OpenGL Programming Guide

Appendix FCalculating Normal Vectors

Finding Normals from Polygonal Data

As mentioned previously, you often want to find normals for surfaces that are described with polygonal data such that the surfaces appear smooth rather than faceted. In most cases, the easiest way for you to do this (though it might not be the most efficient way) is to calculate the normal vectors for each of the polygonal facets and then to average the normals for neighboring facets. Use the averaged normal for the vertex that the neighboring facets have in common. Figure F-2 shows a surface and its polygonal approximation. (Of course, if the polygons represent the exact surface and aren't merely an approximation - if you're drawing a cube or a cut diamond, for example - don't do the averaging. Calculate the normal for each facet as described in the following paragraphs, and use that same normal for each vertex of the facet.)

Figure F-2 : Averaging Normal Vectors

To find the normal for a flat polygon, take any three vertices v1, v2, and v3 of the polygon that do not lie in a straight line. The cross product

[v1 - v2] × [v2 - v3]

is perpendicular to the polygon. (Typically, you want to normalize the resulting vector.) Then you need to average the normals for adjoining facets, to avoid giving too much weight to one of them. For instance, in the example shown in Figure F-2 , if n1, n2, n3, and n4 are the normals for the four polygons meeting at point P, calculate n1+n2+n3+n4 and then normalize it. The resulting vector can be used as the normal for point P.

Sometimes, you need to vary this method for particular situations. For instance, at the boundary of a surface (for example, point Q in Figure F-2 ), you might be able to choose a better normal based on your knowledge of what the surface should look like. Sometimes the best you can do is to average the polygon normals on the boundary as well. Similarly, some models have some smooth parts and some sharp corners (point R is on such an edge in Figure F-2 ). In this case, the normals on either side of the crease shouldn't be averaged. Instead, polygons on one side of the crease should be drawn with one normal, and polygons on the other side with another.

OpenGL Programming Guide