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

To re-position the two overlapping circles with constant velocities, all you need to do is find the time at which the collision occurred, and add that factor of their velocities to their positions.

First, instead of two circles moving, we will consider one circle with combined radius and relative position and velocity. Let the input circles have positions P1 and P2, velocities V1 and V2, and radii r1 and r2. Let the combined circle have position P = P2 - P1, velocity V = V2 - V1, and radius r = r1 + r2.

We have to find the time at which the circle crosses the origin, in other words find the value of t for which r = |P + tV|. There should be 0, 1, or 2 values depending on whether the circle does not pass through the origin, flies tangent to it, or flies through it.

r^2 = ||P + tV|| by squaring both sides.

r^2 = (P + tV)*(P + tV) = t^2 V*V + 2tP*V + P*P using the fact that the L2-norm is equivalent to the dot product of a vector with itself, and then distributing the dot product.

t^2 V*V + 2tP*V + P*P - r^2 = 0 turning it into a quadratic equation.

If there are no solutions, then the discriminant b^2 - 4ac will be negative. If it is zero or positive, then we are interested in the first solution so we will subtract the discriminant.

a = V*V
b = 2 P*V
c = P*P - r^2
t = (-b - sqrt(b^2 - 4ac)) / (2a)

So t is the time of the collision.