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Motion design with Euler–Rodrigues frames of quintic Pythagorean-hodograph curves

2012
*
Mathematics and Computers in Simulation
*

., interpolation of data points and rotations at the points, with

doi:10.1016/j.matcom.2012.04.003
fatcat:3iwejsdntvdphkjnlt3z3qrsoa
*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] . ...##
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Rational rotation-minimizing frames—Recent advances and open problems

2016
*
Applied Mathematics and Computation
*

The simplest non-trivial RRMF

doi:10.1016/j.amc.2015.04.122
fatcat:jjvw3jg67zfshieq46h2kdikhy
*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*. ...##
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Parametric curves with Pythagorean binormal

2014
*
Advances in Computational Mathematics
*

The PB

doi:10.1007/s10444-014-9387-7
fatcat:xfoza35surgv3lmcz25rb6ez2i
*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*. ...##
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Rational frames of minimal twist along space curves under specified boundary conditions

2018
*
Advances in Computational Mathematics
*

Employing the

doi:10.1007/s10444-018-9599-3
fatcat:gnujsuy6kzct7ax5slndadiduu
*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. ...##
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Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces
[chapter]

2005
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Lecture Notes in Computer Science
*

Finally we exploit the rational

doi:10.1007/11537908_22
fatcat:nbhodtgxp5b57oe2ji3uf6l5ka
*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] . ...##
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C1 and C2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

2018
*
Computer Aided Geometric Design
*

The

doi:10.1016/j.cagd.2018.07.005
fatcat:aeietmjwonhy5pz7foz4rut7p4
*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*. ...##
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Construction of Rational Curves with Rational Rotation-Minimizing Frames via Möbius Transformations
[chapter]

2010
*
Lecture Notes in Computer Science
*

Based

doi:10.1007/978-3-642-11620-9_2
fatcat:brc5u6ilorg73l5432ivdr46dq
*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] . ...##
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Quintic space curves with rational rotation-minimizing frames

2009
*
Computer Aided Geometric Design
*

The existence of polynomial space

doi:10.1016/j.cagd.2009.01.005
fatcat:6upt7dzz2bgbtgbael7gxzkzma
*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) . ...##
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A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

2016
*
Advances in Computational Mathematics
*

Polynomial

doi:10.1007/s10444-016-9473-0
fatcat:r7f6rk46dnhxnbjpg6t4uawrvq
*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)| ...##
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Rational Framing Motions and Spatial Rational Pythagorean Hodograph Curves
[article]

2021
*
arXiv
*
pre-print

We propose a new method for constructing rational

arXiv:2111.04600v1
fatcat:oiogpmksbvgo5fwt5warrkaida
*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] . ...##
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Rational rotation-minimizing frames on polynomial space curves of arbitrary degree

2010
*
Journal of symbolic computation
*

Polynomial

doi:10.1016/j.jsc.2010.03.004
fatcat:qmzns23qsbfcjowrarslijyrim
*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). ...##
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Rotation-minimizing Euler-Rodrigues rigid-body motion interpolants

2013
*
Computer Aided Geometric Design
*

A characterization for

doi:10.1016/j.cagd.2013.03.001
fatcat:glth6xu3yjgpncq7hfdeenhw44
*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] . ...##
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Page 6909 of Mathematical Reviews Vol. , Issue 2003i
[page]

2003
*
Mathematical Reviews
*

*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*. ...

##
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Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves

2009
*
Advances in Computational Mathematics
*

Simple criteria for the existence of rational rotation-minimizing

doi:10.1007/s10444-009-9138-3
fatcat:uxf4lpmapbdpfieaxbb6x4aa2y
*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). ...##
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Rational minimal-twist motions on curves with rotation-minimizing Euler–Rodrigues frames

2019
*
Journal of Computational and Applied Mathematics
*

We consider construction of

doi:10.1016/j.cam.2018.12.012
fatcat:nfvt64cvbbgulliunmzgbxywzu
*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*. ...
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