Resolving a Circle-Circle Collision

Question

I am writing software that extends Circle-Rectangle collision detection (intersection) to include responses to the collision. Circle-edge and circle-rectangle are rather str

Answer 1

"This then is the problem: how do I make the move."

It is likely that you want to know how "to position the DPs back to the CPs before adjusting the circles' velocity components."

So there are two issues, how to determine the CPs (where the collision occurs) and how to adjust the circles' motion going forward from that point. The first part has a rather easy solution (allowing for different radii and velocity components), but the second part depends on whether an elastic or inelastic response is modelled. In a Comment you write:

The collision will be modeled as elastic. The math for the exchange of inertia is already in place. The problem is where to position the circles.

Given that I'm going to address only the first issue, solving for the exact position where the collision occurs. Assuming uniform motion of both circles, it is sufficient to know the exact time at which collision occurs, i.e. when does the distance between the circles' centers equal the sum of their radii.

With uniform motion one can treat one circle (red) as motionless by subtracting its velocity from that of the other circle (green). In effect we treat the center of the first circle as fixed and consider only the second circle to be in (uniform) motion.

Now the exact time of collision is found by solving a quadratic equation. Let V = (GVx-RVx, GVy-RVy) be the relative motion of the circles, and let P = (GIPx-RIPx,GIPy-RIPy) their relative positions in the "instant" prior to collision. We "animate" a linear path for the relative position P by defining:

P(t) = P + t*V

and ask when this straight line intersects the circle around the origin of radius Rr+Gr, or when does:

(Px + t*Vx)^2 + (Py + t*Vy)^2 = (Rr + Gr)^2

This is a quadratic equation in unknown time t, all other quantities involved being known. The circumstances are such that (with collision occurring at or before position CP) a positive real solution will exist (typically two solutions, one before CP and one after, but possibly a grazing contact giving a "double root"). The solution (root) t you want is the earlier one, the one where t (which is zero at "instant" RIP,GIP positions) is smaller.