Quadric splines

Surface rendering or point location on a surface can easier be accomplished in an implicit rather than parametric representation This observation has been the key motivation for developing piecewise algebraic splines In particular Dahmen and Guo used triangular segments of quadrics to build tangent plane continuous surfaces interpolating the vertices of a trian gular net with prescribed normals Their construction is based on the implicit B ezier representation introduced by Sederberg and employs the idea of the Powell Sabin split for bivariate C piecewise quadratics While Dahmen s and Guo s approach is completely algebraic the objective of this paper is to derive their quadric splines solely geometrically in projective space The geometric approach has several bene ts It provides a geometric meaning for certain parameters chosen to be the same constant by Dahmen and Guo Furthermore it facilitates the classi cation of the quadrics avoids the global dependencies of Dahmen s and Guo s transversal system and renders the Powell Sabin interpolant as a special case