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?

You've stepped through this with the debugger? I imagine your input values are not what you think they are. — House
– House, Commented Jun 18, 2012 at 15:23

Daniel Carlsson · Accepted Answer · 2012-06-18 15:49:27Z

6

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

edited Jun 18, 2012 at 15:49

answered Jun 18, 2012 at 15:22

Daniel Carlsson

2,37112 silver badges17 bronze badges

\$\begingroup\$ Yes. Thanks for the answer. If you have some time for me :) I still have a big doubt: if I normalize velocity Vector2.Reflect = (velocity, v) v is equal to 0.71 o.71 1.1 or always 1,1? and then if I want to reflect (-5, +2) into (5, -2) which equation should I use? \$\endgroup\$

xdevel2000
– xdevel2000

2012-06-19 07:45:32 +00:00
Commented Jun 19, 2012 at 7:45
\$\begingroup\$ Normalize divides the vector with the length of the vector: [1,1] / Sqrt(1^2 + 1^2) = [0.707..., 0.707...] The formula is as I mentioned in the answer, but you can simply use Vector2.Reflect instead which uses that exact formula \$\endgroup\$

Daniel Carlsson
– Daniel Carlsson

2012-06-19 19:48:53 +00:00
Commented Jun 19, 2012 at 19:48

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