Specular Reflection for 3-D Particle Collision

[Disinterred]

I am writing code for a particle simulation and I have this question:

Homework Statement

A particle, with initial coordinates (x_i,y_i,z_i) w.r.t a fixed origin in a global rectangular coordinate system, is traveling toward one of the faces of a cube with a speed whose direction can be described by the standard spherical polar angles (θ_i,φ_i) w.r.t a local origin at the initial coordinates (x_i,y_i,z_i).

Now, say it hits the cubic face at a point (x_c,y_c,z_c). If the particle is to undergo specular reflection, what are the spherical polar coordinates (θ_c,φ_c) w.r.t a local origin at the point (x_c,y_c,z_c) which describe the direction of the velocity it reflects at? (in terms of the original angles)

Essentially, for a 2-D collision, I know that the particle must satisfy the usual law of reflection if we want specular reflection and thus we can get the new angle easily. But now in 3-D, there are two angles, so I am not sure what specular reflection would entail?

Homework Equations

θ and φ are the usual spherical polar angles,

0° ≤ θ ≤ 180° (pi rad)
0° ≤ φ < 360° (2pi rad)

The Attempt at a Solution

The only solution that makes sense to me is that the θ angle will obey the usual law of reflection, but the φ angle has no restrictions and can be anything from 0 to 2pi

 

[Disinterred]

Hmmm I typed all of that out and I think that helped me figure it out.

Phi would be the same, theta would have to satisfy the law of reflection along the plane formed by the incoming ray vector and the normal vector. This is of course a general solution and would work for any object (cube, triangle, sphere, etc). But if anyone thinks this is wrong/thinks it is right please tell me.

EDIT: On second thought, I do not think this is correct. I think what I need to do is find the angle between the normal of the face and the incoming ray and then find the corresponding spherical polar angles to describe the outgoing ray. This seems rather obvious and trivial, not sure why I never thought of it before.