A biarc based subdivision scheme for space curve interpolation

This paper presents a biarc-based subdivision scheme for space curve interpolation. Given a sequence of space points, or a sequence of space points and tangent vectors, the scheme produces a smooth curve interpolating all input points by iteratively inserting new points and computing new tangent vectors. For each step of subdivision, the newly inserted point corresponding to an existing edge is a specified joint point of a biarc curve which interpolates the two end points and the tangents. A provisional tangent is also assigned to the newly inserted point. Each of the tangents for the following subdivision step is further updated as a linear blending of the provisional tangent and the tangent at the respective point of a circle passing through the local point and its two adjacent points. If adjacent four initial points and their initial reference tangent vectors are sampled from a spherical surface, the limit curve segment between the middle two initial points exactly lies on the same spherical surface. The biarc based subdivision scheme is proved to be G 1 continuous with a nice convexity preserving property. Numerical examples also show that the limit curves are G 2 continuous and fair. Several examples are given to demonstrate the excellent properties of the scheme. This article presents a biarc based interpolatory subdivision scheme.The scheme produces convexity preserving spatial limit curves in 3D.The limit curves are proved to be G 1 continuous, while numerical examples show that they are also G 2 smooth and fair.The scheme reproduces circular arcs and spherical curves with control data sampled on a circle and a sphere, respectively.

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

[2]  Josef Hoschek.  Circular splines , 1992, Comput. Aided Des..

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

[4]  T. J. Sharrock.  Biarcs in three dimensions , 1987 .

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

[6]  Tim N. T. Goodman,et al.  Shape preserving interpolation by space curves , 1997, Comput. Aided Geom. Des..

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

[8]  Panagiotis D. Kaklis,et al.  Shape-preserving interpolation in R3 , 1997 .

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

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

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

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

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

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

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

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

[17]  Neil A. Dodgson,et al.  A Circle-Preserving Variant of the Four-Point Subdivision Scheme , 2012 .

[18]  Francesca Pelosi,et al.  Shape-Preserving Approximation by Space Curves , 2001, Numerical Algorithms.

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