# Find center of rotation after object rotated by known angle (2D)

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

• This rotation is taking place in the plane? If so, why does the image make it look like $P$ and $P'$ get a lot closer together after the rotation? A real rotation would not do that. $P$ and $P'$ are given points? What does the "center of an object" even mean? It could be pretty weirdly shaped, after all... – rschwieb Nov 18 '14 at 20:46
• didn't you consider rotating several times to make $180^o$ and solve it like in the first case? P.S: rotate 12 times by $15^o$, makes a $180^o$ rotation – chouaib Nov 19 '14 at 0:29

## 3 Answers

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:

• $d = \frac{PP'}{2}$
• $\theta = 15^o$ (rotation angle).

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 ;)

The center of the rotation is on the mediatrix of [P,P']. Now, using the half of the angle, and the half of [P,P'], you can determine the position of the center on the mediatrix thx to the formula

tg(alpha/2) = PP'/2 / (position of center).


Apologies for brief response and a rough sketch.

The coordinates of P,P' and $\alpha$ are given

As mentioned in the other posts, we can find the radius R and inclination between radius/chord.

$$d= PP' ;\, R= \frac{d/2}{\sin \alpha/2} ;$$

Inclination to horizontal is known (blue)

Also angle CPP' is known (green), $=\cos^{-1} \frac{PP'/2}{R}$ complement of $\alpha/2$

The difference angle as marked (red) say $=\gamma$.

Subtract $(R \cos \gamma, R \sin \gamma)$ from corresponding coordinates of P.

There are two solutions on each side of PP', D can also be found similarly. 