Vectorized Solution to Rotate Multiple Points each at a Different Angle

Question

John deGrassie 2013-3-7

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https://ww2.mathworks.cn/matlabcentral/answers/66294-vectorized-solution-to-rotate-multiple-points-each-at-a-different-angle

评论： Fvieira 2021-4-30

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!

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Answer 1

Teja Muppirala 2013-3-8

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https://ww2.mathworks.cn/matlabcentral/answers/66294-vectorized-solution-to-rotate-multiple-points-each-at-a-different-angle#answer_77837

编辑：Teja Muppirala 2013-3-8

在 MATLAB Online 中打开

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,:)];

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Fvieira 2021-4-30

在 MATLAB Online 中打开

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')