2 -splines X1. Introduction Most Methods Known for Building G X2. P-patches

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 8 which minimize some energy integral. The B ezier nets can be computed eeciently form the spline control net by some xed masks, i.e. matrix multiplications. k-free-form surfaces need polynomials of relatively high degree, namely O(k 2), see for example 1,2]. Only recently in 1995 this high degree was beaten by two methods giving G k-free-form surfaces of bidegree 2k + 2 with singular 5] and regular 3] parametrizations, respectively. These low degree surfaces can be represented by a control net 3] or a quasi control net 5] and can be designed so as to allow for subdivision. In this paper we will transfer the method given in 3] to triangular box splines. Here we restrict ourselves to G 2-surfaces which are the most important for practical applications besides G 1-surfaces. Further details and the general case are presented in 4,6]. This paper is organized as follows. In paragraph 2 we introduce n-sided G 2-patches. These patches are used together with generalized C 2-box spline surfaces to build surfaces of arbitrary topology. How the free parameters in the construction can be used to generate G 2-splines that minimize certain energy functionals and how these G 2-splines can be generated eeciently will be discussed in paragraph 3. The simplest C 2-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 12 vertices (called control points) which are arranged as in Fig. 1. Furthermore, we can identify in any triangular net regular subnets of the form of Fig. 1. These subnets determine patches forming a generalized box spline surface. A generalized box spline surface has holes corresponding to the