Applications of Laguerre geometry in CAGD

Pottmann, Helmut; Peternell, Martín

doi:10.1016/s0167-8396(97)00023-x

Cited by 110 publications

(73 citation statements)

References 13 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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“…For a general and more in depth discusion on the cyclographic model and its applications to computer aided geometric design see Hoffmnann [5], and Pottmann and Peternell [13].…”

Section: The Cyclographic Modelmentioning

confidence: 99%

Revisiting variable radius circles in constructive geometric constraint solving

Chiang

Joan-Arinyo

2004

Computer Aided Geometric Design

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Variable-radius circles are common constructs in planar constraint solving and are usually not handled fully by algebraic constraint solvers. We give a complete treatment of variable-radius circles when such a circle must be determined simultaneously with placing two groups of geometric entities. The problem arises for instance in solvers using triangle decomposition to reduce the complexity of the constraint problem.This work offers a set of basic constructive methods that permits to determine variable radius circles simultaneously with placing two rigid geometric objects when geometric constraints are defined on both the circumference and center point of the constraint circle. The problem has been classified by the geometric entities in two groups, one is fixed and the other has translational and rotational movement, so that the variable radius circles satisfy the constraints on the geometric entities in these two groups. The number of solutions for each problem is also given.

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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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Applications of Laguerre geometry in CAGD

Cited by 110 publications

References 13 publications

On rational Minkowski Pythagorean hodograph curves

On rational Minkowski Pythagorean hodograph curves

Revisiting variable radius circles in constructive geometric constraint solving

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

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