Journal ArticleDOI
Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures
TLDR
The problem of specifying the two free parameters that arise in spatial Pythagorean-hodograph (PH) quintic interpolants to given first-order Hermite data is addressed and conditions on the data that identify when the ''ordinary'' cubic interpolant becomes a PH curve are formulated.About:
This article is published in Computer Aided Geometric Design.The article was published on 2008-05-01. It has received 70 citations till now. The article focuses on the topics: Quintic function & Hermite interpolation.read more
Citations
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Journal ArticleDOI
Theory of Equations. By J. V. Uspensky. Pp. vii, 353. 27s. 1948. (McGraw-Hill)
Journal ArticleDOI
Cooperative Trajectory Generation Using Pythagorean Hodograph Bézier Curves
TL;DR: In this paper, a cooperative trajectory generation framework was proposed to efficiently compute a set of spatial trajectories for a team of cooperating vehicles, in particular for UAVs. But this method does not guarantee spatial deconfliction between the trajectories.
Journal ArticleDOI
Design of rational rotation–minimizing rigid body motions by Hermite interpolation
TL;DR: Modulation of the hodograph by a scalar polynomial is proposed as a means of introducing additional degrees of freedom, in cases where solutions to the end–point interpolation problem are not found.
Journal ArticleDOI
Construction of G1 planar Hermite interpolants with prescribed arc lengths
TL;DR: The problem of constructing a plane polynomial curve with given end points and end tangents, and a specified arc length, and an algorithm to construct interpolants to planar G1 Hermite data, with exact prescribed arc lengths is presented.
Journal ArticleDOI
Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves
TL;DR: Simple criteria for the existence of rational rotation-minimizing frames (RRMFs) on quintic space curves are determined, in terms of both the quaternion and Hopf map representations for Pythagorean-hodograph curves in ℝ3, which should help facilitate the development of algorithms for their construction, analysis, and practical use in applications such as animation, spatial motion planning, and swept surface constructions.
References
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Book
Theory of Equations.
Abstract: If f(x) represents the function a1x n + a2x n −1 + … + an + 1 and f(a) denotes the value of the function when x is replaced by the value a, then if
$$\matrix{ {f\left( x \right) = 2{x^3} - 3{x^2} + 5x - 6} \cr {f\left( 1 \right) = 2 - 3 + 5 - 6 = - 2} \cr {f\left( { - 1} \right) = - 2 - 3 - 5 - 6 = - 16} \cr } $$
and
$$f\left( {2.5} \right) = \ldots \ldots \ldots .$$
for
$$\matrix{ {f\left( {2.5} \right) = 2{{\left( {2.5} \right)}^3} - 3{{\left( {2.5} \right)}^2} + 5\left( {2.5} \right) - 6} \cr { = 31.25 - 18.75 + 12.5 - 6 = \underline {19} } \cr } $$
Book
Handbook of Computer Aided Geometric Design
TL;DR: This book is meant to be a reference text for researchers in the field as well as an introduction to graduate students wishing to get some exposure to this subject.
Journal ArticleDOI
Theory of Equations. By J. V. Uspensky. Pp. vii, 353. 27s. 1948. (McGraw-Hill)
Journal ArticleDOI
Hermite interpolation by Pythagorean hodograph quintics
Rida T. Farouki,C. A. Neff +1 more
TL;DR: In this article, the Pythagorean hodograph (PH) curves are formulated as complex-valued functions of a real parameter and a compact Hermite interpolation algorithm is proposed to identify the good interpolant.
Journal ArticleDOI
The conformal map z → z 2 of the hodograph plane
TL;DR: A simple conformal map of the hodograph plane serves to elucidate the relationship between Pythagorean-hodograph curves and polynomial curves in general, and relationships concerning the tangents, curvatures, inflections, arc lenghts, and other intrinsic properties of corresponding curves are identified.