Reducibility of offsets to algebraic curves

Computing offset curves and surfaces is a fundamental operation in many technical applications. This paper discusses some issues that are encountered during the process of designing offsets, especially the problems of their reducibility and rationality (which are closely related). This study is crucial especially for formulating subsequent algorithms when the number and quality of offset components must be revealed. We will formulate new algebraic and geometric conditions on reducibility of offsets and demonstrate how they can be applied. In addition, we will present that our investigations can also serve to better understand the varieties fulfilling the Pythagorean conditions (PH curves/PN surfaces). A certain analogy of the PH condition for parameterized curves (or general parameterized hypersurfaces) will be presented also for implicitly given (not necessarily rational) curves (or hypersurfaces).

[1]  J. Rafael Sendra,et al.  Genus formula for generalized offset curves , 1999 .

[2]  R.T. Farouki,et al.  The approximation of non-degenerate offset surfaces , 1986, Comput. Aided Geom. Des..

[3]  Binh Pham,et al.  Offset curves and surfaces: a brief survey , 1992, Comput. Aided Des..

[4]  Miroslav Lávicka,et al.  Algebraic Curves of Low Convolution Degree , 2010, Curves and Surfaces.

[5]  W. Fulton.  Algebraic curves , 1969 .

[6]  Bert Jüttler,et al.  Hermite interpolation by piecewise polynomial surfaces with rational offsets , 2000, Comput. Aided Geom. Des..

[7]  Jan Vrek,et al.  On convolutions of algebraic curves , 2010 .

[8]  Bert Jüttler,et al.  Computing exact rational offsets of quadratic triangular Bézier surface patches , 2008, Comput. Aided Des..

[9]  Takis Sakkalis,et al.  Pythagorean-hodograph space curves , 1994, Adv. Comput. Math..

[10]  Helmut Pottmann,et al.  A Laguerre geometric approach to rational offsets , 1998, Comput. Aided Geom. Des..

[11]  Takashi Maekawa,et al.  An overview of offset curves and surfaces , 1999, Comput. Aided Des..

[12]  T. Sakkalis,et al.  Pythagorean hodographs , 1990 .

[13]  Gershon Elber,et al.  Comparing Offset Curve Approximation Methods , 1997, IEEE Computer Graphics and Applications.

[14]  Rida T. Farouki,et al.  Rational space curves are not "unit speed" , 2007, Comput. Aided Geom. Des..

[15]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[16]  Bohumír Bastl,et al.  Rational hypersurfaces with rational convolutions , 2007, Comput. Aided Geom. Des..

[17]  J. Sendra.  Algebraic analysis of offsets to hypersurfaces , 2000 .

[18]  Zbynek Sír,et al.  Rational Pythagorean-hodograph space curves , 2011, Comput. Aided Geom. Des..

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

[20]  Bert Jüttler,et al.  Rational surfaces with linear normals and their convolutions with rational surfaces , 2006, Comput. Aided Geom. Des..

[21]  J. Stillwell,et al.  Plane Algebraic Curves , 1986 .

[22]  J. Rafael Sendra,et al.  Rationality Analysis and Direct Parametrization of Generalized Offsets to Quadrics , 2000, Applicable Algebra in Engineering, Communication and Computing.

[23]  Helmut Pottmann,et al.  Rational curves and surfaces with rational offsets , 1995, Comput. Aided Geom. Des..

[24]  Martin Peternell,et al.  The Convolution of a Paraboloid and a Parametrized Surface , 2003 .

[25]  J. Rafael Sendra,et al.  Parametric Generalized Offsets to Hypersurfaces , 1997, J. Symb. Comput..

[26]  Bohumír Bastl,et al.  PN surfaces and their convolutions with rational surfaces , 2008, Comput. Aided Geom. Des..

[27]  H. M. Reimann.  Rigidity of H–type groups , 2001 .

[28]  Bert Jüttler,et al.  C HERMITE INTERPOLATION BY PYTHAGOREAN HODOGRAPH SPACE CURVES , 2007 .

[29]  Wei Lü,et al.  Rational parameterization of quadrics and their offsets , 1996, Computing.

[30]  Gershon Elber,et al.  Offsets, sweeps, and Minkowski sums , 1999, Comput. Aided Des..

[31]  Bert Jüttler,et al.  Euclidean and Minkowski Pythagorean hodograph curves over planar cubics , 2005, Comput. Aided Geom. Des..

[32]  Helmut Pottmann,et al.  Polynomial and Rational Pythagorean-Hodograph Curves Reconciled , 1994, IMA Conference on the Mathematics of Surfaces.

[33]  Zbynek Sír,et al.  Reparameterization of Curves and Surfaces with Respect to Their Convolution , 2008, MMCS.

[34]  Bert Jüttler,et al.  Volumes with piecewise quadratic medial surface transforms: Computation of boundaries and trimmed offsets , 2010, Comput. Aided Des..