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Motion design with Euler–Rodrigues frames of quintic Pythagorean-hodograph curves
2012
Mathematics and Computers in Simulation
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., interpolation of data points and rotations at the points, with spatial quintic Pythagorean-hodograph curves so that the Euler-Rodrigues frame of the curve coincides with the rotations at the points. ...
The approximation error is of order four and the Euler-Rodrigues frame differs from the ideal rotation minimizing frame with the order three. ...
There exists a special adapted frame defined particularly on spatial PH curves, so called Euler-Rodrigues frame (ERF frame) that has been introduced in [1] . ...
doi:10.1016/j.matcom.2012.04.003
fatcat:3iwejsdntvdphkjnlt3z3qrsoa
Rational rotation-minimizing frames—Recent advances and open problems
2016
Applied Mathematics and Computation
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The simplest non-trivial RRMF curves are the quintics, characterized by a scalar condition on the angular velocity of the Euler-Rodrigues frame (ERF). ...
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. ...
Spatial Pythagorean-hodograph curves We focus here on polynomial PH curves. ...
doi:10.1016/j.amc.2015.04.122
fatcat:jjvw3jg67zfshieq46h2kdikhy
Parametric curves with Pythagorean binormal
2014
Advances in Computational Mathematics
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The PB curve construction proposed is based upon the dual curve representation and the Euler-Rodrigues frame obtained from quaternion polynomials. ...
The construction significantly simplifies if the curve is a polynomial one. ...
In particular, the Euler-Rodrigues frame (E-R frame) that has been introduced in [1] can be assigned as a conformal frame to the derived PB curve. ...
doi:10.1007/s10444-014-9387-7
fatcat:xfoza35surgv3lmcz25rb6ez2i
Rational frames of minimal twist along space curves under specified boundary conditions
2018
Advances in Computational Mathematics
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Employing the Euler-Rodrigues frame (ERF) -a rational adapted frame defined on spatial Pythagorean-hodograph curves -as an intermediary, an exact expression for an MTF with Ω 1 = constant is derived. ...
An adapted orthonormal frame (f 1 (ξ), f 2 (ξ), f 3 (ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent f 1 (ξ) = r ′ (ξ)/|r ′ (ξ)| and two unit vectors f 2 (ξ), f 3 (ξ) that span the ...
The Euler-Rodrigues frame (ERF) on spatial Pythagorean-hodograph (PH) curves offers a convenient point of departure for such constructions, since it is inherently rational and non-singular. ...
doi:10.1007/s10444-018-9599-3
fatcat:gnujsuy6kzct7ax5slndadiduu
Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces
[chapter]
2005
Lecture Notes in Computer Science
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Finally we exploit the rational frames associated with any space PH curve (Euler-Rodrigues frame) in order to obtain a simple rational approximation of pipe surfaces with a piecewise analytical spine curve ...
As observed by Farouki et al. [9], any set of C 1 space boundary data (two points with associated first derivatives) can be interpolated by a Pythagorean hodograph (PH) curve of degree 5. ...
Approximation of pipe surfaces PH curves possess a simple low degree rational adapted frame, which has been called the Euler-Rodrigues frame in [3] . ...
doi:10.1007/11537908_22
fatcat:nbhodtgxp5b57oe2ji3uf6l5ka
C1 and C2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames
2018
Computer Aided Geometric Design
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The Euler-Rodrigues frame (ERF) is a rational adapted frame defined on any spatial Pythagorean-hodograph curve [2] . ...
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. ...
Figure 7 : 7 (a) The Euler-Rodrigues Frame of the spatial quintic PH curve. ...
doi:10.1016/j.cagd.2018.07.005
fatcat:aeietmjwonhy5pz7foz4rut7p4
Construction of Rational Curves with Rational Rotation-Minimizing Frames via Möbius Transformations
[chapter]
2010
Lecture Notes in Computer Science
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Based on these observations we derive an algorithm for G 1 Hermite interpolation by rational Pythagorean-hodograph curves with rational rotation-minimizing frames. ...
In addition, these transformations are known to preserve the class of rational Pythagorean-hodograph curves and rational frames. ...
These curves are equipped with rational frames [14] , which are called the Euler-Rodrigues frames [5, 6] . ...
doi:10.1007/978-3-642-11620-9_2
fatcat:brc5u6ilorg73l5432ivdr46dq
Quintic space curves with rational rotation-minimizing frames
2009
Computer Aided Geometric Design
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The existence of polynomial space curves with rational rotation-minimizing frames (RRMF curves) is investigated, using the Hopf map representation for PH space curves in terms of complex polynomials α( ...
β(t) that are sufficient and necessary for any PH quintic to admit a rational rotationminimizing frame. ...
Choi and Han (2002) observed that the spatial PH curves always admit a rational adapted frame, the so-called Euler-Rodrigues frame (ERF) . ...
doi:10.1016/j.cagd.2009.01.005
fatcat:6upt7dzz2bgbtgbael7gxzkzma
A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames
2016
Advances in Computational Mathematics
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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 ...
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 ...
The Euler-Rodrigues frame (ERF) is a rational orthonormal frame for R 3 , defined [4] on any spatial PH curve by (e 1 (ξ), e 2 (ξ), e 3 (ξ)) = (A(ξ) i A * (ξ), A(ξ) j A * (ξ), A(ξ) k A * (ξ)) |A(ξ)| ...
doi:10.1007/s10444-016-9473-0
fatcat:r7f6rk46dnhxnbjpg6t4uawrvq
Rational Framing Motions and Spatial Rational Pythagorean Hodograph Curves
[article]
2021
arXiv
pre-print
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We propose a new method for constructing rational spatial Pythagorean Hodograph (PH) curves based on determining a suitable rational framing motion. ...
While the spherical component of the framing motion is arbitrary, the translation part is determined be a modestly sized and nicely structured system of linear equations. ...
The spherical component has been called Euler-Rodrigues motion or, equivalently referring to the images of a suitable orthogonal tripod, Euler-Rodrigues frame [4, 21] . ...
arXiv:2111.04600v1
fatcat:oiogpmksbvgo5fwt5warrkaida
Rational rotation-minimizing frames on polynomial space curves of arbitrary degree
2010
Journal of symbolic computation
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Polynomial curves with rational rotation-minimizing frames (RRMF curves) are necessarily Pythagorean-hodograph (PH) curves -since only the PH curves possess rational unit tangents -and they may be characterized ...
A rotation-minimizing adapted frame on a space curve r(t) is an orthonormal basis (f 1 , f 2 , f 3 ) for R 3 such that f 1 is coincident with the curve tangent t = r /|r | at each point and the normal-plane ...
As an alternative to invoking the Frenet frame as a reference, Choi and Han (2002) defined an adapted frame on spatial PH curves called the Euler-Rodrigues frame (ERF). ...
doi:10.1016/j.jsc.2010.03.004
fatcat:qmzns23qsbfcjowrarslijyrim
Rotation-minimizing Euler-Rodrigues rigid-body motion interpolants
2013
Computer Aided Geometric Design
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A characterization for spatial Pythagorean-hodograph (PH) curves of degree 7 with rotation-minimizing Euler-Rodrigues frames (ERFs) is determined, in terms of one real and two complex constraints on the ...
curve coefficients. ...
The Euler-Rodrigues frame (ERF) is a rational adapted rational frame, defined on spatial PH curves [2] . ...
doi:10.1016/j.cagd.2013.03.001
fatcat:glth6xu3yjgpncq7hfdeenhw44
Page 6909 of Mathematical Reviews Vol. , Issue 2003i
[page]
2003
Mathematical Reviews
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frames on spatial Pythagorean-hodograph curves. ...
The curves in question are special polynomial curves in 3-space useful in the creation of ruled surfaces swept out by an axis of a frame moving along a curve. ...
Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves
2009
Advances in Computational Mathematics
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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 ...
The identification of these constraints is based on introducing a "canonical form" for spatial PH curves and judicious transformations between the quaternion and Hopf map descriptions. ...
Choi and Han [2] defined a rational adapted frame on spatial PH curves, the so-called Euler-Rodrigues frame (ERF). ...
doi:10.1007/s10444-009-9138-3
fatcat:uxf4lpmapbdpfieaxbb6x4aa2y
Rational minimal-twist motions on curves with rotation-minimizing Euler–Rodrigues frames
2019
Journal of Computational and Applied Mathematics
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We consider construction of curves with rational minimal twist frames, based on the Pythagoreanhodograph curves of degree 7 that have rational rotation-minimizing Euler-Rodrigues frames (e 1 (ξ), e 2 ( ...
A minimal twist frame (f 1 (ξ), f 2 (ξ), f 3 (ξ)) on a polynomial space curve r(ξ), ξ ∈ [ 0, 1 ] is an orthonormal frame, where f 1 (ξ) is the tangent and the normal-plane vectors f 2 (ξ), f 3 (ξ) have ...
The Euler-Rodrigues frame (ERF) is a rational adapted frame defined on spatial PH curves [5] , that serves as a "reference frame" for the identification of other rational adapted frames on PH curves. ...
doi:10.1016/j.cam.2018.12.012
fatcat:nfvt64cvbbgulliunmzgbxywzu
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