MATLAB: Vectorized Solution to Rotate Multiple Points each at a Different Angle

Question

I want to rotate a matrix of points, i.e. column vectors. However, I want to rotate each vector in the matrix by a different angle. For example:

pointMatrix  = [v1,v2,v3,v4]; %vi is a column vector
rotateAngles = [10,20,30,40]; %degrees

Say I want to rotate these points around the z-axis. Therefore, for one point I could do something like the following:

Rz = [[cos(rotateAngles(1)) -sin(rotateAngles(1)) 0];...
      [sin(rotateAngles(1))  cos(rotateAngles(1)) 0];...
      [       0                      0            1]];
v1Rotated = Rz*v1;

Is there a non-loop way to rotate all the vectors in my pointMatrix by each one's unique rotation angle? Something like this…

   allRotatedPoints = superRotationMatrix*pointMatrix;

where the superRotationMatrix "magically" rotates each column by the corresponding angle.

Thanks!

Answer 1

This vectorized solution uses complex exponentials and works about 2 orders of magnitude faster for large vectors.

M = exp(rotateAngles*1i) .* ([1 1i 0]*pointMatrix);
allRotatedPoints = [real(M); imag(M); pointMatrix(3,:)];