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Rational rotation-minimizing frames on polynomial space curves: recent advances and open problems Farouki, Rida
Description
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 reviewed—including construction of swept surfaces, rigid-body motion planning, 5-axis CNC machining, and camera orientation control.
Item Metadata
Title |
Rational rotation-minimizing frames on polynomial space curves: recent advances and open problems
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2016-08-10T16:15
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Description |
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 reviewed—including construction
of swept surfaces, rigid-body motion planning, 5-axis CNC machining,
and camera orientation control.
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Extent |
42 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: UC Davis
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Series | |
Date Available |
2017-02-09
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0342701
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International