Help with Transform Path

Question

I have a 2D square. It has an initial position translation T0. It rotates about the Z axis, about an arbitrary point, say 45 degrees, then moves (translates) along the +X axis some distance.

I'm trying to put together the 4x4 homogeneous transform (this example is only 2D but the problem is really in 3D) to move the square but I seem to be missing something, and I get odd results.

T0 := initial translation to some point T1 := center of rotation point Rz := rotation Matrix about Z and about point located at T1 T2 := additional translation TF := final transform p := a vertex of the square to rotate If I use the following sequence -

TF = T2 * T1 * Rz * T1^-1 * T0

I can rotate about T1 as long as T2 is (0,0).

If I use only T0 - by combining T0 and T2 - it rotates correctly but it moves along the rotated X axis rather than the original X axis.

So, it seems I can

Transform first and get the rotation correct but not the translation OR Transform last and get the translation correct but not the rotation.

example:

move to 10,10 rotate about 9.5, 9.5 move -5,0

It seems there should be a way to get my desired result, but I seem to be missing something.

Thanks

Rick

Answer 1

is this the transform you require for
move to 10,10 rotate about 9.5, 9.5 move -5,0:

rot = 45 degrees
trans1 = vec(10.0,10.0,0.0)
trans2 = vec(-5.0,0.0,0.0)
rot_point = vec(9.5,9.5,0.0)

translate(trans2) *
translate(rot_point) *
rotatez(rot) *
translate(trans1 - rot_point)