Geometric continuity of parametric curves: three equivalent characterizations

@article{Barsky1989GeometricCO,
  title={Geometric continuity of parametric curves: three equivalent characterizations},
  author={Brian A. Barsky and Tony DeRose},
  journal={IEEE Computer Graphics and Applications},
  year={1989},
  volume={9},
  pages={60-69}
}
  • B. Barsky, T. DeRose
  • Published 1989
  • Mathematics, Computer Science
  • IEEE Computer Graphics and Applications
Some of the important basic results on geometric continuity of curves are presented in a self-contained manner. The paper covers parametric representation and smoothness, parametric continuity, reparameterization and equivalent parameterization, beta-constraints, and arc-length parameterization.<<ETX>> 

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References

SHOWING 1-10 OF 47 REFERENCES
Projectively invariant classes of geometric continuity for CAGD
Abstract A new classification of geometric continuity is presented. It is based on the simultaneous application of the concept of ‘contact of order r” to curves (or surfaces) and to their tangentExpand
Geometric continuity of parametric curves: constructions of geometrically continuous splines
  • B. Barsky, T. DeRose
  • Mathematics, Computer Science
  • IEEE Computer Graphics and Applications
  • 1990
TLDR
A geometrically continuous subclass of Catmull-Rom splines based on geometric continuity and possessing shape parameters is discussed, which leads to the development of geometric constructions for quadratic G/sup 1/ and cubic G/Sup 2/ Beta-splines. Expand
Piecewise polynomial spaces and geometric continuity of curves
SummaryWe say that a curve has geometric continuity if its curvatures and Frenet frame are continuous. In this paper we introduce spaces of piecewise polynomials which can be used to model spaceExpand
Basic functions for rational continuity
The parametric or geometric continuity of a rational polynomial curve has often been obtained by requiring the homogeneous polynomial curve associated with the rational curve to possess parametric orExpand
Curve and surface constructions using rational B-splines
Abstract This paper presents the non-uniform rational B-spline approximation form as a unified approach to representing free-form as well as standard analytic curves and surfaces commonly used inExpand
Parametric Curves, Part Two
TLDR
It is described how Bezier curve segments can be stitched together with G/Sup 1/ or G/sup 2/ continuity, using G/ Sup 1/ continuity and some observations are made concerning the source and nature of shape parameters. Expand
Rational B-Splines for Curve and Surface Representation
  • W. Tiller
  • Computer Science
  • IEEE Computer Graphics and Applications
  • 1983
Nonuniform, rational B-splines, capable of representing both precise quadric primitives and free-form curves and surfaces, offer an efficient mathematical form for geometric modeling systems.
Rational continuity: parametric, geometric, and Frenet frame continuity of rational curves
TLDR
This paper derives constraints on the associated homogeneous curve that are both necessary and sufficient to ensure that the rational curve is either parametrically, geometrically, or Frenet frame continuous. Expand
An Intuitive Approach to Geometric Continuity for Parametric Curves and Surfaces (Extended Abstract)
The notion of geometric continuity is extended to an arbitrary order for curves and surfaces, and an intuitive development of constraints equations is presented that are necessary for it. TheExpand
Geometric Continuity of Parametric Curves
Parametric spline curves are typically constructed so that the first n parametric derivatives agree where the curve segments abut. This type of continuity condition has become known as Cn or nthExpand
...
1
2
3
4
5
...