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Comment by bawana | Hacker News Reader

I thought I knew what rotation matrices were until I read this article. Now I am confused. I used to think I needed a 3x3 matrix to rotate a point (with x,y,z coordinates) in 3 D space. But the article refers to Rx,Ry and Rz EACH of which are a 3x3 matrix.Why are there 3 matrices? To move an x coordinate to a new coordinate system, I only need to displace it among the the existing three axes. So each coordinate only needs a 3x1 vector to displace it . That a makes a single 3x3 matrix.

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Any number of successive rotations can be represented as a single matrix. Both are right at the same time: an arbitrary rotation (represented by a single matrix) is a combination of 3 rotations about the axes of the reference frame (represented by 3 matrices).

R_x, R_y and R_z are the basis to generate all rotations. Think of them as the equivalent of the basis vectors in 3d euclidian space: e_x = (1, 0, 0), e_y=(0, 1, 0) and e_z = (0, 0, 1)

> I only need to displace it among the the existing three axes.

That is exactly what these three matrices do. R_x depends on a single parameter and tells you how much you rotate around the x-axis. R_y tells you how much you rotate around the y-axis and R_z for the z-axis. Any rotation can be written as a product of these three (which then is a single 3x3 matrix depending on three parameters).

Thank you to all who replied. My clue should have been the use of the same theta instead of alpha, beta and gamma.

Those 3 mats Rx Ry Rz are just 3 separate examples from Wikipedia of how to construct a rotation matrix where the rotation axis is in the x, y, or z direction. They are separate rotations that don’t have to be combined.

Those three matrices describe three different rotations. Rx is a rotation about the x axis, Ry about the y axis, Rz about the z axis. They're parameterized by an angle theta that describes how much to rotate.

Rx rotates about the x-axis (in the yz plane), Ry about the y-axis (in the xz plane), and Rz about the z-axis (in the xy plane).

We need all three because none of them can be made from the other two, but any other rotation can be made from a combination of these three.

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