A Laguerre geometric approach to rational offsets

Peternell, Martín; Pottmann, Helmut

doi:10.1016/s0167-8396(97)00024-1

Cited by 92 publications

(55 citation statements)

References 19 publications

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“…We note that x ± are also known as components of the cyclographic image of y in the context of Laguerre geometry, see e.g. Peternell and Pottmann (1998); Pottmann and Peternell (1998).…”

Section: Medial Axis Transforms With Rational Domain Boundariesmentioning

confidence: 99%

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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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“…We note that x ± are also known as components of the cyclographic image of y in the context of Laguerre geometry, see e.g. Peternell and Pottmann (1998); Pottmann and Peternell (1998).…”

Section: Medial Axis Transforms With Rational Domain Boundariesmentioning

confidence: 99%

“…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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“…[19,20,21,12,22]. Later, the concept of polynomial planar PH curves was generalized to space PH curves ( [23,24,25,26]) and to rational PH curves ( [27,28,29]). Analogously, the notion of rational surfaces with rational offsets, the so called Pythagorean Normal vector (PN) surfaces, was introduced in [27].…”

Section: Introductionmentioning

confidence: 99%

Volumes with piecewise quadratic medial surface transforms: Computation of boundaries and trimmed offsets

Bastl

Jüttler

Kosinka

et al. 2010

Computer-Aided Design

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MOS surfaces are rational surfaces in R 3,1 which provide rational envelopes of the associated two-parameter family of spheres. Moreover, all the offsets admit rational parameterizations as well. Recently, it has been proved that quadratic triangular Bézier patches in R 3,1 are MOS surfaces. Following this result, we describe an algorithm for computing an exact rational envelope of a two-parameter family of spheres given by a quadratic patch in R 3,1 . The main focus of this paper is given to geometric aspects of the algorithm. Since these patches are capable of producing C 1 smooth approximations of medial surface transforms of spatial domains, we use this algorithm to generate rational approximations of envelopes of general medial surface transforms. One of the main advantages of this approach to offsetting is the fact that the trimming procedure becomes considerably simpler.

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“…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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A Laguerre geometric approach to rational offsets

Cited by 92 publications

References 19 publications

On rational Minkowski Pythagorean hodograph curves

On rational Minkowski Pythagorean hodograph curves

Volumes with piecewise quadratic medial surface transforms: Computation of boundaries and trimmed offsets

On quadratic two-parameter families of spheres and their envelopes

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