3D rays

adriank

Also, the Pythagorean Theorem has to hold as well: x² + y² + z² = r².

x² + y² + z²
= (r sin θ cos ϕ)² + (r sin θ sin ϕ)² + (r cos θ)²
= r² sin² θ cos² ϕ + r² sin² θ sin² ϕ + r² cos² θ
= r² (sin² θ (cos² ϕ + sin² ϕ) + cos² θ)
= r² (sin² θ + cos² θ)
= r².

Also, as a general note, observe that x and y depend on both angles, so they both better be in the equations for them.