Studying cyclides with Laguerre geometry

Krasauskas, Rimvydas; Mäurer, Christoph

doi:10.1016/s0167-8396(99)00008-4

Cited by 46 publications

(34 citation statements)

References 17 publications

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“…We recall a concept of the so called isotropic surface in R 2,1 (Krasauskas and Mäurer, 2000;Peternell and Pottmann, 1998;Pottmann and Peternell, 1998) studied in connection with Laguerre geometry and cyclographic mapping. Starting from a curve x(t) ⊂ R 2 , the corresponding isotropic surface Γ ⊂ R 2,1 is described as…”

Section: Theoremmentioning

confidence: 99%

On rational Minkowski Pythagorean hodograph curves

Kosinka

Lávička

2010

Computer Aided Geometric Design

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Minkowski Pythagorean hodograph curves are polynomial curves with polynomial speed, measured with respect to Minkowski norm. Curves of this special class are particularly well suited for representing medial axis transforms of planar domains. In the present paper we generalize this polynomial class to a rational class of curves in Minkowski 3-space. We show that any rational Minkowski Pythagorean hodograph curve can be obtained in terms of its associated planar rational Pythagorean hodograph curve and an additional rational function. Moreover, both in the original polynomial and new rational case, we investigate the close relationship between these associated curves in Euclidean plane and Minkowski space.

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Section: Theoremmentioning

confidence: 99%

On rational Minkowski Pythagorean hodograph curves

Kosinka

Lávička

2010

Computer Aided Geometric Design

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“…These two-parameter family of lines i 1 , i 2 form the isotropic hyper-surface Γ(Φ) passing through Φ. This method of computing the cyclographic image has already been proposed in [20].…”

Section: Envelope Construction Using the Projective Extension Inmentioning

confidence: 99%

“…The rationality of envelopes of one-and two-parameter families of spheres has attracted the interest of several researchers, see [19,20,21,22,23,24,26,33]. Unfortunately, the direct computation does in general not lead to rational parameterizations in case of rational envelopes and thus it is necessary to study these objects in more detail in order to gain insight to their geometric properties.…”

Section: Introductionmentioning

confidence: 99%

On quadratic two-parameter families of spheres and their envelopes

Peternell

Odehnal

Sampoli

2008

Computer Aided Geometric Design

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In the present paper we investigate rational two-parameter families of spheres and their envelope surfaces in Euclidean R 3 . The four dimensional cyclographic model of the set of spheres in R 3 is an appropriate framework to show that a quadratic triangular Bézier patch in R 4 corresponds to a two-parameter family of spheres with rational envelope surface. The construction shows also that the envelope has rational offsets. Further we outline how to generalize the construction to obtain a much larger class of surfaces with similar properties.

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“…The curvature lines of F are the generator lines and the circles in planes orthogonal to A. From decomposition (19) we know that the bisector surface of O and F is…”

Section: Point-circular Cone and Point-circular Cylinder Bisectorsmentioning

confidence: 99%

Geometric Properties of Bisector Surfaces

Peternell¹

2000

Graphical Models

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This paper studies algebraic and geometric properties of curve-curve, curvesurface, and surface-surface bisectors. The computation is in general difficult since the bisector is determined by solving a system of nonlinear equations. Geometric considerations will help us to determine several distinguished curve and surface pairs which possess elementary computable bisectors. Emphasis is on lowdegree rational curves and surfaces, since they are of particular interest in surface modeling.

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Studying cyclides with Laguerre geometry

Cited by 46 publications

References 17 publications

On rational Minkowski Pythagorean hodograph curves

On rational Minkowski Pythagorean hodograph curves

On quadratic two-parameter families of spheres and their envelopes

Geometric Properties of Bisector Surfaces

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