Trying to understand the rotation matrix

[dibyendumajumdar]

Hi Mike

I was checking what happens in this line of code:

self.rot_mat = t3d.euler.euler2mat(*np.deg2rad(convertl2r(self)))

So I stepped in the euler2mat implementation.
Unfortunately the calculations done here don't match what I get from other implementations - nor does it seem to match https://mathworld.wolfram.com/EulerAngles.html or the Spencer / Murty paper.

I was wondering if you could shed some light on the treatment of euler angles and what conventions are being used.

[dibyendumajumdar]

It seems that the rotation matrix matches the one described here:
https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions

represents a rotation whose yaw, pitch, and roll angles are α, β and γ, respectively. More formally, it is an intrinsic rotation whose Tait–Bryan angles are α, β, γ, about axes z, y, x, respectively. 

If so the euler array is [gamma, beta, alpha] instead of [alpha, beta, gamma]?

[mjhoptics]

The system of euler angles commonly used in optical design programs (Zemax, CODE V, Oslo, etc) is comprised of rotations in this order:

1. alpha: rotation around x axis
2. beta: rotation around y axis
3. gamma: rotation about z axis

Interestingly, Spencer and Murty have alpha and beta reversed. Spencer was the author of ACCOS V: Oslo is a "next generation" version of ACCOS and CODE V adopted the coordinate systems used by ACCOS in the late '70s. But I digress...

The author of the transforms3d package that I use to build the matrix from the angles differentiates between extrinsic and intrinsic euler angles. The former reference the rotations to a fixed XYZ axis, the latter has the coordinate system moving after each rotation about an axis. Optical design software use the intrinsic, moving coordinate system with euler angles.

Given that, I have a question about the call euler2mat. It uses the default axes sequence which is 'sxyz'. This is for the explicit euler rotations. The implicit ones would be 'rxyz'. Fortunately all models I've seen that have tilts and decenters have tilts that are alpha tilts, i.e. always in the YZ plane. For this case of only alpha tilt, the euler2mat function generates the same rotation matrix. So an axes='rxyz' needs to be added to the call to euler2mat in surface.py.

The convertl2r function converts the angle sign convention used by CODE V (and documented in their Prompting Guide) to the right-handed convention used by transforms3d. Both the rotation about x and y axes are left-handed in CODE V.

It is a little known aspect of rotation matrices that an intrinsic matrix of 3 angles is equal to the extrinsic matrix with the angles taken in the reverse order, i.e.
rxyz = szyx

[dibyendumajumdar]

Thank you for the detailed reply. I am still trying to get my head around rotations; this is one area that I struggle with.

[mjhoptics]

Rotations and transformations in general are very picky - you have to code them right! - but very powerful when you get your head around them. When we were first writing LightTools, I found a robotics book that had a really good introductory chapter on transformations, and how to notate them to really show what was going on. That really helped us implementing a geometric model.

[dibyendumajumdar]

Thank you for the reference i will see if I can get hold of a copy.