Construction of Rational Curves with Rational Rotation-Minimizing Frames via Möbius Transformations

Bartoň, Michael; Jüttler, Bert; Wang, Wenping

doi:10.1007/978-3-642-11620-9_2

Cited by 13 publications

(15 citation statements)

References 22 publications

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“…Now from the definition (49) of η, the conditions (41) allow us to conclude that η A ∈ [ 0,η ] and η B ∈ [ −π, −π +η ] ⊆ [ −π, −η ]. Thus, it follows also that β A ∈ [ 0, 1 2 π ] and β B ∈ − π, − 1 2 π . The proofs for the other three possibilities, depending on the signs of o f · f i and o f · g i , are analogous.…”

Section: Appendixmentioning

confidence: 86%

“…Hence 0, 1 2 π ⊂ (−β * , π − β * ) and − π, − 1 2 π ∩ (−β * , π − β * ) = ∅. So it suffices to show that the conditions (41) imply that β A ∈ 0, 1 2 π and β B ∈ [ −π, − 1 2 π ].…”

Section: Appendixmentioning

confidence: 93%

“…Proposition 1 addresses the generic case with o f · f i = 0 and o f · g i = 0, while the special cases o f · g i = 0 and o f · f i = 0 are treated in Propositions 3 and 4 below. It is convenient to express these results in terms of the angleη ∈ (0, 1 2 π ] defined by (40), equal to the right-hand side of (51) when β = 1 2 π . Note that, when β = − 1 2 π , the angle on the right in (51) is equal to −η.…”

Section: Closurementioning

confidence: 99%

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An interpolation scheme for designing rational rotation-minimizing camera motions

2011

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When a moving (real or virtual) camera images a stationary object, the use of a rotation-minimizing directed frame (RMDF) to specify the camera orientation along its path yields the least apparent rotation of the image. The construction of such motions, using curves that possess rational RMDFs, is considered herein. In particular, the construction entails interpolation of initial/final camera positions and orientations, together with an initial motion direction. To achieve this, the camera path is described by a rational space curve that has a rational RMDF and interpolates the prescribed data. Numerical experiments are used to illustrate implementation of the method, and sufficient conditions on the two end frame orientations are derived, to ensure the existence of exactly one interpolant. By specifiying a sequence of discrete camera positions/orientations and an initial motion direction, the method can be used to construct general rotation-minimizing camera motions.

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Section: Appendixmentioning

confidence: 86%

Section: Appendixmentioning

confidence: 93%

Section: Closurementioning

confidence: 99%

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An interpolation scheme for designing rational rotation-minimizing camera motions

2011

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“…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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“…x ′2 (ξ) + y ′2 (ξ) + z ′2 (ξ) ≡ σ 2 (ξ) (1) for some polynomial σ(ξ). The solutions to (1) can be characterized [5,10] in terms of the quaternion algebra H = R + Ri + Rj + Rk. We identify with R 3 the vector subspace Ri + Rj + Rk ⊂ H, whose elements are called pure vectors.…”

Section: Introductionmentioning

confidence: 99%

A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

et al. 2016

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A rotation-minimizing frame (f 1 , f 2 , f 3 ) on a space curve r(ξ) defines an orthonormal basis for R 3 in which f 1 = r ′ /|r ′ | is the curve tangent, and the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about f 1 . Polynomial curves that admit rational rotation-minimizing frames (or RRMF curves) form a subset of the Pythagorean-hodograph (PH) curves, specified by integrating the form r ′ (ξ) = A(ξ) i A * (ξ) for some quaternion polynomial A(ξ). By introducing the notion of the rotation indicatrix and the core of the quaternion polynomial A(ξ), a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms.

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Construction of Rational Curves with Rational Rotation-Minimizing Frames via Möbius Transformations

Cited by 13 publications

References 22 publications

An interpolation scheme for designing rational rotation-minimizing camera motions

An interpolation scheme for designing rational rotation-minimizing camera motions

Rational rotation-minimizing frames—Recent advances and open problems

A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

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