Computing Isophotes on Free-form Surfaces based on Approximate Dual Implicitization

The support function of a free-form–surface is closely related to the implicit equation of the dual surface, and the process of computing both the dual surface and its support function can be seen as dual implicitization. The support function can be used to parameterize a surface by its inverse Gauss map. This map makes it relatively simple to study offset surfaces and isophotes (which are simply images of spherical circles). We present several classes of surfaces which admit a particularly simple computation of the dual surfaces and of the support function. These include quadratic polynomial surfaces, ruled surfaces with direction vectors of low degree and polynomial translational surfaces of bidegree (3,2). In addition, we use an approximation method for bivariate quadratic splines over criss-cross triangulations in order to formulate a method for approximate dual implicitization. The inverse Gauss maps of the bivariate quadratic spline surfaces are computed and used for approximate isophote computation. The approximation order of the isophote approximation is shown to be 2.