Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

Three Characterizations of Geometric Continuity for Parametric Curves

Brian A. Barsky and Anthony D. DeRose

EECS Department
University of California, Berkeley
Technical Report No. UCB/CSD-88-417
May 1988

http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/CSD-88-417.pdf

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 C^n or n^th order geometric continuity. It has previously been shown that the use of parametric continuity disallows many parameterizations which generate geometrically smooth curves.

A relaxed form of nth order parametric continuity has been developed and dubbed nth order geometric continuity and denoted G^n. These notes explore three characterizations of geometric continuity. First, the concept of equivalent parametrizations is used to view geometric continuity as a measure of continuity that is parametrization independent, that is, a measure that is invariant under reparametrization. The second characterization develops necessary and sufficient conditions, called Beta-constraints, for geometric continuity of curves. Finally, the third characterization shows that two curves meet with G^n continuity if and only if their arc length parameterizations meet with C^n continuity.

G^n continuity provides for the introduction of n quantities known as shape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices.

Several applications of geometric continuity are present. First, composite Bezier curves are stitched together with G^1 and G^2 continuity using geometric constructions. Then, a subclass of the Catmull-Rom splines based on geometric continuity and possessing shape parameters is discussed. Finally, quadratic G^1 and cubic G^2 Beta-splines are developed using the geometric constructions for the geometrically continuous Bezier segments.


BibTeX citation:

@techreport{Barsky:CSD-88-417,
    Author = {Barsky, Brian A. and DeRose, Anthony D.},
    Title = {Three Characterizations of Geometric Continuity for Parametric Curves},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {1988},
    Month = {May},
    URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/5276.html},
    Number = {UCB/CSD-88-417},
    Abstract = {Parametric spline curves are typically constructed so that the first <i>n</i> parametric derivatives agree where the curve segments abut. This type of continuity condition has become known as <i>C^n</i> or <i>n^th</i> order geometric continuity. It has previously been shown that the use of parametric continuity disallows many parameterizations which generate geometrically smooth curves. <p>A relaxed form of <i>nth</i> order parametric continuity has been developed and dubbed <i>nth</i> order geometric continuity and denoted <i>G^n</i>. These notes explore three characterizations of geometric continuity. First, the concept of equivalent parametrizations is used to view geometric continuity as a measure of continuity that is parametrization independent, that is, a measure that is invariant under reparametrization. The second characterization develops necessary and sufficient conditions, called Beta-constraints, for geometric continuity of curves. Finally, the third characterization shows that two curves meet with <i>G^n</i> continuity if and only if their arc length parameterizations meet with <i>C^n</i> continuity. <p><i>G^n</i> continuity provides for the introduction of <i>n</i> quantities known as shape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices. <p>Several applications of geometric continuity are present. First, composite Bezier curves are stitched together with <i>G^1</i> and <i>G^2</i> continuity using geometric constructions. Then, a subclass of the Catmull-Rom splines based on geometric continuity and possessing shape parameters is discussed. Finally, quadratic <i>G^1</i> and cubic <i>G^2</i> Beta-splines are developed using the geometric constructions for the geometrically continuous Bezier segments.}
}

EndNote citation:

%0 Report
%A Barsky, Brian A.
%A DeRose, Anthony D.
%T Three Characterizations of Geometric Continuity for Parametric Curves
%I EECS Department, University of California, Berkeley
%D 1988
%@ UCB/CSD-88-417
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1988/5276.html
%F Barsky:CSD-88-417