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

, 1988

http://www2.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},
    Year= {1988},
    Month= {May},
    Url= {http://www2.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://www2.eecs.berkeley.edu/Pubs/TechRpts/1988/5276.html
%F Barsky:CSD-88-417