Differential Geometry of Implicit Surfaces -- A Primer


Differential geometry is typically presented for parametric surfaces of the form (u,v) -> (X(u,v), Y(u,v), Z(u,v)). But in computer graphics, we often encounter smooth surfaces defined implicitly, by some equation of the form G(X,y,z) = 0; for example, the unit sphere in 3-space can be defined by x^2 + y^2 + z^1 -1 = 0. If (x,y,z) is a point of such an implicit surface, how can be compute the differential geometry of the surface near that point -- its surface normal, its curvature in any tangent direction, its principle curvatures, its mean and gaussian curvatures? Fortunately, all of these questions are answered by a somewhat obscure paper by Dombrowski. Unfortunately for many readers, it is in German, and it treats the subject in the full generality of n dimensions. In this note, we state some of the results of that paper, and give their proofs. The reader should have some familiarity with differential geometry already; we use the notation of Millman and Parker.