I’m trying to understand how 3D vector rotation work, and I will understand it best if I have a chance to observe the actual process of how the correct calculations are supposed to be.
With default value (you can change the values if you want! I just want to see the process):
Rotating vector = v = (-2,-2,-2)
Rotational vector = n = (3,3,3)
Pivot point = P = (1,1,1)
Angle of rotation = θ = 60 degrees
Please do the step by step calculation, and please provide illustrated diagram for each step (I’ve already made the diagram in this link below, you just need to edit it to fit in each step).
Link: https://www.mathcha.io/editor/1Y2E0fYzIDmI9oHm4VXkyt91QYO7sVr5Y6mh59wLLV
Formulas for the calculation:
Rodigues’ Rotation formula:
- v’ = (1-cos(θ))[(v·n^)n^]+cos(θ)v+sin(θ)n^×v (I’m interested in seeing how this formula work, so I don’t prefer using 3D rotation matrix at the moment)
Rotation around pivot point formula (2D, I haven‘t seen any confirmed 3D version of the formula, please write the 3D one if you know the correct formula):
v.x’ = [(v.x-P.x)*cos(θ)-(v.y-P.y)*sin(θ)]+P.x
v.y’ = [(v.x-P.x)*sin(θ)+(v.y-P.y)*cos(θ)]+P.y
My other questions regarding the same subject:
- Can P (the pivot point) be considered as a vector? If it is, is it correct to say ‘’Rotation around a vector formula’’ instead of ‘’Rotation around pivot point formula’’?
- Is there exist the 3D version of the ‘’Rotation around pivot point formula’’ above? Can you please provide it if there are any?
- Is Rodigues’ Rotation formula supposed to control the ‘’direction’’ of the rotation and ‘’Rotation around pivot point formula’’ supposed to control the ‘’position’’ of the rotation? Can you please provide proof if possible? I don’t have any better way to word these mathematical terms, if you do, please word them for me.
Thank you. Any help would be greatly appreciated!
Also, I really need to observe the process to understand, but if you don’t do it, you can also answer the 3 questions above! Thank you!