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I need to calculate the rotation matrix and the translation vector between 2 given triangles in Euclidean space. This is really just a rotation and a translation, the lengths of sides of the triangle are equal. Coordinates of the points are real. $P_1, P_2, P_3$ are the 3 points of the 1st triangle, $Q_1, Q_2, Q_3$ are the 3 points of the 2nd.

My idea was: I translate the center of the triangles to the point of origin with $t_1 := \frac{P_1+P_2+P_3}{3}$, $P'_i = P_i-t_1$ and $t_2 := \frac{Q_1+Q_2+Q_3}{3}$, $Q'_i = Q_i-t_1$ and then I only need to solve the set of 9 linear equations with 9 unknown (the 9 entries of the rotation matrix $R$): $$ Q'_i = R.P'_i $$ Afterwards I could calculate the translation with $$ t = t_2 - R.t_1 $$ The problem is: after I translate the triangles to the point of origin the points are linearly dependent ($P_3 = -P_1 - P_2$) So in fact I only have 6 equation with 9 unknown, which allows multiple solutions for $R$. But I need "one" precise solution, because the triangles are just parts of two identical models, which I need the rotation for.

I think the approach will work, if I use 4 points instead of 3. But can I somehow calculate the rotation matrix with "only" 3 points?

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If you have three linearly independent points $P_1, P_2, P_3$ being transformed to three linearly independent points $Q_1,Q_2,Q_3$ then your matrix is already uniquely determined. If your points are not linearly independent then it gets a little more complicated. –  process91 Aug 16 '12 at 12:46

1 Answer 1

up vote 2 down vote accepted

In principle you need two rotations and a matrix-multiplication. Call the recentered matrices Q ad S and P as T.
Then find the rotation s, which rotates S to "triangular" form of its coordinates, so that the coordinates of the first point becomes $[x_{s,1},0,0]$, of the second becomes $[x_{s,2},y_{s,2},0]$. This can be done with your ansatz of using unknowns, if you assume three rotations (for each pair of planes 1 rotation, and gives a 3x3-matrix whose 3'rd column is zero and describes thus a triangle in a plane)

Then do the same with another rotation t which rotes T in the same manner.
Then $T = S \cdot s \cdot t^{-1} = S \cdot r$ where r is the matrix for the complete rotation.


A bit of pseudocode (code in my proprietary MatMate-tool):

  S = Q - Meanzl(Q)        // do the translation to the origin
  T = P - Meanzl(P)      

    // get the rotation-matrices which rotate some matrix to triangular shape 
  ts = gettrans(S,"tri") 
  tt = gettrans(T,"tri") 

    // make one rotation-metrix. the quote-symbol means transposition
  tr = ts * tt'  

    // difference should be zero      
  CHK = T - S*tr 


Example:
We assume S and T being centered. Let
$ \small \text{ S =} \begin{bmatrix} 14.469944&22.964690&-7.581631\\ -15.275348&5.923432&23.720255\\ 0.805404&-28.888122&-16.138624 \end{bmatrix} $
and
$ \small \text{ T =} \begin{bmatrix} 22.808501&2.515200&16.361035\\ 8.393637&-5.071089&-27.109127\\ -31.202138&2.555889&10.748092 \end{bmatrix} $
Now you can solve for a rotation in y/z-plane, which makes the entry in $S_{1,3}=0$. The rotation-parameters are some cos/sin-values. Apply this and you get
$\small \text{ S}^{(1)} = \begin{bmatrix} 14.469944&24.183840&0.000000\\ -15.275348&-1.811476&24.381471\\ 0.805404&-22.372364&-24.381471 \end{bmatrix}$ Now you can solve for a rotation in x/y-plane, which makes the entry in $S_{1,2}=0$. The rotation-parameters are some other cos/sin-values. Apply this and you get
$\small \text{S}^{(2)} = \begin{bmatrix} 28.182218&0.000000&0.000000\\ -9.397482&12.178056&24.381471\\ -18.784736&-12.178056&-24.381471 \end{bmatrix}$
After that a third set of rotation-parameters cos/sin-values make S triangular. It looks then like this
$\small \text{ S}^{(3)} = \begin{bmatrix} 28.182218&0.000000&0.000000\\ -9.397482&27.253645&0.000000\\ -18.784736&-27.253645&0.000000 \end{bmatrix}$

Because the center of your original triangle was moved to the origin, the last column (the z-coordinates) are zero/not needed, since 3 points can always be placed in a plane.
From the three rotation with their cos/sin-values you can construct a 3x3-rotation-matrix, say s.

The same can be done using the matrix T leading to a rotation-matrix t. If S and T describe in fact the same triangle, only rotated, the results are equal: $T^{(3)}=S^{(3)}$. Then you can use the nice fact, that the inverse of a rotation-matrix is just its transpose, such that with
$\small \text{ tr =} \begin{bmatrix} 0.205215&0.860645&0.466022\\ 0.946329&-0.295966&0.129867\\ 0.249696&0.414359&-0.875190 \end{bmatrix}$ we get $ T = S \cdot tr $


In principle this can also be solved using the concept of pseudoinverses: we demand
$ tr = S^{-1} \cdot T $ . But because S has reduced rank the inverse means to divide by zero. Using SVD-decomposition (see wikipedia) the pseudoinverse can be computed if the inverse of the diagonal matrix of the SVD-factors is used (where zeros are simply left zero). This should lead to the same solution.

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I'm not sure if I understand you correctly. If I use my ansatz I get 9 equations with more than 9 unknowns, (9 for $s$ and $x_{\epsilon,1}$,$...$ Also, if the 3rd column of the matrix t is zero, there exists no inverse of t. –  user38034 Aug 16 '12 at 9:55
    
Ok, I'll add an example; it'll need some time for editing, please wait –  Gottfried Helms Aug 16 '12 at 10:38
    
That would be great, thank you! –  user38034 Aug 16 '12 at 11:02
    
Thanks, I really had my wires crossed. –  user38034 Aug 16 '12 at 13:25

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