Dual representation of spatial rational Pythagorean-hodograph curves

Kozak, Jernej; Krajnc, Matjaž; Vitrih, Vito

doi:10.1016/j.cagd.2013.12.001

Cited by 13 publications

(18 citation statements)

References 6 publications

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“…Exploiting the property that every spatial curve has an associated tangent developable surface, the PH curve is determined from its tangent developable as the singular locus, or edge of regression. In [16,21] the first step was modified and the tangent indicatrix was described by a quaternion valued polynomial. In this way a comparatively simpler formula was provided (c.f.…”

Section: Introductionmentioning

confidence: 99%

“…Other results on rational spatial PH curves include [1] where spherical rational curves with rational rotation minimizing frame are constructed by applying Möbius transformations in R 3 to piecewise planar PH cubics. In the series of papers [21][22][23] the authors exploit the dual representation to interpolate with rational spatial PH curves of low class. In [13] a special form of the rational hodograph is used to construct planar rational PH curves with rational arc-length function.…”

Section: Introductionmentioning

confidence: 99%

“…When composed with the translation along the curve, it is a framing motion of the curve. The spherical component has been called Euler-Rodrigues motion or, equivalently referring to the images of a suitable orthogonal tripod, Euler-Rodrigues frame [4,21].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Rational Framing Motions and Spatial Rational Pythagorean Hodograph Curves

Kalkan¹,

Scharler²,

Schröcker³

et al. 2021

Preprint

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We propose a new method for constructing rational spatial Pythagorean Hodograph (PH) curves based on determining a suitable rational framing motion. While the spherical component of the framing motion is arbitrary, the translation part is determined be a modestly sized and nicely structured system of linear equations. Rather surprisingly, generic input data will only result in polynomial PH curves. We provide a complete characterization of all cases that admit truly rational (non-polynomial) solutions. Examples illustrate our ideas and relate them to existing literature.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rational Framing Motions and Spatial Rational Pythagorean Hodograph Curves

Kalkan¹,

Scharler²,

Schröcker³

et al. 2021

Preprint

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show abstract

“…The study of curves with such frames was extended in [42] to the rational case, based on the dual representation [30,41], and it was shown that rational curves with rational osculating RMFs of degree ≥ 6 exist, and the minimum degree of polynomial curves with this property is 7. The use of osculating RMFs to define ruled surfaces, with tangent planes matching the osculating planes of a given space curve, and rulings having the least rotation consistent with this constraint, was also discussed in [20,42].…”

Section: Other Rotation-minimizing Framesmentioning

confidence: 99%

“…Rational PH curves employ entirely different methods for their construction [30,41] and, in general, do not admit rational arc lengths. A discussion of the use of the Möbius transformation in R 3 to generate rational curves with rational RMFs may be found in [1].…”

Section: Spatial Pythagorean-hodograph Curvesmentioning

confidence: 99%

Rational rotation-minimizing frames—Recent advances and open problems

Farouki

2016

Applied Mathematics and Computation

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Recent developments in the basic theory, algorithms, and applications for curves with rational rotation-minimizing frames (RRMF curves) are reviewed, and placed in the context of the current state-of-the-art by highlighting the many significant open problems that remain. The simplest non-trivial RRMF curves are the quintics, characterized by a scalar condition on the angular velocity of the Euler-Rodrigues frame (ERF). Two different classes of RRMF quintics can be identified. The first class of curves may be characterized by quadratic constraints on the quaternion coefficients of the generating polynomials; by the root structure of those polynomials; or by a certain polynomial divisibility condition. The second class has a strictly algebraic characterization, less well-suited to geometrical construction algorithms. The degree 7 RRMF curves offer more shape freedoms than the quintics, but only one of the four possible classes of these curves has been satisfactorily described. Generalizations of the adapted rotation-minimizing frames, for which the angular velocity has no component along the tangent, to directed and osculating frames (with analogous properties relative to the polar and binormal vectors) are also discussed. Finally, a selection of applications for rotation-minimizing frames are briefly reviewedincluding construction of swept surfaces, rigid-body motion planning, 5-axis CNC machining, and camera orientation control.

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New Developments in Theory, Algorithms, and Applications for Pythagorean–Hodograph Curves

Farouki

Giannelli

Sestini

2019

Advanced Methods for Geometric Modeling and Numerical Simulation

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This series will publish textbooks, multi-authors books, thesis and monographs in English language resulting from workshops, conferences, courses, schools, seminars, doctoral thesis, and research activities carried out at INDAM -Istituto Nazionale di Alta Matematica, http://www.altamatematica.it/en. The books in the series will discuss recent results and analyze new trends in mathematics and its applications.

show abstract

Dual representation of spatial rational Pythagorean-hodograph curves

Cited by 13 publications

References 6 publications

Rational Framing Motions and Spatial Rational Pythagorean Hodograph Curves

Rational Framing Motions and Spatial Rational Pythagorean Hodograph Curves

Rational rotation-minimizing frames—Recent advances and open problems

New Developments in Theory, Algorithms, and Applications for Pythagorean–Hodograph Curves

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