Is There an Easier Way to Calculate 3D Rotation Matrices?

[kent davidge]

While resolving a problem in mechanics I discovered a beautiful and easy way for finding out what the rotation matrices in 3 dimensions are! And I'm surprised that I do not find this method anywhere on the internet! Would it be because it's not technically correct? Anyways, here it is:

It's all about realising that a rotation through one axis happens on a plane. For example, rotation about the y-axis happens on the z-x plane. Now pick the normalized polar vectors for that plane and put them into the matrix such that their components along the axis we are rotating about are zero (for them to be confined into the plane) and such that they are distributed along the rows of the matrix. Also, the corresponding component of the vector along the axis of rotation should be preserved, so the corresponding rotation vector should have only one non vanishing component along that axis and it should be equal to 1. That's it.

Using what I said above, for instance, rotation along the x-axis is given by $$\begin{pmatrix}\cos\beta&\sin\beta&0\\-\sin\beta&\cos\beta&0\\0&0&1\end{pmatrix}$$

Isn't this the easiest way ever? Why we don't find this across the literature?

 

[fresh_42]

You meant something like this: https://en.wikipedia.org/wiki/Rotation_group_SO(3)

 

[kent davidge]

You meant something like this: https://en.wikipedia.org/wiki/Rotation_group_SO(3)

Not really, obtaining the matrices as presented there is much more complicated.

 

[fresh_42]

These three rotation matrices are well-known and standard. That the entire group is generated by them looks as a bit of work to do, but the Wiki article has it.

 

[A.T.]

Why we don't find this across the literature?

https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations