Spline Approximations of Real Algebraic Surfaces

We use a combination of both symbolic and numerical techniques to construct several degree boundedG0andG1continuous, piecewise spline approximations of real implicit algebraic surfaces for both computer graphics and geometric modeling. These approximations are based upon an adaptive triangulation (aG0planar approximation) of the real components of the algebraic surface, and include both singular points and singular curves on the surface. A curvilinear wireframe is also constructed using minimum bending energy, parametric curves with additionally normals varying along them. The spline approximations over the triangulation or curvilinear wireframe could be one of several forms: either low degree, implicit algebraic splines (triangular A-patches) or multivariate functional B-splines (B-patches) or standardized rational Bernstein?Bezier patches (RBB), or triangular rational B-Splines. The adaptive triangulation is additionally useful for a rapid display and animation of the implicit surface.

[1]  Jindong Chen,et al.  Modeling with cubic A-patches , 1995, TOGS.

[2]  Vaughan R. Pratt,et al.  Direct least-squares fitting of algebraic surfaces , 1987, SIGGRAPH.

[3]  Mark Hall,et al.  Adaptive polygonalization of implicitly defined surfaces , 1990, IEEE Computer Graphics and Applications.

[4]  Gary Herron.  A Characterization of Certain $C^1 $ Discrete Triangular Interpolants , 1985 .

[5]  Ahmad H. Nasri,et al.  Surface interpolation on irregular networks with normal conditions , 1991, Comput. Aided Geom. Des..

[6]  Pat Hanrahan,et al.  Ray tracing algebraic surfaces , 1983, SIGGRAPH.

[7]  Charles A. Micchelli,et al.  Computing surfaces invariant under subdivision , 1987, Comput. Aided Geom. Des..

[8]  Jules Bloomenthal,et al.  Polygonization of implicit surfaces , 1988, Comput. Aided Geom. Des..

[9]  Thomas W. Sederberg,et al.  Scan line display of algebraic surfaces , 1989, SIGGRAPH.

[10]  Joe D. Warren,et al.  Blending algebraic surfaces , 1989, TOGS.

[11]  James F. Blinn,et al.  A Generalization of Algebraic Surface Drawing , 1982, TOGS.

[12]  Wolfgang Dahmen,et al.  Cubicoids: modeling and visualization , 1993, Comput. Aided Geom. Des..

[13]  Stefan Gnutzmann,et al.  Simplicial pivoting for mesh generation of implicity defined surfaces , 1991, Comput. Aided Geom. Des..

[14]  Insung Ihm,et al.  Smoothing polyhedra using implicit algebraic splines , 1992, SIGGRAPH.

[15]  F. S. Macaulay,et al.  The Algebraic Theory of Modular Systems , 1972 .

[16]  Chandrajit L. Bajaj,et al.  Geometric computations with algebraic varieties of bounded degree , 1990, SCG '90.

[17]  Guoliang Xu,et al.  Piecewise Rational Approximations of Real Algebraic Curves , 1997 .

[18]  E. Allgower,et al.  An algorithm for piecewise lin-ear approximation of implicitly de?ned two-dimensional surfaces , 1987 .

[19]  Devendra Kalra,et al.  Guaranteed ray intersections with implicit surfaces , 1989, SIGGRAPH.

[20]  Carl S. Petersen.  Adaptive contouring of three-dimensional surfaces , 1984, Comput. Aided Geom. Des..

[21]  Chandrajit L. Bajaj,et al.  The GANITH algebraic geometry toolkit , 1990, DISCO.

[22]  Chandrajit L. Bajaj,et al.  NURBS approximation of surface/surface intersection curves , 1994, Adv. Comput. Math..

[23]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[24]  A. F. Adams,et al.  The Survey , 2021, Dyslexia in Higher Education.

[25]  B. Guo,et al.  Surface generation using implicit cubics , 1991 .

[26]  Tony DeRose,et al.  8. A Survey of Parametric Scattered Data Fitting Using Triangular Interpolants , 1992, Curve and Surface Design.

[27]  C. Bajaj.  The Emergence of Algebraic Curves and Surfaces in Geometric Design , 1992 .

[28]  Jörg Peters,et al.  Local cubic and bicubic C1 surface interpolation with linearly varying boundary normal , 1990, Comput. Aided Geom. Des..

[29]  Gary J. Herron,et al.  Smooth closed surfaces with discrete triangular interpolants , 1985, Comput. Aided Geom. Des..