The perils of problematic parameterization

No matter how or when one uses polygons to approximate a curved surface, one needs to be very sure that they match that surface closely. The author can think of two ways to check this match: visually and mathematically. A visual check requires looking at the rendered imagery with a critical eye-if the surfaces look free of polygonal artifacts, then the approximation is good enough for that image. This is obviously pretty tough to quantize and implement in software. The mathematical approach suggests that one finds some descriptive measurement that one can apply to both the original model and the polygonal approximation. He evaluates the quality of a polygonal approximation to the original curved object by comparing their surface areas. If the areas are way off, something is wrong with the approximation. But even if the areas are the same, one can't say much. For example, one could start with a curved model of a lion and create a polygonal approximation of a bat. If they both have the same surface area, this measure won't tell one that there's any difference at all. On the other hand, if the areas are way off, one might be prompted to take a closer look. A folk theorem in graphics says if one wants to improve the quality of a polygonal approximation, one should use triangles, reduce their size, and use more of them, making sure of course that their vertices always lie on the surface. The resulting model can't help but get better and better.