174 Free Form Surface Design with A-Patches· Chandrajit L. Bajaj Jindon Chen Guoliang Xu De partme nt of Comput er Sciences Purdu e University West Lafayett e, Indiana {bajaj , jdc ,xuguo }@cs.purdue.edu Fax : 317-494-0739, Tel: 317-494-6531 Abstract We present a sufficient criterion for the Bernstein-Bezier (BB) form of a trivariate polynomial within a tetrahedron, such that the real zero contour of the polynomial defines a smooth and single sheeted algebraic surface patch. We call this an A-patch. We present algorithms to build a mesh of cubic A-patches to interpolate a given set of scattered point data in three dimensions, respecting the topology of any surface triangulation T of the gi ven point set. In these algorithms we first specify "normals" on the data points, then build a simplicial hull consisting of tetrahedra surrounding the surface triangulation T and finally construct cubic A-patches within each tetrahedron. The resulting surface constructed is Cl (tangent plane) continuous and single sheeted in each of the tetrahedra. We al so show how to adjust the free parameters of the A-patches to achieve both local and global shape control. Keywords: Algebraic Surface Patches, Interpolation, Approximation, Cl Continuity 1 Introduction The importance of implicit surface representation in mod- eling geometric objects or reconstructing the image to scattered data has been described in various papers (see for e.g. [2, 7, 9, I I , 16]). The main shortcoming held against the popular use of implicit surfaces is that the representation being multi valued may cause the real zero contour surface to have mUltiple sheets, self-intersections and several other undesirable singularities. In section 3 of this paper, we present a sufficient cri- *This wo rk was supported in part by AFOSR grants F49620-93- 10 138. F49620-94-I-0080, NSF grants CCR 92-22467, DMS 91 - 0 1424, NASA grant NAG- I-1473 a nd ONR gra nt NOOO I4-94- 1-0370 terion for the Bernstein-Bezier (BB) form of a trivariate polynomial within a tetrahedron such that the real zero- contour of the polynomial is smooth (non-singular) and a single sheeted algebraic surface. We call this an A -patch. In section 4, we describe how to build a simplicial hull consisting of tetrahedra surrounding a surface triangula- tion T of the set of scattered data points in 3D. We then show in section 5 how a mesh of cubic A-patches can be used to construct a Cl interpolatory surface, respecting the topology of the surface triangulation T. In section 6, we show how to adjust the free parameters of the A- patches to achieve both local and global shape control. This Cl cubic A-patch fitting algorithm is quite appropri- ate for free form design. In analogy to the final smoothing of an artist's rough sketches, complicated smooth models can be directly formed by first creating a rough polyhe- dral model of the desired object and then using the fitting algorithms to produce a Cl smooth solid with extra local and global parameters for fine shape control. Proofs of all theorems and lemmas are given in the full version of this paper [3]. Related Prior Work: The work of characterizing the BB form of polynomi- als within a tetrahedron such that the zero contour of the polynomial is a single sheeted surface within the tetrahe- dron, has been attempted in the past. In [16], Sederberg showed that if the coefficients of the BB form of the trivariate polynomial on the lines that parallel one edge, say L, ofthe tetrahedron, all increase (or decrease) mono- tonically in the same direction, then any line parallel to L will intersect the zero contour algebraic surface patch at most once. In [9], Guo treats the same problem by en- forcing monotonicity conditions on a cubic polynomial along the direction from one vertex to a point of the oppo- site face of the vertex. From this he derives a condition a>' -e,+ e4 - a>. 2: 0 for all A = ( AI , A 2, A 3, A4f with AI 2: I, where a>. are the coefficients of the cubic in BB form and ei is the i-th unit vector. This condition Graphics Interface '9 4