Rational frames of minimal twist along space curves under specified boundary conditions
type: Item
instance of: scholarly article
title: Rational frames of minimal twist along space curves under specified boundary conditions
zbMATH Open document ID: 1419.14046
author: Rida T. Farouki
author: Hwan Pyo Moon
published in: Advances in Computational Mathematics
publication date: 2018-12-18T00:00:00Z
full work available at URL: 4435w73c
review text: To describe the spacial motion of a rigid body, one must specify the variation of its position and orientation with respect to time. The path of a point is employed to specify the variation of position as a parametric curve, to describe the variation of orientation one can use an orthonormal frame embedded within the body. In many contexts, a frame whose first component is the curve tangent and the second and third components span the curve normal plane at each point is desirable. An example is the Frenet frame and the rotation minimizing adapted frame called the Bishop frame, useful in computer animation, robotics, spatial motion planning, the construction of swept surfaces, and related applications. In this article, the authors address the problem of constructing an adapted orthonormal frame on a space curve that exhibits minimal ``twisting'' between prescribed initial and final instances. Their construction uses the Euler-Rodrigues frame (ERF) on spatial Pythagorean-hodograph (PH) curves. One of the advantages of this initial frame is that it is inherently rational and non-singular. In order to ensure monotonicity of the rational minimal twist frame, or of further suppressing variations of the tangent angular velocity component, the authors propose a strategy of subdividing the curve at the points that correspond to inflections and/or extrema of the ERF tangent angular velocity component. The article includes an algorithm that yields (rational) solutions to the problem of smoothly varying the orientation of a rigid body along a given spatial path, compatible with prescribed initial and final orientations.
reviewed by: Nelly Villamizar
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 68U05
Mathematics Subject Classification ID: 68U07
Mathematics Subject Classification ID: 65D18
zbMATH DE Number: 6995454
zbMATH Keywords: twist
zbMATH Keywords: space curves
zbMATH Keywords: Pythagorean-hodograph curves
zbMATH Keywords: Euler-Rodrigues frames
zbMATH Keywords: adapted orthonormal frames
zbMATH Keywords: angular velocity
zbMATH Keywords: minimal twist frames
describes a project that uses: BRENT
MaRDI profile type: Publication
OpenAlex ID: W2790892740
cites work: There is More than One Way to Frame a Curve
cites work: Q5657612
cites work: Euler-Rodrigues frames on spatial Pythagorean-hodograph curves.
cites work: Geometry of Călugăreanu's theorem
cites work: Q4256595
cites work: Pythagorean-hodograph curves. Algebra and geometry inseparable
cites work: Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves
cites work: Rational rotation-minimizing frames -- recent advances and open problems
cites work: A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames
cites work: Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures
cites work: Design of rational rotation–minimizing rigid body motions by Hermite interpolation
cites work: Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves
cites work: Rotation-minimizing Euler-Rodrigues rigid-body motion interpolants
cites work: Algorithms for polynomials in Bernstein form
cites work: Rational rotation-minimizing frames on polynomial space curves of arbitrary degree
cites work: A complete classification of quintic space curves with rational rotation-minimizing frames
cites work: Q4251366
cites work: Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling.
cites work: Conformational analysis of stiff chiral polymers with end-constraints
cites work: Two moving coordinate frames for sweeping along a 3D trajectory
cites work: Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves
cites work: Q3260965
cites work: Q4948049
cites work: Mathematics of Surfaces XI
cites work: Q4779819
Wikidata QID: Q130157856
DOI: 10.1007/S10444-018-9599-3
Recommended article: Rational minimal-twist motions on curves with rotation-minimizing Euler-Rodrigues frames
Recommended article: On new types of rational rotation-minimizing frame space curves
Recommended article: Construction of Rational Curves with Rational Rotation-Minimizing Frames via Möbius Transformations
Recommended article: Rational rotation-minimizing frames on polynomial space curves of arbitrary degree
Recommended article: Quintic space curves with rational rotation-minimizing frames
Recommended article: A complete classification of quintic space curves with rational rotation-minimizing frames
Recommended article: Mapping rational rotation-minimizing frames from polynomial curves on to rational curves
Recommended article: Rotation minimizing frames and quaternionic rectifying curves
Recommended article: A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames
Recommended article: Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves
description: scientific article; zbMATH DE number 6995454