2.3.2. Projected Angles

For the reconstruction of the 3D orientation from projected angles, giving a closed description of all 64 possible axis-plane projection combinations is not feasible in this article. Only a brief overview of the derivation of the equations will be given here. Every given projected angle can be used to determine a semicircle containing all the unit vectors that would lead to this angle if projected. If the j-th axis is projected onto the femoral (or patellar) coordinate plane, the parametrized semicircle can be set to equal the j-th column (or row, in the case of the patellar plane) vector of the rotation matrix. The orthogonality of rotation matrices (pairwise orthogonality of rows and columns, Euclidian norm of every row and column equals 1, third row/column is cross product of first and second row/column) gives the conditions to determine the parameters and the missing entries of R.
