A geometric constraint on curve networks suitable for smooth interpolation

A key problem when interpolating a network of curves occurs at vertices: an algebraic condition, called the vertex enclosure constraint, must hold wherever an even number of curves meet. This paper recasts the vertex enclosure constraint in terms of the local geometry of the curve network. This allows formulating a new geometric constraint, related to Euler's Theorem on local curvature. The geometric constraint implies the vertex enclosure constraint and is equivalent to it where four curve segments meet without forming an X. Also the limiting case of collinear curve tangents is analyzed.

[1]  Helmut Pottmann,et al.  Interpolation on surfaces using minimum norm networks , 1992, Comput. Aided Geom. Des..

[2]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .

[3]  Gerald Farin,et al.  Geometric modeling : algorithms and new trends , 1987 .

[4]  J. Peters,et al.  C 1 -surface splines , 1995 .

[5]  Jörg Peters,et al.  Patching Catmull-Clark meshes , 2000, SIGGRAPH.

[6]  Franz-Erich Wolter,et al.  Geometrical Criteria to Guarantee Curvature Continuity of Blend Surfaces , 1992 .

[7]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[8]  M. A. Watkins,et al.  Problems in geometric continuity , 1988 .

[9]  J. A. Gregory.  Smooth interpolation without twist constraints , 1974 .

[10]  Ulrich Reif,et al.  Degenerate Bézier patches with continuous curvature , 1997, Comput. Aided Geom. Des..

[11]  J. Peters.  Smooth interpolation of a mesh of curves , 1991 .

[12]  Ramon F. Sarraga,et al.  Errata: G1 interpolation of generally unrestricted cubic Bézier curves , 1989, Comput. Aided Geom. Des..

[13]  Jarke J. van Wijk,et al.  Bicubic patches for approximating non-rectangular control-point meshes , 1986, Comput. Aided Geom. Des..

[14]  J. A. Gregory.  Geometric continuity , 1989 .

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

[16]  Francis Schmitt,et al.  G1 Smooth Connection Between Rectangular and Triangular Bézier Patches at a Common Corner , 1991, Curves and Surfaces.

[17]  Marian Neamtu,et al.  Degenerate polynomial patches of degree 4 and 5 used for geometrically smooth interpolation in 3 , 1994, Comput. Aided Geom. Des..

[18]  W. Boehm,et al.  Bezier and B-Spline Techniques , 2002 .

[19]  Thomas Hermann,et al.  G2 interpolation of free form curve networks by biquintic Gregory patches , 1996, Comput. Aided Geom. Des..

[20]  G. Renner.  Polynomial N-sided Patches , 1991, Curves and Surfaces.

[21]  Josef Hoschek,et al.  Handbook of Computer Aided Geometric Design , 2002 .