Matching admissible G2 Hermite data by a biarc-based subdivision scheme

Spirals are curves with single-signed, monotone increasing or decreasing curvature. A spiral can only interpolate certain G^2 Hermite data that is referred to as admissible G^2 Hermite data. In this paper we propose a biarc-based subdivision scheme that can generate a planar spiral matching an arbitrary set of given admissible G^2 Hermite data, including the case that the curvature at one end is zero. An attractive property of the proposed scheme is that the resulting subdivision spirals are also offset curves if the given input data are offsets of admissible G^2 Hermite data. A detailed proof of the convergence and smoothness analysis of the scheme is also provided. Several examples are given to demonstrate some excellent properties and practical applications of the proposed scheme.

[1]  William H. Frey,et al.  Designing Bézier conic segments with monotone curvature , 2000, Comput. Aided Geom. Des..

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

[3]  Dereck S. Meek,et al.  A Pythagorean hodograph quintic spiral , 1996, Comput. Aided Des..

[4]  WangGuozhao,et al.  Incenter subdivision scheme for curve interpolation , 2010 .

[5]  Bruce R. Piper,et al.  Interpolation with cubic spirals , 2004, Comput. Aided Geom. Des..

[6]  Dereck S. Meek,et al.  An involute spiral that matches G2 Hermite data in the plane , 2009, Comput. Aided Geom. Des..

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

[8]  Neil A. Dodgson,et al.  An interpolating 4-point C2 ternary stationary subdivision scheme , 2002, Comput. Aided Geom. Des..

[9]  M. Sabin.  The use of piecewise forms for the numerical representation of shape , 1976 .

[10]  K. M. Bolton.  Biarc curves , 1975, Comput. Aided Des..

[11]  Guozhao Wang,et al.  Incenter subdivision scheme for curve interpolation , 2010, Comput. Aided Geom. Des..

[12]  W. H. Frey,et al.  1. Approximation with Aesthetic Constraints , 1994, Designing Fair Curves and Surfaces.

[13]  Nira Dyn,et al.  Interpolatory convexity-preserving subdivision schemes for curves and surfaces , 1992, Comput. Aided Des..

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

[15]  Xunnian Yang.  Normal based subdivision scheme for curve design , 2006, Comput. Aided Geom. Des..

[16]  Nira Dyn,et al.  A 4-point interpolatory subdivision scheme for curve design , 1987, Comput. Aided Geom. Des..

[17]  Zulfiqar Habib,et al.  G 2 PH QUINTIC SPIRAL TRANSITION CURVES AND THEIR APPLICATIONS , 2004 .

[18]  D. Walton,et al.  Clothoid spline transition spirals , 1992 .

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

[20]  Dereck S. Meek,et al.  A generalisation of the Pythagorean hodograph quintic spiral , 2004 .

[21]  Zulfiqar Habib,et al.  G 2 Pythagorean hodograph quintic transition between two circles , 2003 .

[22]  P. Bézier.  Numerical control : mathematics and applications , 1972 .

[23]  Pierre Vandergheynst,et al.  Non-linear subdivision using local spherical coordinates , 2003, Comput. Aided Geom. Des..

[24]  Said M. Easa,et al.  State of the Art of Highway Geometric Design Consistency , 1999 .

[25]  Rida T. Farouki,et al.  Pythagorean-hodograph quintic transition curves of monotone curvature , 1997, Comput. Aided Des..

[26]  Zulfiqar Habib,et al.  On PH quintic spirals joining two circles with one circle inside the other , 2007, Comput. Aided Des..

[27]  Larry L. Schumaker,et al.  Curve and Surface Fitting: Saint-Malo 1999 , 2000 .

[28]  Zulfiqar Habib,et al.  QUINTIC SPIRAL TRANSITION CURVES AND THEIR APPLICATIONS , 2004 .

[29]  Dereck S. Meek,et al.  Planar G 2 transition with a fair Pythagorean hodograph quintic curve , 2002 .

[30]  Nira Dyn,et al.  Geometrically Controlled 4-Point Interpolatory Schemes , 2005, Advances in Multiresolution for Geometric Modelling.

[31]  Kimon P. Valavanis,et al.  Using a biarc filter to compute curvature extremes of NURBS curves , 2009, Engineering with Computers.

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

[33]  G. D. Sandel.  Zur Geometrie der Korbbgen . , 1937 .

[34]  Nira Dyn,et al.  Geometric conditions for tangent continuity of interpolatory planar subdivision curves , 2012, Comput. Aided Geom. Des..

[35]  Shinji Mukai,et al.  Interpolating Involute Curves , 2000 .

[36]  D. Walton,et al.  Spiral arc spline approximation to a planar spiral , 1999 .