Solution for optimal alignment of 3d polygonal mesh

Question

I am trying to implement a geometry modeling mechanism. One of the parts occupies a prototype polygonal mesh and aligns the instance with some points in a larger object.

Thus, the problem is this: if three-dimensional position positions are specified for some (possibly all) vertices in a polygonal grid, find a scaled rotation that minimizes the difference between the converted vertices and these points. I also have a center that can stay fixed if that helps. The correspondence between vertices and three points is fixed.

I think this can be done by solving for the coefficients of the transformation matrix, but I'm a little unsure how to create a system for the solution.

An example of this is a cube. The prototype would be a unit cube centered at the origin, with vert indices:

4----5
|\    \
| 6----7
| |    |
0 |  1 |
 \|    |
  2----3

An example of the location of the vertices:

• v0: 1.243,2.163, -3.426
• v1: 4.190, -0.408, -0.485
• v2: -1.974, -1.525, -3.426
• v3: 0.974, -4.096, -0.485
• v5: 1.974.1.525.3.426
• v7: -1.243, -2.163,3.426

So, given this prototype and these points, how can I find a single scale factor and rotation around x, y and z, which minimizes the distance between the vertices and these positions? It would be best if the method were generalized to an arbitrary grid, and not just to a cube.

Answer 1

, , , :

minimize Norm(T*V-M)

T - , , V - , M - . . M V - 3xN, 3- . T - 3x3. , , inverse(V*transpose(V))*V*transpose(M). , , ( ), Procrustes, SVD.

, , , , , . . ., , , , , .

Answer 2

, , .

octree. , 3D-, .

, , , - . "" , . , , , , .