Bi-quadratic B -spline surfaces generated from arbitrary polyhedral meshes: a constructive approach

This paper describes an algorithm for generating piecewise parametric curved surfaces from arbitrary topological meshes. The algorithm is developed by extending and modifying Riesenfeld's constructive method for determining points on B -spline curves. The surfaces produced near extraordinary points are shown to be slope continuous and compatible with bi-quadratic B -spline surfaces generated from adjacent rectangular regions. In fact, applying this algorithm to rectangular meshes gives bi-quadratic B -spline surfaces. The implementation of this algorithm is simple and the evaluation of position vectors, tangent vectors and normal vectors of points on the surfaces is straightforward.

[1]  Richard F. Riesenfeld,et al.  A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces , 1980, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  Hiroaki Chiyokura,et al.  Design of solids with free-form surfaces , 1983, SIGGRAPH.

[3]  A. R. Forrest,et al.  On coons and other methods for the representation of curved surfaces , 1972, Comput. Graph. Image Process..

[4]  W. Boehm.  Inserting New Knots into B-spline Curves , 1980 .

[5]  M. Sabin,et al.  Behaviour of recursive division surfaces near extraordinary points , 1978 .

[6]  J. A. Gregory,et al.  A C1 triangular interpolation patch for computer-aided geometric design , 1980 .

[7]  Gerald E. Farin,et al.  A construction for visualC1 continuity of polynomial surface patches , 1982, Comput. Graph. Image Process..

[8]  Tom Lyche,et al.  Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics , 1980 .

[9]  Fumihiko Kimura,et al.  Geomap-III: Designing Solids With Free-Form Surfaces , 1984, IEEE Computer Graphics and Applications.

[10]  Requicha,et al.  Solid Modeling: A Historical Summary and Contemporary Assessment , 1982, IEEE Computer Graphics and Applications.

[11]  P. Bézier.  Numerical control : mathematics and applications , 1972 .

[12]  H. Voelcker,et al.  Solid modeling: current status and research directions , 1983, IEEE Computer Graphics and Applications.

[13]  R. Barnhill.  SMOOTH INTERPOLATION OVER TRIANGLES , 1974 .

[14]  Richard Franklin Riesenfeld,et al.  Applications of b-spline approximation to geometric problems of computer-aided design. , 1973 .

[15]  R. J. McDermott.  GRAPHICAL REPRESENTATION OF SURFACES OVER TRIANGLES AND RECTANGLES , 1974 .

[16]  E. Catmull,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[17]  W. J. Gordon,et al.  B-SPLINE CURVES AND SURFACES , 1974 .