Triangular G Splines

We introduce curvature continuous regular free form surfaces with triangular control nets These surfaces are composed of quartic box spline surfaces and are piecewise polynomial multisided patches of total degree which minimize some energy integral The B ezier nets can be computed e ciently form the spline control net by some xed masks i e matrix multiplications x Introduction Most methods known for building G free form surfaces need polynomials of relatively high degree namely O k see for example Only recently in this high degree was beaten by two methods giving G free form surfaces of bidegree k with singular and regular parametrizations respec tively These low degree surfaces can be represented by a control net or a quasi control net and can be designed so as to allow for subdivision In this paper we will transfer the method given in to triangular box splines Here we restrict ourselves to G surfaces which are the most important for practical applications besides G surfaces Further details and the general case are presented in This paper is organized as follows In paragraph we introduce n sided G patches These patches are used together with generalized C box spline surfaces to build surfaces of arbitrary topology How the free parameters in the construction can be used to generate G splines that minimize certain energy functionals and how these G splines can be generated e ciently will be discussed in paragraph x P Patches The simplest C box splines are those over the three directional grid of total polynomial degree four In this paper we consider only these box splines A quartic box spline surfaces has a regular triangular control net and each of its polynomial patches is determined by vertices called control points which are arranged as in Fig Furthermore we can identify in any triangular net regular subnets of the form of Fig These subnets determine patches forming a generalized box spline surface A generalized box spline surface has holes corresponding to the H Prautzsch G Umlauf Fig Schematic illustration of a quartic box spline patch gray and its control net Fig A triangular net with a vertex of valence left and the cor responding generalized quartic box spline surface with an sided hole right irregular vertices in the net An example is shown in Fig The control net left contains an irregular vertex of valence and the generalized box spline surface right has an sided hole If every irregular vertex is surrounded by at least three rings of regular vertices every irregular vertex corresponds to exactly one hole in the general ized quartic box spline surface In this case an n sided hole is surrounded by a complete surface ring consisting of n patches How to ll such holes with regular G surfaces is described in the follow ing First for any n n we de ne a special generalized box spline surface that lies in the xy plane and has the control net shown in Fig left for n Its control points are the points cijk j ci si k ci si for i n and j and k j where ci cos i n and si sin i n Thus this surface consists of n patches say xn x n which are shown schematically in Fig right Second we construct n patches x xn lling the hole left by the patches xn x n see Fig right Let xl u v w X blijkB ijk u v w Triangular G Splines