Euler–Rodrigues frames on spatial Pythagorean-hodograph curves

Choi, Hyeong In; Han, Chang Yong

doi:10.1016/s0167-8396(02)00165-6

Cited by 64 publications

(76 citation statements)

References 21 publications

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“…PH curves possess a simple low degree rational adapted frame, which has been called the Euler-Rodrigues frame in [3]. Based on this construction, Farouki proposed a rational approximation of the rotation minimizing frame for any space PH curve [13].…”

Section: Approximation Of Pipe Surfacesmentioning

confidence: 99%

Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces

Šı́r

Jüttler

2005

Mathematics of Surfaces XI

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Abstract. As observed by Farouki et al. [9], any set of C 1 space boundary data (two points with associated first derivatives) can be interpolated by a Pythagorean hodograph (PH) curve of degree 5. In general there exists a two dimensional family of interpolants. In this paper we study the properties of this family in more detail. We introduce a geometrically invariant parameterization of the family of interpolants. This parameterization is used to identify a particular solution, which has the following properties. Firstly, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Secondly, it has the best possible approximation order (4). Thirdly, it is symmetric in the sense that the interpolant of the "reversed" set of boundary data is simply the "reversed" original interpolant. These observations lead to a fast and precise algorithm for converting any (possibly piecewise) analytical curve into a piecewise PH curve of degree 5 which is globally C 1 . Finally we exploit the rational frames associated with any space PH curve (Euler-Rodrigues frame) in order to obtain a simple rational approximation of pipe surfaces with a piecewise analytical spine curve and we analyze its approximation order.

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Section: Approximation Of Pipe Surfacesmentioning

confidence: 99%

Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces

Šı́r

Jüttler

2005

Mathematics of Surfaces XI

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“…Now if we desire rational adapted frames, we may consider only PH curves -since condition (2) is necessary for f 1 (t) to be rational. A rational adapted frame (e 1 (t), e 2 (t), e 3 (t)) known (Choi and Han, 2002) …”

Section: Adapted Frames On Spatial Ph Curvesmentioning

confidence: 99%

“…In general, however, both Frenet frames and rotation-minimizing frames are not rational -even for PH curves. Choi and Han (2002) observed that the spatial PH curves always admit a rational adapted frame, the so-called Euler-Rodrigues frame (ERF). Although the ERF does not have an intuitive geometric significance, and is dependent upon the chosen Cartesian coordinates, it has the advantage over the Frenet frame of being non-singular at inflection points.…”

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confidence: 99%

“…These facts have motivated recent interest in two special classes of PH curves -the double Pythagorean-hodograph (DPH) curves (Beltran and Monterde, 2007;Farouki et al, 2009aFarouki et al, , 2009b which have rational Frenet frames (and thus might also be called RFF curves), and the set of PH curves with rational RMFs (Choi and Han, 2002;Han, 2008). For brevity, we shall call the latter RRMF curves -bearing in mind that they are necessarily PH curves.…”

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confidence: 99%

“…Choi and Han (2002) studied conditions under which the ERF of a PH curve coincides with an RMF, and showed that, for PH cubics, the ERF and Frenet frame are equivalent; for PH quintics, the ERF can be rotation-minimizing only in the degenerate case of planar curves; and the simplest non-planar PH curves for which the ERF can be an RMF are of degree 7. More recently, Han (2008) presented an algebraic criterion characterizing RRMFs of any (odd) degree, and showed that RRMF cubics are degenerate -i.e., they are either planar PH curves, or PH curves with non-primitive hodographs.…”

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confidence: 99%

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Quintic space curves with rational rotation-minimizing frames

Farouki

Giannelli

Manni

et al. 2009

Computer Aided Geometric Design

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The existence of polynomial space curves with rational rotation-minimizing frames (RRMF curves) is investigated, using the Hopf map representation for PH space curves in terms of complex polynomials α(t), β(t). The known result that all RRMF cubics are degenerate (linear or planar) curves is easily deduced in this representation. The existence of nondegenerate RRMF quintics is newly demonstrated through a constructive process, involving simple algebraic constraints on the coefficients of two quadratic complex polynomials α(t), β(t) that are sufficient and necessary for any PH quintic to admit a rational rotationminimizing frame. Based on these constraints, an algorithm to construct RRMF quintics is formulated, and illustrative computed examples are presented. For RRMF quintics, the Bernstein coefficients α 0 , β 0 and α 2 , β 2 of the quadratics α(t), β(t) may be freely assigned, while α 1 , β 1 are fixed (modulo one scalar freedom) by the constraints. Thus, RRMF quintics have sufficient freedoms to permit design by the interpolation of G 1 Hermite data (end points and tangent directions). The methods can also be extended to higher-order RRMF curves.

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Algorithms for spatial Pythagoreanhodograph curves

Farouki

Han

Geometric Properties for Incomplete Data

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Euler–Rodrigues frames on spatial Pythagorean-hodograph curves

Cited by 64 publications

References 21 publications

Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces

Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces

Quintic space curves with rational rotation-minimizing frames

Algorithms for spatial Pythagoreanhodograph curves

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