Spiral fat arcs - Bounding regions with cubic convergence

A bounding region for spiral curve segments shaped by two circular arcs, parts of the osculating circles at the spiral's endpoints, and two lines is introduced. This bounding region, denoted spiral fat arc (SFA) is simple to construct and process, and shows a cubic approximation order to a given spiral curve. Given a general planar parametric curve, it can be split at curvature extrema (and inflection points), solving for the parametric locations for which @k'=0 (and @k=0), @k being the signed curvature field, to yield a set of spiral curves. Each of the spirals is then fitted with a bounding SFA. Finding the intersection locations of two free-form planar curves is a fundamental task in geometric computing and computer aided design, and can immediately benefit from this new SFA bounding region. A recursive curve-curve intersection (CCI) algorithm that efficiently computes the intersection location of two parametric curves using SFAs is also introduced.

[1]  Norimasa Yoshida,et al.  Log-aesthetic space curve segments , 2009, Symposium on Solid and Physical Modeling.

[2]  A. Kurnosenko.  Applying inversion to construct rational spiral curves , 2009 .

[3]  Fujio Yamaguchi,et al.  Computer-Aided Geometric Design , 2002, Springer Japan.

[4]  Yves Mineur,et al.  A shape controled fitting method for Bézier curves , 1998, Comput. Aided Geom. Des..

[5]  Bruce R. Piper,et al.  Rational cubic spirals , 2008, Comput. Aided Des..

[6]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[7]  Tomoyuki Nishita,et al.  Curve intersection using Bézier clipping , 1990, Comput. Aided Des..

[8]  Dereck S. Meek,et al.  Planar spirals that match G2 Hermite data , 1998, Comput. Aided Geom. Des..

[9]  Zhiyang Yao,et al.  A Novel Cutter Path Planning Approach to High Speed Machining , 2006 .

[10]  Carlo H. Séquin,et al.  Interpolating Splines: Which is the fairest of them all? , 2009 .

[11]  Gershon Elber,et al.  Symbolic and Numeric Computation in Curve Interrogation , 1995, Comput. Graph. Forum.

[12]  Thomas W. Sederberg,et al.  Fat arcs: A bounding region with cubic convergence , 1989, Comput. Aided Geom. Des..

[13]  A. I. Kurnosenko.  Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data , 2010, Comput. Aided Geom. Des..

[14]  Gerald E. Farin,et al.  Class A Bézier curves , 2006, Comput. Aided Geom. Des..

[15]  Raphael L. Levien,et al.  From Spiral to Spline: Optimal Techniques in Interactive Curve Design , 2009 .

[16]  M. Barton,et al.  ROOTS OF SYSTEMS OF POLYNOMIALS BY LINEAR CLIPPING , 2007 .

[17]  Gershon Elber,et al.  Subdivision termination criteria in subdivision multivariate solvers using dual hyperplanes representations , 2007, Comput. Aided Des..

[18]  Michael Bartoň.  Quadratic Clipping and its Generalization for Polynomial Systems , 2007 .

[19]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[20]  Christian Schulz.  Bézier clipping is quadratically convergent , 2009, Comput. Aided Geom. Des..

[21]  Dereck S. Meek,et al.  G2 curve design with a pair of Pythagorean Hodograph quintic spiral segments , 2007, Comput. Aided Geom. Des..

[22]  Ayellet Tal,et al.  3D Euler spirals for 3D curve completion , 2010, Comput. Geom..

[23]  Thomas W. Sederberg,et al.  Loop detection in surface patch intersections , 1988, Comput. Aided Geom. Des..

[24]  Bert Jüttler,et al.  A Quadratic Clipping Step with Superquadratic Convergence for Bivariate Polynomial Systems , 2011, Math. Comput. Sci..

[25]  Bert Jüttler,et al.  Fat Arcs for Implicitly Defined Curves , 2008, MMCS.