A Note on Robust Biarc Computation

A new robust algorithm for the numerical computation of biarcs, i.e. $G^1$ curves composed of two arcs of circle, is presented. Many algorithms exist but are based on geometric constructions, which must consider many geometrical configurations. The proposed algorithm uses an algebraic construction which is reduced to the solution of a single $2$ by $2$ linear system. Singular angles configurations are treated smoothly by using the pseudoinverse matrix when solving the linear system. The proposed algorithm is compared with the Matlab's routine \texttt{rscvn} that solves geometrically the same problem. Numerical experiments show that Matlab's routine sometimes fails near singular configurations and does not select the correct solution for large angles, whereas the proposed algorithm always returns the correct solution. The proposed solution smoothly depends on the geometrical parameters so that it can be easily included in more complex algorithms like splines of biarcs or least squares data fitting.

[1]  Les A. Piegl,et al.  Data Approximation Using Biarcs , 2002, Engineering with Computers.

[2]  G. Stewart.  On the Continuity of the Generalized Inverse , 1969 .

[3]  Hyungjun Park.  Optimal Single Biarc Fitting and its Applications , 2004 .

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

[5]  Masaaki Sibuya,et al.  Numerical algorithms for the moore-penrose inverse of a matrix: Iterative methods , 1972 .

[6]  Josef Stoer,et al.  Curve Fitting With Clothoidal Splines. , 1982, Journal of research of the National Bureau of Standards.

[7]  Bahattin Koc,et al.  Max-fit biarc fitting to STL models for rapid prototyping processes , 2001, SMA '01.

[8]  Weiyin Ma,et al.  Matching admissible G2 Hermite data by a biarc-based subdivision scheme , 2012, Comput. Aided Geom. Des..

[9]  D. Walton,et al.  G 1 interpolation with a single Cornu spiral segment , 2008 .

[10]  Enrico Bertolazzi,et al.  On the G2 Hermite Interpolation Problem with clothoids , 2018, J. Comput. Appl. Math..

[11]  Weiyin Ma,et al.  A biarc based subdivision scheme for space curve interpolation , 2014, Comput. Aided Geom. Des..

[12]  E. Bertolazzi,et al.  Interpolating clothoid splines with curvature continuity , 2018 .

[13]  Jianmin Zheng.  C1 NURBS representations of G1 composite rational Bézier curves , 2009, Computing.

[14]  Les A. Piegl,et al.  Biarc approximation of NURBS curves , 2002, Comput. Aided Des..

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

[16]  Zezhong C. Chen,et al.  A practicable approach to G1 biarc approximations for making accurate, smooth and non-gouged profile features in CNC contouring , 2006, Comput. Aided Des..

[17]  Biarcs and bilens , 2013, Comput. Aided Geom. Des..

[18]  Georg Maier.  Optimal arc spline approximation , 2014, Comput. Aided Geom. Des..

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

[20]  Rida T. Farouki,et al.  Algorithm 952 , 2015 .

[21]  Guozhao Wang,et al.  Planar point set fairing and fitting by arc splines , 2001, Comput. Aided Des..

[22]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[23]  D. Walton,et al.  Approximating smooth planar curves by arc splines , 1995 .

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

[25]  Bert Jüttler,et al.  Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics , 2006, Comput. Aided Des..

[26]  Dereck S. Meek,et al.  The family of biarcs that matches planar, two-point G1 Hermite data , 2008 .

[27]  Smita Narayan,et al.  Approximating Cornu spirals by arc splines , 2014, J. Comput. Appl. Math..

[28]  Sanjeev Kumar,et al.  Reconstruction of free-form space curves using NURBS-snakes and a quadratic programming approach , 2015, Comput. Aided Geom. Des..

[29]  Emil Zagar,et al.  Curvature approximation of circular arcs by low-degree parametric polynomials , 2016, J. Num. Math..

[30]  Gershon Elber,et al.  Efficient offset trimming for planar rational curves using biarc trees , 2012, Comput. Aided Geom. Des..

[31]  Vito Vitrih,et al.  Pythagorean-hodograph cycloidal curves , 2015, J. Num. Math..