Estimating 3d transformation of two triangles

[nimmie]

I have two triangles in 3d. I need to calculate transformation matrix(3X3) between two triangles in 3D.

1)How can I calculate the transformation matrix(rigid) while fixing one of the points to the origin(to get rid of the translation part)?

2)How does it affect if the deformation is non rigid?

 

[chiro]

I have two triangles in 3d. I need to calculate transformation matrix(3X3) between two triangles in 3D.

1)How can I calculate the transformation matrix(rigid) while fixing one of the points to the origin(to get rid of the translation part)?

2)How does it affect if the deformation is non rigid?

1) You can, in the general case solve a chunk of linear systems where the determinant of the matrix is 1, given the constraints of initial and final points that satisfy the transformation. If you can calculate something that makes sense, that is your answer (you may also have free parameters depending on your initial/final points as well the dimension of your space).

2) If the deformation is non-rigid then you won't have a pure-rotation, but some thing a lot more general and you will have to provide the relaxed constraints (i.e. det <> 1).

For finding the matrix given two points I recommend you find two orthogonal rotations in the way that one rotation transforms triangle so one point is positioned and then the next rotation axis is aligned with the vertex that has the proper position and the angle corresponds to rotating the other vertex into position.

The idea of this is that if you try and rotate a vertex around an axis that is parallel to the vertex itself, it won't change. Because you have two vertices, I think this idea is optimal for your situation.

If you need to know how to create axis-angle rotation matrices look here:

http://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle

 

[haruspex]

I assume you mean you've normalised each triangle to have a vertex at the origin.
If it's rigid there are only 3 degrees of freedom, and only the direction vectors are needed as inputs.
If it's non-rigid then there will be a subspace of possibilities. There will be 9 unknowns but only 6 equations. And that's assuming the association of the vertices of one to those of the other is given.

 

[haruspex]

Suppose you want to rotate vector u₁ to u₂ and v₁ to v₂.
The axis of rotation is given by w where w.u₁ = w.u₂ and w.v₁ = w.v₂. w might as well be a unit vector, so that's two equations, two unknowns.
If w = (x, y, z), the matrix which rotates space an angle theta around w is (from http://www.fastgraph.com/makegames/3drotation/) R =
tx²+c txy-sz txz+sy
txy+sz ty²+c tyz-sx
txz-sy tyz+sx tz²+c
where c = cos(theta), s = sin(theta), t = 1-c.
It remains to determine theta from R u₁ = u₂.

 

[CellCoree]

1) To calculate the rigid transformation matrix between two triangles in 3D, you can use a combination of translation, rotation, and scaling. First, you will need to choose a point on one of the triangles to serve as the origin point. Then, you can calculate the translation vector from this chosen point to the origin. Next, you can use the known points of the two triangles to calculate the rotation and scaling factors. Once you have these values, you can construct the 3x3 transformation matrix by combining the translation, rotation, and scaling components.

2) If the deformation is non-rigid, meaning that the triangles are not simply being translated, rotated, or scaled, the calculation of the transformation matrix becomes more complex. In this case, you will need to use advanced mathematical techniques such as least squares optimization or finite element analysis to accurately determine the transformation between the two triangles. Additionally, the resulting transformation matrix may not be a 3x3 matrix, but rather a higher dimensional matrix that takes into account the non-rigid deformation. This type of analysis is often used in computer graphics and animation to simulate realistic movements and deformations of objects in 3D space.