Vector reflect problem

Question

I'm testing some vector reflection and I want to check what happens when a ball collides with a paddle.

So if I have:

Vector2 velocity = new Vector2(-5, 2); 
position_ball += velocity;

if (position_ball.X < 10) 
{
    Vector2 v = new Vector2(1,0); // or Vector2.UnitX   
    velocity = Vector2.Reflect(velocity, v); 
}

then, correctly, velocity is (5,2) after Reflect, but if I do:

if (position_ball.X < 10)
{
  Vector2 v = new Vector2(1,1); 
  velocity = Vector2.Reflect(velocity, v);
}

then velocity is (1,8) and not (5, -2) that is the solution of reflection equation R = V - 2 * (V . N)

Why is that?

Answer 1

The reflection equation is R = V - 2 * (V . N) * N This formula also assumes that N is a normal vector, which it isn't in your case new Vector2(1,1).Length() == 1.414...

Try this instead:

if (position_ball.X < 10)
{
  Vector2 v = new Vector2(1,1); 
  v.Normalize();
  velocity = Vector2.Reflect(velocity, v);
}

The reflection of [-5,2] will actually be [-2,5] (or [1,8] if not normalized) and not [5,-2] as you assumed:

R = V - 2 * (V.N) * N
N = [1,1] * 2^-0.5
V = [-5,2]

Since Nx = Ny in this particular case we can refer to either of them as Nc

R = V - 2 * (v.N) * N
  = V - 2 * (Vx*Nc + Vy*Nc) * N
  = V - 2*Nc(Vx+Vy) * N
  = V - 2*Nc(-5+2) * N
  = V - 2*Nc(-3) * N
  = V + 6*Nc * N
  = V + [6*Nx*Nx, 6*Ny*Ny]
  = V + [6*Nx^2, 6*Ny^2]
  = [6*Nx^2-5, 6*Ny^2+2]
  = [6*(2^-1)-5, 6*(2^-1)+2]
  = [-2, 5]

(N^a)^b = N^(a*b) which is used in the above proof (2^-0.5)^2 = 2^-1