Geometric Hermite interpolation by logarithmic arc splines

This paper considers the problem of G 1 curve interpolation using a special type of discrete logarithmic spirals. A "logarithmic arc spline" is defined as a set of smoothly connected circular arcs. The arcs of a logarithmic arc spline have equal angles and the curvatures of the arcs form a geometric sequence. Given two points together with two unit tangents at the points, interpolation of logarithmic arc splines with a user specified winding angle is formulated into finding the positive solutions to a vector equation. A practical algorithm is developed for computing the solutions and construction of interpolating logarithmic arc splines. Compared to known methods for logarithmic spiral interpolation, the proposed method has the advantages of unbounded winding angles, simple offsets and NURBS representation. A special kind of discrete logarithmic spiral, logarithmic arc spline, is defined.A practical algorithm for G 1 curve interpolation by logarithmic arc splines is given.The winding angles of the interpolating curves are unbounded.Logarithmic arc splines are compatible with NURBS and their offsets are simple to compute.

[1]  Norimasa Yoshida,et al.  Fitting G2 multispiral transition curve joining two straight lines , 2012, Comput. Aided Des..

[2]  Benjamin B. Kimia,et al.  Euler Spiral for Shape Completion , 2003, International Journal of Computer Vision.

[3]  N. Yoshida,et al.  Quasi-Log-Aesthetic Curves in Polynomial Bézier Form , 2013 .

[4]  A. I. Kurnosenko.  Two-point G2 Hermite interpolation with spirals by inversion of hyperbola , 2010, Comput. Aided Geom. Des..

[5]  Walter Gautschi,et al.  Spirals: From Theodorus to Chaos , 1993 .

[6]  Shuchun Wang,et al.  Designing fair curves using monotone curvature pieces , 2004, Comput. Aided Geom. Des..

[7]  Takafumi Saito,et al.  Compound-Rhythm Log-Aesthetic Curves , 2009 .

[8]  Jun-Hai Yong,et al.  Geometric Hermite curves with minimum strain energy , 2004, Comput. Aided Geom. Des..

[9]  Gerald E. Farin,et al.  Approximation of logarithmic spirals , 1997, Comput. Aided Geom. Des..

[10]  Xunnian Yang,et al.  Curve fitting and fairing using conic splines , 2004, Comput. Aided Des..

[11]  Ling Xu,et al.  Magnetic Curves: Curvature-Controlled Aesthetic Curves Using Magnetic Fields , 2009, CAe.

[12]  Norimasa Yoshida,et al.  Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions , 2012, Comput. Aided Geom. Des..

[13]  Ayellet Tal,et al.  The Natural 3D Spiral , 2011, Comput. Graph. Forum.

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

[15]  K. Miura.  A General Equation of Aesthetic Curves and its Self-Affinity , 2006 .

[16]  D. Walton,et al.  An arc spline approximation to a clothoid , 2004 .

[17]  Hailing Zhou,et al.  Euler arc splines for curve completion , 2012, Comput. Graph..

[18]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[19]  Kenjiro T. Miura.  Unit quaternion integral curve: A new type of fair free-form curves , 2000, Comput. Aided Geom. Des..

[20]  Norimasa Yoshida,et al.  Planar two-point G1 Hermite interpolating log-aesthetic spirals , 2012, J. Comput. Appl. Math..

[21]  Derek S. Meek,et al.  Curve design with more general planar Pythagorean-hodograph quintic spiral segments , 2013, Comput. Aided Geom. Des..

[22]  Norimasa Yoshida,et al.  Interactive aesthetic curve segments , 2006, The Visual Computer.

[23]  I. Coope.  Curve interpolation with nonlinear spiral splines , 1993 .

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