If I have two sets of points in 3-dimensional space, each sets of points are the coordinates of vertexs of a Polyhedron. The two Polyhedrons 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 Polyhedrons such that the more the two Polyhedrons 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 Tetrahedrons (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)

than:

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 Polyhedrons.

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.

If I have two sets of points in 3-dimensional space, each sets of points are the coordinates of vertexs of a Polyhedron. The two Polyhedrons 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 Polyhedrons such that the more the two Polyhedrons congruent the more they have high degree of congruence i.e. if one can be transformed into the other by a sequence of rotations, translations, reflections but forbid scaling, than they have high degree of congruence.

For example, there are three Tetrahedrons (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)

than:

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 Polyhedrons.

If I have two sets of points in 3-dimensional space, each sets of points are the coordinates of vertexs of a Polyhedron. The two Polyhedrons 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 Polyhedrons such that the more the two Polyhedrons 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 Tetrahedrons (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)

than:

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 Polyhedrons.