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Describe the difference in appearance you would expect between a Phong Illumination model that uses (N.H)ⁿ and one that uses (R.V)ⁿ , where H is the halfway vector between the light and the viewer.
Kenny Hoff
2/25/96

You should see a "spreading" of the highlight, when using the (N.H)ⁿ term, since the angle between N and H is generally about half of the angle between R and V (exactly if V, L, and R are coplanar). Why would the highlight "spread" if the angle is cut in half? Because the as the angle between the viewing direction (V) and the reflected light direction (R) increases, the angle between the surface normal (N) and the halfway vector (H) would only increase by half as much resulting in a higher computed intensity further around the highlight.

Why does the angle increase only by half? Well, imagine the given L, N, and V arranged such that the V and R coincide (the viewing direction is exactly the same as the reflected direction of the light - maximum highlight intensity). So the halfway vector (H) should coincide with the normal to the surface (N). Now if we begin to rotate the viewing direction (V) (in the same plane to be exact), the angle between V and R will begin to increase directly, and the angle between N and H will only increase by half as much! Why? Because V and R are being spaced apart directly as a result of the rotation of the viewing direction, but H is recomputed as being halfway between the viewer (V) and the light source (L); however, L hasn't moved so it still contributes to half of the computation of H (H=(L+V)/(||L+V||).

We can show analytically that in the coplanar case that ANG(N,H) (radian angle between N and H) is 1/2 * ANG(R,V):

Since H is the halfway vector between L and V, we can define A equal to ANG(L,H) and ANG(H,V) since both of these angles must be equal.

(1) A = ANG(L,H) = ANG(H,V)

We also know that since R is the vector L reflected about the normal N, ANG(L,N) must be equal to ANG(N,R) by the definition of reflection.

(2) ANG(L,N) = ANG(N,R)

We can write ANG(L,N) in terms of A and ANG(N,H):

(3) ANG(L,N) = A - ANG(N,H)

ANG(N,R) can be written in terms of A, ANG(R,V), and ANG(N,H):

(4) ANG(N,R) = (A-ANG(R,V)) + ANG(N,H)

So now we can obtain an expression containing ANG(N,H) and ANG(R,V) it terms of the given vectors by substituting 3 and 4 into 2:

(5) A - ANG(N,H) = (A-ANG(R,V)) + ANG(N,H)

Finally, we can easily solve for ANG(N,H):

 A - ANG(N,H) = A - ANG(R,V) + ANG(N,H)
    -ANG(N,H) = -ANG(R,V) + ANG(N,H)
-2 * ANG(N,H) = -ANG(R,V)
     ANG(N,H) = 1/2 * ANG(R,V)

Is this slower tapering really a problem? No, since you can always adjust the exponent. Besides, both illumination models are just based on observation and experiment, not on what actually happens.