Biarcs and bilens

doi:10.1016/j.cagd.2012.12.002

Computer Aided Geometric Design

Volume 30, Issue 3, March 2013, Pages 310-330

https://doi.org/10.1016/j.cagd.2012.12.002 Get rights and content

Abstract

Biarc curves are considered from the standpoints of the theory of spirals and Möbius maps. Parametrization and reference formulae, covering the whole variety of biarcs, are proposed. A region is constructed, enclosing all spirals with common circles of curvature at the endpoints. This region, named bilens, is bounded by two biarcs.

Graphical abstract

Highlights

► Simplest spirals, biarcs curves, are explored from the viewpoints of the theory of spirals and Möbius maps. ► Parametrization is proposed for the whole variety of biarcs. ► Region is constructed, bounded by two biarcs and enclosing all spirals with given $G^{2}$ Hermite data.

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A circular biarc can be defined by using two points K and L with their (oriented) tangents $g_{K}$ and $g_{L}$ as input. It is well-known that one can determine a one parametric set of circular arc pairs $k, ℓ$ such that k starts at K with tangent $g_{K}$ , ℓ ends at L with tangent $g_{L}$ and k and ℓ meet with a common tangent in an intermediate point P. In this paper we investigate a similar construction where we replace the circle biarcs by pairs of conic arcs. It turns out that in this case we can prescribe a conic $k_{0}$ with a point K on it, another conic $ℓ_{0}$ with a point L on it, and moreover an intermediate point P to obtain a unique pair $k, ℓ$ of conics such that k osculates $k_{0}$ in K, ℓ osculates $ℓ_{0}$ in L and k and ℓ osculate each other in P. This also confirms a result of H. Pottmann from 1991. We use our method to solve an interpolation task of Hermite type whose input consists of a series of points with their curvature circles and another series of intermediate points. The output is a $G C^{2}$ spline curve with conic arc segments.
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Citation Excerpt :
Using a dynamic bounding volume hierarchy of bounding circular arcs (BCAs) efficiently generated for deformable spiral curves, Lee et al. (2015a) accelerated the offset curve trimming for planar freeform curves. There are also other types of tight bounding volumes such as spiral fat arcs (Barton and Elber, 2011) and bilens (Kurnosenko, 2013) which have great potential in accelerating other types of geometric computations. Spiral segmentation also provides a handy tool for proving the cubic convergence of biarc approximation to freeform planar curves (Meek and Walton 1995, 1999).
We present a real-time algorithm for computing the Voronoi diagram of planar freeform piecewise-spiral curves. The efficiency and robustness of our algorithm is based on a simple topological structure of Voronoi cells for spirals, which also enables us a direct construction of Voronoi structure without relying on intermediate polygonal or biarc approximations to the given planar curves. Using a Möbius transformation, we provide an efficient search for maximal disks. The correct topology of Voronoi diagram is computed by sampling maximal disks systematically, which entails subdividing spirals until each belongs to a pair/triple of spirals under a certain matching condition. The matching pairs and triples serve as the basic building blocks for bisectors and bifurcations, and their connectivity implies the Voronoi structure. We demonstrate a real-time performance of our algorithm using experimental results including the medial axis computation for planar regions under deformation with non-trivial self-intersections and the Voronoi diagram construction for disconnected planar freeform curves.
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^☆: This paper has been recommended for acceptance by H. Prautzsch.

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Biarcs and bilens☆

Abstract

Graphical abstract

Highlights

Graph. Models

Comput.-Aided Design

Comput. Aided Geom. Design

Comput. Aided Geom. Design

J. Comput. Appl. Math.

Comput.-Aided Design