Calculate the position of a Point B, so point B, with rotation point A, looks at a point C with always same distance to A

Question

I have 3 points and one point's direction in 2D space.

Point A and point C are free moveable points.

Point B has always the same distance to point A and point B has a look direction and should always rotate around point A, so point B looks to point C.

I have always given point A's position and point C's position. The problem is to find point B's position with the according rules above.

I illustrated the problem for more clarification: Point B's position update illustrated

What would be the formula to get the proper position?

Answer 1

Let

• $d = |\overline{AB}|$
• $\vec{OA}$, $\vec{OC}$ the position vectors of points $A$ and $C$
• $\vec{CA} = \vec{OA} - \vec{OC}$

Then the position vector of $B$ is

$$\vec{OB} = \vec{OA} + \frac{d}{|\vec{CA}|}\vec{CA}$$