# Phong lighting model - R vector

In Phong lighting model a vector R is used to calculate the spectral component of light. The equation used to obtain it is

R = 2(N•L)N - L


I can't understand why we use this equation. Could someone give me a demonstration of how we obtain this equation?

• For anybody arriving here from a search, I have a more detailed explanation of the derivation of the formula in question - without requiring prior knowledge of vector algebra - on the math sister site: math.stackexchange.com/a/4335556/6928 Commented Dec 21, 2021 at 16:51

The equation is a consequence of the symmetry of the reflection of the light about the normal:

We know from the law of reflection that the angle between R and N is the same as the one between L and N. Therefore it's also clear that:

R•N = L•N (assume all are unit vectors)


From symmetry, it's also evident that

R - (R•N)N = - (L - (L•N)N)


Now we can replace (R•N)N with (L•N)N from the previous equation. So we have:

R - (L•N)N = - (L - (L•N)N)


Adding (L•N)N to both sides gives:

R = - (L - (L•N)N) + (L•N)N
= - L + (L•N)N + (L•N)N
= 2(L•N)N - L