# 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