How to define and compute the degree of congruence of two rigid polyhedra in same type with knowing vertex coordinates? If I have two sets of points in 3-dimensional space, each sets of points are the coordinates of vertices of a polyhedron. The two polyhedra have same type, so we don't need to consider the topological property of them. Now I want to define and compute the degree of congruence of these two polyhedra such that the more the two polyhedra congruent the more they have high degree of congruence i.e. if one can be transformed into the other as much as possible by a sequence of rotations, translations, reflections but forbid scaling, than they have high degree of congruence.
For example, there are three tetrahedra $(A,B,C)$ with the coordinates:
$$A:(0,0,0),(10,0,0),(0,10,0),(0,0,10)$$
$$B:(0,0,0),(1,0,0),(0,1,0),(0,0,1)$$
$$C:(0,0,0),(10,0,0),(0,10,0),(0,0,9)$$
then:
$A$ and $B$ have low degree of congruence
$A$ and $C$ have high degree of congruence
Is there any mathematical theory could define and compute this degree of congruence?
By the way, we don't know the vertex correspondence between two polyhedra.
 A: Here is an idea, though I have to make some assumptions.
Suppose you have two sets of points $p_1,...,p_n\in\Bbb R^d$ and $q_1,...,q_n\in\Bbb R^d$ (for example, the vertices of your polyhedra, but with a fixed order).
Assume that they are translated to be centered at the origin, i.e. $p_1+\cdots +p_n=0$, and respectively for the $q_i$, so that we can ignore translations.
In a first step you could compute the covariance matrices of both point clouds and compare them. That is
$$C_p:=\sum_{i=1}^n p_ip_i^\top,\quad C_q:=\sum_{i=1}^n q_i q_i^\top.$$
These are positive semi-definite matrices, and you can compare their lists of eigenvalues, say $\lambda_i^p$ and $\lambda_i^q$ for all $i\in\{1,...,n\}$, sorted in descending order. They tell you about how unevenly these points clouds are distributed direction-wise.
The next step is to remove this unevenness from the point clouds. If we assume that the point clouds are full-dimensional (i.e. $\mathrm{span}(p_1,...,p_n)=\Bbb R^d$), then we can define
$$p_i':=C_p^{-1/2} p_i,\qquad q_i':=C_q^{-1/2} q_i.$$
Both point sets can now no longer be distinguished by translations or directional unevenness.
The last step is to consider the correlation matrix
$$C_{pq}:=\sum_{i=1}^n p_i'q_i^{\prime \top}.$$
You could e.g. compute $\delta:=\det(C_{pq})$.
This values lies between $-1$ and $1$.
We can use it as follows:

*

*if $\delta=\pm1$, then the point clouds are just reorientations of each other, that is, there exists an orthogonal matrix $X\in\mathrm{O}(\Bbb R^d)$ with $\det(X)=\delta$ and $p_i=X q_i$ for all $i\in\{1,...,n\}$.

*if $\delta=0$, then these point sets are as distinct as possible.

*in general, the smaller the value of $|\delta|$, the more different these point sets are.

In the end you have to somehow use the numbes $\delta,\lambda_i^p,\lambda_i^q$ for $i\in\{1,...,n\}$ to quantify the difference between the point sets. I do not have a recipe for this. All I can tell you is, that if $\lambda_i^p=\lambda_i^q$ for  all $i\in\{1,...,n\}$ and if $\delta=\pm 1$, then these point sets are the same up to some (possibly orientation-reversing) orthogonal transformation.
This of course assumes that your point sets have a predefined order (which might be given by the isomorphism between your polyhedra).
A: There are two phrases that may help in your search:

*

*Point-set
registration.
The link is to a (long) Wikipedia article, which includes "rigid
registration," which seems closest to your case.

*Geometric shape matching. For example:


Alt, Helmut, and Leonidas J. Guibas. "Discrete geometric shapes: Matching, interpolation, and approximation." In Handbook of Computational Geometry, pp. 121-153. North-Holland, 2000. Handbook link.

In the early 1990's it was established that exact matching under rigid motions could be solved in polynomial time in the number of points $n$, but the algorithms were
impractically complicated. The recent emphasis has been on fast approximation algorithms.
Here is an algorithm specifically for convex polytopes under rigid motion, which
guarantees (under certain conditions)
achieving within $(1-\epsilon)$ of the optimal volume overlap,
with high probability:

Ahn, Hee-Kap, Siu-Wing Cheng, Hyuk Jun Kweon, and Juyoung Yon. "Overlap of convex polytopes under rigid motion." Computational Geometry 47, no. 1 (2014): 15-24. Journal link.

