I need to be able to calculate and find the true center of rotation (x,y) of an object after it has been rotated by a known angle.
Previously I would simply find the center of the object, rotate 180 degrees, find the new center of the object, and calculate the midpoint of the line that connects the 2 centers to find the true center of rotation. Now, I don't have the luxury of being able to rotate 180 degrees, instead it will be only 10-15 degrees, but it will be known.
I've searched for a while and haven't found the appropriate equation to solve my particular problem.
Limitation: Only know center of object, cannot use multiple points found on object.
Knowns: Center of object before and after rotation, and angle of rotation
Unknown: True center of rotation
Hopefully the image explains what I'm shooting for
Thanks
Best Answer
Let point $C$ be the center of rotation.
It should belong to the perpendicular bisector of the segment $[P, P']$.
So if we imagine it as a triangle, $CPP'$ should be a equilateral triangle where the angle $C = 15^0$ (Or whatever your rotation angle below 180). Let's assemble it in a drawing:
Your task can be minimized to computing $M$!:
Knowns:
Solution:
Notice that: $\frac{d}{M} = tan(\frac{\theta}{2})$
So $M = \frac{d}{tan(\frac{\theta}{2})}$
Now you have all the necessary information to find your center of rotation $C$, I will left to you to transform everything to $(x,y)$ coordinates as an exercise ;)