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

Marjeta Krajnc, Vito Vitrih
2012 Mathematics and Computers in Simulation
., 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] .  ...

### Rational rotation-minimizing frames—Recent advances and open problems

Rida T. Farouki
2016 Applied Mathematics and Computation
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.  ...

### Parametric curves with Pythagorean binormal

Jernej Kozak, Marjeta Krajnc, Vito Vitrih
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.  ...

### Rational frames of minimal twist along space curves under specified boundary conditions

Rida T. Farouki, Hwan Pyo Moon
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.  ...

### Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces [chapter]

Zbyněk Šír, Bert Jüttler
2005 Lecture Notes in Computer Science
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] .  ...

### C1 and C2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

Hwan Pyo Moon, Rida T. Farouki
2018 Computer Aided Geometric Design
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.  ...

### Construction of Rational Curves with Rational Rotation-Minimizing Frames via Möbius Transformations [chapter]

Michael Bartoň, Bert Jüttler, Wenping Wang
2010 Lecture Notes in Computer Science
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] .  ...

### Quintic space curves with rational rotation-minimizing frames

Rida T. Farouki, Carlotta Giannelli, Carla Manni, Alessandra Sestini
2009 Computer Aided Geometric Design
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) .  ...

### A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

Rida T. Farouki, Graziano Gentili, Carlotta Giannelli, Alessandra Sestini, Caterina Stoppato
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)|  ...

### Rational Framing Motions and Spatial Rational Pythagorean Hodograph Curves [article]

Bahar Kalkan, Daniel F. Scharler, Hans-Peter Schröcker, Zbyněk Šír
2021 arXiv   pre-print
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] .  ...

### Rational rotation-minimizing frames on polynomial space curves of arbitrary degree

Rida T. Farouki, Takis Sakkalis
2010 Journal of symbolic computation
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).  ...

### Rotation-minimizing Euler-Rodrigues rigid-body motion interpolants

Rida T. Farouki, Chang Yong Han, Petroula Dospra, Takis Sakkalis
2013 Computer Aided Geometric Design
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] .  ...

### 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.  ...

Rida T. Farouki