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I have two tetrahedra in $\mathbb R^3$, $T_1$ and $T_2$, and access to the coordinates of their vertices. $T_1$ and $T_2$ are tetrahedra in the sense that they each have four vertices, each vertex is connected to every other vertex, and no three edges lie along the same line. There are no further restrictions on their geometry.

Is there a simple method of determining whether the two tetrahedra are identical up to some rotation and translation operation? What if I wish to test whether $T_1$ and $T_2$ have some reflection (mirror) symmetry? One can, of course, exhaustively test for these things by attempting every possible mapping of $T_2$ onto $T_1$, which is my current strategy. However, I'm sure a better method must exist.

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Tetrahedra are determined up to congruence by the edge lengths, so you can just solve for the isometry sending one to the other and check if it preserves orientation. – Louis Dec 31 '11 at 13:29
Just to simplify things, how about translating and rotating both tetrahedra such that one vertex is the origin, the longest edge lies along one axis, and the face with largest area lies in a coordinate plane? – J. M. Dec 31 '11 at 13:31
See to note that edge length congruence is not enough. – bossylobster Jul 1 '14 at 16:47
up vote 1 down vote accepted

Let's assume you already know that the two tetrahedra are congruent, and you just want to test whether they're identical or whether one is a mirror image version of the other. (You can test for congruence by looking at the edge lengths, as Louis' comment says.)

Now, call the vertices of the first tetrahedron A, B, C, and D. The signed volume of this tetrahedron is

$\frac{1}{6}(\vec{AB} \times \vec{AC}) \bullet \vec{AD}$.

Naturally, signed volume will reverse its sign when you mirror-reverse an object, but it will stay invariant under translation and rotation. So, if the signed volumes of the two tetrahedra are the same, then they have the same orientations; otherwise, if one is negative the other, they're mirror-reflections.

One note: make sure to label the tetrahedrons' vertices isomorphically. In other words, for the purposes of the formula, if the side $AB$ has length $7$ on one tetrahedron, then the corresponding side should have length $7$ on the other tetrahedron too.

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