New velocity after circle collision

Question

On a circular billiard-table, the billiard-ball collides with the boundary of that table with some velocity v1. This collision is detected as follows:

double s = sqrt( (p.x-a)*(p.x-a) + (p.y-b)*(p.y-b) );
if (s<r)        // point lies inside circle
                    // do nothing
else if (s==r)  // point lies on circle
                    // calculate new velocity
else if (s>r)   // point lies outside circle 
                    // move point back onto circle (I already have that part)
                    // calculate new velocity

Now how can the new velocity v2 after the collision be calculated, such that angle of incidence = angle of reflection (elastic collision)?

PS: The billiard-ball is represented by a point p(x,y) with a velocity-vector v(x,y). The simulation is without friction.

This question does not seem to be C++-specific. In fact, it seems more like a math-question than a programming question. — Björn Pollex
Where's your initial velocity and how are you representing it? — Tim
Something like v_new = coeff*(v_old - 2*dot(v_old, boundary_normal)*boundary_normal); for some seriously simplified physics? — Bart
That's a mechanics question, and it would help if you stated what each symbol represents. Also, are the table boundaries aligned along X and Y or not? Do they absorb any energy or are they perfectly elastic? In the simplest case you flip Vx or Vy respectively and calculate the new magnitude. Which should be the same as the one before the collision... — juanchopanza
Basic physics - see: en.wikipedia.org/wiki/Coefficient_of_restitution — Paul R

Bart Bart · Accepted Answer · 2011-12-08T10:50:03

Assuming you're making some simple (game-like) billiards simulation you could use something like:

v_new = coeff*(v_old - 2*dot(v_old, boundary_normal)*boundary_normal);

Here v_old is your current velocity vector and boundary_normal is the inward pointing normal of your circular billiards table at the point of impact. If you know the center c of your circular table and you have the point of impact p then the normal is simply normalize(c-p). That is, the normalized vector you obtain when subtracting p from c.

Now I have taken coeff to be a fudge factor between 0 (no velocity at all anymore after impact) and 1 (same velocity after impact). You can make this more physically plausible by determining a correct coefficient of restitution.

In the end all the formula above is, is simple reflection as you might have seen in a basic ray tracer for example. As said, it's a fairly crude abstraction from an accurate physics simulation, but will most likely do the job.

New velocity after circle collision

2 Answers