Geometric continuity of parametric curves: three equivalent characterizations

Barsky, Brian A.; DeRose, Tony

doi:10.1109/38.41470

Cited by 99 publications

(53 citation statements)

References 17 publications

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“…In [1,16,17], the definition of G k continuity is given. Further, the practical Beta-constraints for the geometric continuity of curves are provided in [16,17]. According to the Beta-constraints, we have the following conclusion.…”

Section: Composite Adjustable Bézier Curvesmentioning

confidence: 99%

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Adjustable Bézier Curves with Simple Geometric Continuity Conditions

Yan

2016

MCA

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This paper aims to simplify the continuity conditions of Bézier curves. For this purpose, a special family of Bézier curves with three parameters, to be called adjustable Bézier curves, is constructed. They have the same structure as the quartic Bézier curves. The newly constructed curves possess some of the basic properties of Bézier curves, such as the convex hull property, symmetry, geometric invariance, etc., and they have shape adjustability. Moreover, under the geometric continuity of order 1 (G 1 ) conditions of the usual Bézier curves, the adjustable Bézier curves can reach geometric continuity of order k (G k ); here, k is one of the parameters of the newly constructed curves. The recursive evaluation algorithm of the new curves is provided. We also discuss how to construct the adjustable Bézier curves with a given tangent polygon. Numerical examples illustrate the correctness and validity of the proposed method.

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Section: Composite Adjustable Bézier Curvesmentioning

confidence: 99%

“…To make the two curves G k -continuous, it is necessary that (taken from the Beta-constraints in [16] where β 1 > 0. Substituting (16) into (17), we obtain…”

Section: Lemma 1 Let T ∈ [0mentioning

confidence: 99%

Adjustable Bézier Curves with Simple Geometric Continuity Conditions

Yan

2016

MCA

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“…Let us consider the notion corresponding to reparametrization. The necessary and su cient conditions can be expressed as replacing the matrix consisting of 0 s in (2) by a matrix that is expressed as the product of an matrix (the same as the one de ned in (2)) and a matrix as de ned in (Barsky and DeRose 1989Barsky and DeRose 1990DeRose 1985. Thus, for second order rational geometric continuity, w e will have v e shape parameters 0 , 1 , 2 , 1 , a n d 2 .…”

Section: Rational Geometric Continuitymentioning

confidence: 99%

“…That is, a curve is said to be geometrically continuous (denoted G n ) if there exists some reparametrization of its segments such that the resulting curve i s C n . The reparametrization criterion on its segments leads to the derivation of Beta-constraints (Barsky 1988bBarsky and DeRose 1989Barsky and DeRose 1990DeRose 1985. The second notion of geometric continuity of parametric curves is based on the continuity o f Frenet Frame and higher order curvatures (Boehm 1985Boehm 1987Dyn and Micchelli 1985Hagen 1985.…”

Section: Introductionmentioning

confidence: 99%

Basis Functions for Rational Continuity

Monocha

Barsky

1990

CG International ’90

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The parametric or geometric continuity of a rational polynomial curve h a s often been obtained by requiring the homogeneous polynomial curve associated with the rational curve to possess parametric or geometric continuity, respectively. Recently this approach has been shown overly restrictive. We m a k e use of the necessary and su cient conditions of rational parametric continuity for de ning basis functions for the homogeneous representation of a rational curve.These functions are represented in terms of shape parameters of rational continuity, which a r e i n troduced due to these exact conditions. The shape parameters may b e v aried globally, a ecting the entire curve, or modi ed locally thereby a ecting only a few segments. Moreover, the local parameters can be represented as continuous or discrete functions. Based on these properties, we i n troduce three classes of basis functions which can be used for the homogeneous representation of rational parametric curves.

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“…is a so-called j8-matrix [1], [3], [5], [22], [23], [33] [53]. Following the standard convention [27], [37] we will assume throughout that the lower triangular connection matrices Cj are nonsingular and totally positive, but otherwise arbitrary.…”

Section: Geometric Continuitymentioning

confidence: 99%

New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree

Seidel¹

1992

ESAIM: M2AN

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Geometric continuity of parametric curves: three equivalent characterizations

Cited by 99 publications

References 17 publications

Adjustable Bézier Curves with Simple Geometric Continuity Conditions

Adjustable Bézier Curves with Simple Geometric Continuity Conditions

Basis Functions for Rational Continuity

New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree

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