Modifying the shape of NURBS surfaces with geometric constraints

NURBS surfaces are among the most commonly used parametric surfaces in CAGD and Computer Graphics. This paper investigates shape modification of NURBS surfaces with geometric constraints, such as point, normal vector, and curve constraints. Two new methods are presented by constrained optimization and energy minimization. The former is based on minimizing changes in control net of surfaces, whereas the latter is based on strain energy minimization. By these two methods, we change control points and weights of an original surface, such that the modified surface satisfies the given constraints. Comparison results and practical examples are also given.

[1]  I. Gibson,et al.  A New Approach for Direct Manipulation of Free‐Form Curve , 1998, Comput. Graph. Forum.

[2]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[3]  Hong Qin,et al.  Dynamic NURBS with geometric constraints for interactive sculpting , 1994, TOGS.

[4]  Eugene Fiume,et al.  Wires: a geometric deformation technique , 1998, SIGGRAPH.

[5]  Matthew Ming-Fai Yuen,et al.  Unified approach to curve shape modification , 1995, Comput. Aided Des..

[6]  Fuhua Cheng,et al.  Energy and B-spline interproximation , 1997, Comput. Aided Des..

[7]  Richard H. Bartels,et al.  Constraint-based curve manipulation , 1993, IEEE Computer Graphics and Applications.

[8]  Andrew P. Witkin,et al.  Variational surface modeling , 1992, SIGGRAPH.

[9]  Caiming Zhang,et al.  Removing local irregularities of NURBS surfaces by modifying highlight lines , 1998, Comput. Aided Des..

[10]  L. Piegl.  Modifying the shape of rational B-splines. part2: surfaces , 1989 .

[11]  Michel Bercovier,et al.  A comparison of invariant energies for free-form surface construction , 1999, The Visual Computer.

[12]  G. Farin.  NURB curves and surfaces , 1994 .

[13]  Jean-Claude Léon,et al.  Parametrically deformed free-form surfaces as part of a variational model , 1998, Comput. Aided Des..

[14]  Javier Sánchez-Reyes,et al.  A Simple Technique for NURBS Shape Modification , 1997, IEEE Computer Graphics and Applications.

[15]  D. Ross.  Computer-aided design , 1961, CACM.

[16]  G. Farin.  NURB curves and surfaces: from projective geometry to practical use , 1995 .

[17]  Yifan Chen,et al.  Highlight-line algorithm for realtime surface-quality assessment , 1994, Comput. Aided Des..

[18]  Masahiro Kimura,et al.  Surface deformation with differential geometric structures , 1996, Comput. Aided Geom. Des..

[19]  George Celniker,et al.  Deformable curve and surface finite-elements for free-form shape design , 1991, SIGGRAPH.

[20]  David C. Gossard,et al.  Fitting 3D curves to unorganized data points using deformable curves , 1992 .

[21]  Imre Juhász,et al.  Weight-based shape modification of NURBS curves , 1999, Comput. Aided Geom. Des..

[22]  Junji Ishida.  The general B-spline interpolation method and its application to the modification of curves and surfaces , 1997, Comput. Aided Des..