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Why does this matter? It matters because, on average, half the polygons in any scene are facing away from the viewer, and hence shouldn’t be drawn. One way to identify such polygons is to see whether they’re facing toward or away from the viewer; that is, whether their normals have negative z values (so they’re visible) or positive z values (so they should be culled). However, we’re talking about screenspace normals here, because the perspective projection can shift a polygon relative to the viewpoint so that although its viewspace normal has a negative z, its screenspace normal has a positive z, and vice-versa, as shown in Figure 61.3. So we need screenspace normals, but those can’t readily be generated by transformation from worldspace.
Figure 61.3 A problem with determining front/back visibility.
The solution is to use the cross product of two of the polygon’s edges to generate a normal. The formula for the cross product is:
(eq. 7)
(Note that the cross product operation is denoted by an X.) Unlike the dot product, the result of the cross product is a vector. Not just any vector, either; the vector generated by the cross product is perpendicular to both of the original vectors. Thus, the cross product can be used to generate a normal to any surface for which you have two vectors that lie within the surface. This means that we can generate the screenspace normals we need by taking the cross product of two adjacent polygon edges, as shown in Figure 61.4.
In fact, we can cull with only one-third the work needed to generate a full cross product; because we’re interested only in the sign of the z component of the normal, we can skip entirely calculating the x and y components. The only caveat is to be careful that neither edge you choose is zero-length and that the edges aren’t collinear, because the dot product can’t produce a normal in those cases. |
Figure 61.4 How the cross product of polygon edge vectors generates a polygon normal.
Perhaps the most often asked question about cross products is “Which way do normals generated by cross products go?” In a left-handed coordinate system, curl the fingers of your left hand so the fingers curl through an angle of less than 180 degrees from the first vector in the cross product to the second vector. Your thumb now points in the direction of the normal.
If you take the cross product of two orthogonal (right-angle) unit vectors, the result will be a unit vector that’s orthogonal to both of them. This means that if you’re generating a new coordinate space—such as a new viewing frame of reference—you only need to come up with unit vectors for two of the axes for the new coordinate space, and can then use their cross product to generate the unit vector for the third axis. If you need unit normals, and the two vectors being crossed aren’t orthogonal unit vectors, you’ll have to normalize the resulting vector; that is, divide each of the vector’s components by the length of the vector, to make it a unit long.
The dot product is the cosine of the angle between two vectors, scaled by the magnitudes of the vectors. Magnitudes are always positive, so the sign of the cosine determines the sign of the result. The dot product is positive if the angle between the vectors is less than 90 degrees, negative if it’s greater than 90 degrees, and zero if the angle is exactly 90 degrees. This means that just the sign of the dot product suffices for tests involving comparisons of angles to 90 degrees, and there are more of those than you’d think.
Consider, for example, the process of backface culling, which we discussed above in the context of using screenspace normals to determine polygon orientation relative to the viewer. The problem with that approach is that it requires each polygon to be transformed into viewspace, then perspective projected into screenspace, before the test can be performed, and that involves a lot of time-consuming calculation. Instead, we can perform culling way back in worldspace (or even earlier, in objectspace, if we transform the viewpoint into that frame of reference), given only a vertex and a normal for each polygon and a location for the viewer.
Here’s the trick: Calculate the vector from the viewpoint to any vertex in the polygon and take its dot product with the polygon’s normal, as shown in Figure 61.5. If the polygon is facing the viewpoint, the result is negative, because the angle between the two vectors is greater than 90 degrees. If the polygon is facing away, the result is positive, and if the polygon is edge-on, the result is 0. That’s all there is to it—and this sort of backface culling happens before any transformation or projection at all is performed, saving a great deal of work for the half of all polygons, on average, that are culled.
Backface culling with the dot product is just a special case of determining which side of a plane any point (in this case, the viewpoint) is on. The same trick can be applied whenever you want to determine whether a point is in front of or behind a plane, where a plane is described by any point that’s on the plane (which I’ll call the plane origin), plus a plane normal. One such application is in clipping a line (such as a polygon edge) to a plane. Just do a dot product between the plane normal and the vector from one line endpoint to the plane origin, and repeat for the other line endpoint. If the signs of the dot products are the same, no clipping is needed; if they differ, clipping is needed. And yes, the dot product is also the way to do the actual clipping; but before we can talk about that, we need to understand the use of the dot product for projection.
Figure 61.5 Backface culling with the dot product.
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