Vectorized Solution to Rotate Multiple Points each at a Different Angle

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

[Teja Muppirala]

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,:)];

[Fvieira]

Notice that this works for 3 coordinate vectors (p = x,y,z).

For 2d one must consider this variation:

M = exp(rotateAngles*1i) .* ([1 1i]*pointMatrix);
allRotatedPoints = [real(M); imag(M)].'

Here is a full example that I did based on Muppirala's answer regarding some small details (angle in radians, for example):

v1 = [1;1]; v2 = [2;2]; v3 = [3;0]; v4 = [4;2];
pointMatrix  = [v1 v2 v3 v4] %vi is a column vector
rotateAngles = [90 90 90 90] *pi/180 % radians

M = exp(rotateAngles*1i) .* ([1 1i]*pointMatrix);
allRotatedPoints = [real(M); imag(M)].'

pointMatrix = pointMatrix.'

figure(1)
plot(pointMatrix(:,1),pointMatrix(:,2),'o')
axis([-5 5 -5 5])
grid on, hold on
plot(allRotatedPoints(:,1),allRotatedPoints(:,2),'x')

Here are the plots: Circles are the points, crosses are them rotated by 90 degrees.

Answer 2

[Matt J]

You can use MTIMESX on the file exchange

http://www.mathworks.com/matlabcentral/fileexchange/25977-mtimesx-fast-matrix-multiply-with-multi-dimensional-support

If your superRotationMatrix is 3x3xN and you reshape your pointMatrix to be 3x1xN, then

mtimesx(superRotationMatrix,pointMatrix)

will give you the rotated vectors in a 3x1xN output.