Moving a point along a circle by a given arclength.

Question

Given:

1. $C$, the center of the circle
2. $r$, the radius of the circle
3. $A$, a point on the circle

I need to find point $B$ on the circle, which is arclength $D$ from point $A$.

The use case is for a character moving along a circular path. The speed of the character needs to be constant regardless of the circle's radius.

Answer 1

I recommend first changing your frame of reference for coordinates to be the center C of your circle. Next, express A in polar coordinates (where the origin is at C). Next, you can determine theta by knowing that the measure of an angle in radians is equal to how many radii of an arc the angle's arc length is. In this case, your arc length and radius is given, and the path that your character follows is indeed an arc of a circle with that radius. Now that you have theta, you know that in polar coordinates B and A have the same radius, and you know the difference in theta. Now that you have B in polar coordinates you can convert back to Cartesian. Of course, these are the Cartesian coordinates relative to C, so convert your frame of reference back to the origin.

Doffing my mathematician hat and donning my programmer hat, I recommend that you use a library for this. In fact, if you're making a game, use an engine (Unity, Unreal). I think it's great for your education in math to write your own math, but it's great for your education as a programmer to not re-invent the wheel.