Rotation – minimizing conformal frames

An orthonormal frame (f1, f2, f3) is rotation–minimizing with respect to fi if its angular velocity ω satisfies ω · fi ≡ 0 or, equivalently, the derivatives of fj , fk are both parallel to fi. The Frenet frame (t,p,b) is rotation–minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation–minimizing with respect to the tangent t have attracted much interest. This study is concerned with conformal frames, that are rotation–minimizing with respect to the binormal b along a space curve. Such a frame (f ,g,b) incorporates osculating–plane vectors f ,g that have no rotation about b, and may be defined through a rotation of t,p by an amount equal to minus the integral of curvature with respect to arc length. In aeronautical terms, a rotation–minimizing conformal frame (RMCF) specifies “yaw–free” rigid–body motion on a curved path. The existence of rational RMCFs on polynomial space curves with rational Frenet frames is investigated, and it is shown that they must be degree 7 at least. The RMCF is also employed to construct a novel type of ruled surface, characterized by tangent planes that coincide with the osculating planes along a given space curve, and rulings that exhibit the least possible rate of rotation consistent with this constraint.

[1]  Hyeong In Choi,et al.  Euler-Rodrigues frames on spatial Pythagorean-hodograph curves , 2002, Comput. Aided Geom. Des..

[2]  Carla Manni,et al.  Design of rational rotation-minimizing rigid body motions by Hermite interpolation , 2011, Math. Comput..

[3]  Rida T. Farouki,et al.  Spatial camera orientation control by rotation‐minimizing directed frames , 2009, Comput. Animat. Virtual Worlds.

[4]  J. Monterde,et al.  A characterization of quintic helices , 2005 .

[5]  Emin Kasap,et al.  Parametric representation of a surface pencil with a common asymptotic curve , 2012, Comput. Aided Des..

[6]  R. R. Martin,et al.  Principal Patches -a New Class of Surface Patch Based on Differential Geometry , 1983 .

[7]  Nathalie Sprynski,et al.  Surface reconstruction via geodesic interpolation , 2008, Comput. Aided Des..

[8]  H. GUGGENHEIMER.  Computing frames along a trajectory , 1989, Comput. Aided Geom. Des..

[9]  Rubén Dorado,et al.  Constrained design of polynomial surfaces from geodesic curves , 2008, Comput. Aided Des..

[10]  Rida T. Farouki,et al.  Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable , 2007, Geometry and Computing.

[11]  Carla Manni,et al.  Characterization and construction of helical polynomial space curves , 2004 .

[12]  Rida T. Farouki,et al.  Helical polynomial curves and double Pythagorean hodographs I. Quaternion and Hopf map representations , 2009, J. Symb. Comput..

[13]  R. Bishop.  There is More than One Way to Frame a Curve , 1975 .

[14]  Bahram Ravani,et al.  Curves with rational Frenet-Serret motion , 1997, Comput. Aided Geom. Des..

[15]  Bert Jüttler,et al.  Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces , 2005, IMA Conference on the Mathematics of Surfaces.

[16]  Rida T. Farouki,et al.  Construction of Bézier surface patches with Bézier curves as geodesic boundaries , 2009, Comput. Aided Des..

[17]  Hwan Pyo Moon,et al.  Clifford Algebra, Spin Representation, and Rational Parameterization of Curves and Surfaces , 2002, Adv. Comput. Math..

[18]  Michael Barton,et al.  Construction of Rational Curves with Rational Rotation-Minimizing Frames via Möbius Transformations , 2008, MMCS.

[19]  Rida T. Farouki,et al.  A complete classification of quintic space curves with rational rotation-minimizing frames , 2012, J. Symb. Comput..

[20]  Bert Jüttler,et al.  Computation of rotation minimizing frames , 2008, TOGS.

[21]  M. V. Cook.  Flight Dynamics Principles , 1997 .

[22]  Renhong Wang,et al.  Parametric representation of a surface pencil with a common line of curvature , 2011, Comput. Aided Des..

[23]  Chang Yong Han.  Nonexistence of rational rotation-minimizing frames on cubic curves , 2008, Comput. Aided Geom. Des..

[24]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[25]  Rida T. Farouki,et al.  Construction of rational surface patches bounded by lines of curvature , 2010, Comput. Aided Geom. Des..

[26]  Marco Paluszny,et al.  Cubic polynomial patches through geodesics , 2008, Comput. Aided Des..

[27]  B. Jüttler,et al.  Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics , 1999 .

[28]  JakličGašPer,et al.  C1 rational interpolation of spherical motions with rational rotation-minimizing directed frames , 2013 .

[29]  Bert Jüttler,et al.  Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling , 1999, Comput. Aided Des..

[30]  Rida T. Farouki,et al.  Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves , 2003, Comput. Aided Geom. Des..

[31]  Rida T. Farouki,et al.  An interpolation scheme for designing rational rotation-minimizing camera motions , 2013, Adv. Comput. Math..

[32]  Alessandra Sestini,et al.  C1 rational interpolation of spherical motions with rational rotation-minimizing directed frames , 2013, Comput. Aided Geom. Des..

[33]  Kai Tang,et al.  Parametric representation of a surface pencil with a common spatial geodesic , 2004, Comput. Aided Des..

[34]  Rida T. Farouki,et al.  Rational rotation-minimizing frames on polynomial space curves of arbitrary degree , 2010, J. Symb. Comput..

[35]  Fopke Klok.  Two moving coordinate frames for sweeping along a 3D trajectory , 1986, Comput. Aided Geom. Des..

[36]  Rida T. Farouki,et al.  Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves , 2009, J. Symb. Comput..

[37]  Juan Monterde,et al.  A characterization of helical polynomial curves of any degree , 2009, Adv. Comput. Math..

[38]  Rida T. Farouki,et al.  Structural invariance of spatial Pythagorean hodographs , 2002, Comput. Aided Geom. Des..

[39]  Rida T. Farouki,et al.  Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves , 2010, Adv. Comput. Math..

[40]  Carla Manni,et al.  Quintic space curves with rational rotation-minimizing frames , 2009, Comput. Aided Geom. Des..

[41]  Barry Joe,et al.  Robust computation of the rotation minimizing frame for sweep surface modeling , 1997, Comput. Aided Des..