Hello all,
I get a problem that needs all you support.
The question is that I need to transform a set of points on an arbitrary plane to 2-D dimensions, Oxy. Also, the distance between points must be the same after transformation.
For example, a set of points are on the plane with normal vector v=(a; b; c). The angle between this normal vector and direction vector of Oz axis k=(0; 0; 1) is Theta calculated by
cos(Theta) = c/ sqrt(a^2+b^2+c^2);
The rotation axis u has to be orthogonal to vector k and vector v.
vetor u = cross product of v,k = (u1,u2, 0).
Finally, the rotation matrix R is represented as the attached file.
According to the characteristic of the rotation matrix, the determinant of R must be equal to 1;
However, when I implemented this formula, the result of R in return is with det(R) !=1 which demonstrate something are incorrect.
I attached a Matlab code file. If I choose the v=(1; 0 ; 1), the det(R) =0.6402.
I attached 2 files ( in Matlab and Python ) for your reference.
Could you spend some time to check it and tell me why?
Thank you.
Have a look at https://au.mathworks.com/help/phased/ref/rotx.html
https://au.mathworks.com/matlabcentral/answers/349969-3d-rotation-matrix
If you mentioned the transformation of matrix 3D dimension to 2D dimension then this task is similar to a tensor sketch, Fast Johnson Lindenstrauss Transform and similar technics to reduce dimension of points set. May be such approach will be useful.
https://en.wikipedia.org/wiki/Tensor_sketch
The problem is that you should use a normalised vector to build the mattrix.
u = v x k / |v x k |
In your code, you should normalise the vector u, explicitely
u1 = b/sqrt(a^2 + b^2)
u2 = - a/sqrt(a^2+b^2)
And the answer becomes the expected one.
The generic matrix can be found here
https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle
If you want to rotate around (a, b, c) = (1,0,1), then (u1, u2, u3) = (sqrt (1/2), 0, sqrt (1/ 2)) However, u3 = 0 in your matrix.
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