Nowadays, I just say “Second Law” and include the pseudo faces.)

In 2005 and 2006, I wrote about these results: “The Laws of Cosines for Non-Euclidean Tetrahedra” was  another early TeXperiment, so I went a little overboard on writing out less-than-efficient steps to derive various formulas. (I’ve grown rather fond of my “Morse code” representation of sine and cosine, however; it’s quite a space-saver.) Also, I note somewhat in passing a “symmetric, face-agnostic” consequence of these Laws, involving all seven areas —four faces and three pseudofaces— in one equation; I’ve since dubbed that the “Third Law of Cosines”.

Someday, I’ll compile a proper primer on the state-of-the-art in non-Euclidean hedronometry, so that I can retire these evolutionary discussions.

Posted 17 July, 2012 by Blue in Hedronometry