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The paper Microfacet Models for Refraction through Rough Surfaces (among others) reminds us the following assumptions about the microfacet distribution function D:

  1. Microfacet density is positive valued
  2. Total microsurface area is at least as large as the corresponding macrosurface’s area
  3. The (signed) projected area of the microsurface is the same as the projected area of the macrosurface for any direction v

I can see why 1) a distribution density is a positive value, and intuitively believe that 2) means that the total area of sloped microfacets cannot be smaller than their projection.
However I am not sure to understand the justification for 3). What does the third condition mean mean?

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It's a geometric assumption like the other two. Consider a flat macrosurface. Its projected area in any direction $v$ is just $v\dot\ \hat N$ times its area (where $\hat N$ is the surface normal). In particular, the case where you're looking at it along the normal is simplest: the projected area is equal to the area of the surface.

Now split the macrosurface into microfacets. The total area of the microfacets is at least as much (assumption 2), but each 'kink' in the surface bends the normals of the separate microfacets away from the original normal. Whatever the shape of the microfacets, the sum of their projected areas doesn't change. In the case where you're looking along the normal, it's easy to see that the total projected area is the same: the surface would have to get larger or smaller for it to change.

For any direction, the microfacet has to cover a portion of the original projected area of the surface. Changing the orientation of the microfacet while still filling that portion doesn't change its projected area.

There's one tricky case, which is where the microfacets overhang each other. In this case, the total area is greater, because some area is covered by more than one microfacet. But in this case, at least one of the microfacets has to end up pointing away from the view direction, back into the surface. In this case, the dot product is negative, so this cancels out the area covered by more than one microfacet. This is why the text is careful to single out that it's the signed projected area.

There's one more tricky case, which is where the microfacets extend past the silhouette of the object. This might happen when you're looking from very glancing angles, or where overhanging facets overhang outside the perimeter of the surface. In this case, the projected area of the microfacets will be greater, violating the third assumption. We don't generally consider this case. Intuitively, it matches up with the fact that techniques like bump-mapping don't change the shape of the silhouette of the object.

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    $\begingroup$ I think even in the silhouette case, using signed projected area (as you noted) means that assumption 3 isn't violated, so long as the microsurface's boundaries match the macrosurface's. Even if there are overhangs beyond the silhouette, the signed projected area of facets on the front and back sides of the overhang will cancel out. $\endgroup$ – Nathan Reed Jan 19 '16 at 18:30
  • $\begingroup$ (Also, maybe this goes without saying, but I think the assumptions also guarantee that the microsurface is a nice, 2-manifold surface without any holes or other weird stuff.) $\endgroup$ – Nathan Reed Jan 19 '16 at 18:51
  • $\begingroup$ @NathanReed That's true, I should have been more precise about that. As for what the assumptions guarantee, I think of it the other way round: the fact that a surface, however faceted, has to be the whole of a boundary between some "inside" and some "outside" forces it to have the three properties. $\endgroup$ – Dan Hulme Jan 20 '16 at 9:58

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