Geometric Interpolation by Quartic Rational Spline Motions

Jüttler, Bert; Krajnc, Matjaž; Žagar, Emil

doi:10.1007/978-90-481-9262-5_40

Cited by 3 publications

(3 citation statements)

References 7 publications

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“…for the interpolation at the right end point. Identifying the equations for the inner control points, i.e., (9) and (14), (10) and (13), (11) and (12), we derive three quaternion equations…”

Section: Analysis Of a Rotational Part Of The Motionmentioning

confidence: 99%

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Construction of G 3 rational motion of degree eight

Ferjančič

Krajnc

Vitrih

2016

Applied Mathematics and Computation

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The paper presents a construction of a rigid body motion with point trajectories being rational spline curves of degree eight joining together with G 3 smoothness. The motion is determined through interpolation of positions and derivative data up to order three in the geometric sense. Nonlinearity in the spherical part of construction results in a single univariate quartic equation which yields solutions in a closed form. Sufficient conditions on the regions for the curvature data are derived, implying the existence of a real admissible solution. The algorithm how to choose appropriate data is proposed too. The theoretical results are substantiated with numerical examples.

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Section: Analysis Of a Rotational Part Of The Motionmentioning

confidence: 99%

“…One of the possible remedies to get motions of a lower degree is to use geometric interpolation techniques instead (see [6], [7], [8], [9], [10], [11]). The first steps in this direction were proposed in [12] and [13]. Geometric interpolation by parabolic splines was used to construct G 1 quartic rational spline motions.…”

Section: Introductionmentioning

confidence: 99%

Construction of G 3 rational motion of degree eight

Ferjančič

Krajnc

Vitrih

2016

Applied Mathematics and Computation

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“…These values are listed in Table 2. The next step is to construct the piecewise-quadratic function w(ξ) = a(ξ) + i b(ξ) with coefficients given by (17) k Table 2: Angular velocity estimates obtained from the cubic spline interpolant.…”

Section: Examplementioning

confidence: 99%

C1 and C2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

Moon

Farouki

2018

Computer Aided Geometric Design

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The problem of constructing a rational adapted frame (f1(ξ), f2(ξ), f3(ξ)) that interpolates a discrete set of orientations at specified nodes along a given spatial Pythagorean-hodograph (PH) curve r(ξ) is addressed. PH curves are the only polynomial space curves that admit rational adapted frames, and the Euler-Rodrigues frame (ERF) is a fundamental instance of such frames. The ERF can be transformed into other rational adapted frame by applying a rationally-parametrized rotation to the normal-plane vectors. When orientation and angular velocity data at curve end points are given, a Hermite frame interpolant can be constructed using a complex quadratic polynomial that parametrizes the normal-plane rotation, by an extension of the method recently introduced to construct a rational minimal twist frame (MTF). To construct a rational adapted spline frame, a representation that resolves potential ambiguities in the orientation data is introduced. Based on this representation, a C 1 rational adapted spline frame is constructed through local Hermite interpolation on each segment, using angular velocities estimated from a cubic spline that interpolates the frame phase angle relative to the ERF. To construct a C 2 rational adapted spline frame, which ensures continuity of the angular acceleration, a complex-valued cubic spline is used to directly interpolate the complex exponentials of the phase angles at the nodal points.

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Hermite G 1 rational spline motion of degree six

Počkaj

2013

Numer Algor

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Geometric Interpolation by Quartic Rational Spline Motions

Cited by 3 publications

References 7 publications

Construction of G 3 rational motion of degree eight

Construction of G 3 rational motion of degree eight

C1 and C2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

Hermite G 1 rational spline motion of degree six

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