The Computational Geometry Algorithms Library Cgal [2] offers severals packages to compute Voronoi diagrams. Euclidean Voronoi diagrams of points and power diagrams are represented through their dual Delaunay and regu-lar triangulations. Cgalprovides Delaunay and regular triangulations inR2 andR3. The implementation is based on a randomized incremental algorithm
Fig. 2.15. Two λ-medial axes of the same shape, with λ increasing from left to right, computed as a subset of the Voronoi diagram of a sample of the boundary (courtesy of Steve Oudot)
using a variant of the Voronoi hierarchy described in Sect. 2.6. Delaunay tri-angulations are also provided in higher dimensions.
The library also contains packages to compute Voronoi diagrams of line segments [215] and Apollonius diagrams in R2 [216]. Those packages ment the incremental algorithm described in Sect. 2.6. A prototype imple-mentation of M¨obius diagrams inR2also exists. This prototype computes the M¨obius diagram as the projection of the intersection of a 3-dimensional power diagram with a paraboloid, as described in Sect. 2.4.1. This prototype also serves as the basis for the developement of aCgalpackage for 3-dimensional Apollonius diagrams, where the boundary of each cell is computed as a 2-dimensional M¨obius diagram, following the results of Sect. 2.4.3 [62]. See Fig. 2.8.
2.9 Applications
Euclidean and affine Voronoi diagrams have numerous applications we do not discuss here. The interested reader can consult other chapters of the book, most notably Chap. 5 on surface meshing and Chap. 6 on reconstruction.
Other applications can be found in the surveys and the textbooks mentionned in the introduction.
Additively and multiplicatively weighted distances arise when modeling growing processes and have important applications in biology, ecology and other fields. Consider a number of crystals, all growing at the same rate, and all starting at the same time : one gets a number of growing circles. As these circles meet, they draw a Euclidean Voronoi diagram. In reality, crystals start
Fig. 2.16. A cell in an Apollonius diagram of spheres
growing at different times. If they still grow at the same rate, they will meet along an Apollonius diagram. This growth model is known as the Johnson-Mehl model in cell biology. In other contexts, all the crystals start at the same time, but grow at different rates. Now we get what is called the multiplicatively weighted Voronoi diagram, a special case of M¨obius diagrams.
Spheres are common models for a variety of objects such as particles, atoms or beads. Hence, Apollonius diagrams have been used in physics, material sciences, molecular biology and chemistry [245, 339, 227, 228]. They have also been used for sphere packing [246] and shortest paths computations [256].
Euclidean Voronoi diagrams of non punctual objects find applications in robot motion planning [237, 197]. Medial axes are used for shape analysis [160], for computing offsets in Computer-Aided Design [118], and for mesh generation [290, 289, 316]. Medial axes are also used in character recogni-tion, road network detection in geographic information systems, and other applications.
Acknowledgments
We thank D. Attali, C. Delage and M. Karavelas with whom part of the research reported in this chapter has been conducted. We also thank F. Chazal and A. Lieutier for fruitful discussions on the approximation of the medial axis.
Algebraic Issues in Computational Geometry
Bernard Mourrain, Sylvain Pion, Susanne Schmitt, Jean-Pierre T´ecourt, Elias Tsigaridas, and Nicola Wolpert
3.1 Introduction
Geometric modeling plays an increasing role in fields at the frontier between computer science and mathematics. This is the case for example in CAGD (Computer-aided Geometric design, where the objects of a scene or a piece to be built are represented by parameterized curves or surfaces such as NURBS), robotics or molecular biology (rebuilding of a molecule starting from the ma-trix of the distances between its atoms obtained by NMR).
The representation of shapes by piecewise-algebraic functions (such as B-spline functions) provides models which are able to encode the geometry of an object in a compact way. For instance, B-spline representations are heavily used in Computed Aided Geometric Design, being now a standard for this area. Recently, we also observe a new trend involving the use of patches of implicit surfaces. This includes in particular the representation by quadrics, which are more natural objects than meshes for the representation of curved shapes.
From a practical point of view, critical operations such as computing in-tersection curves of parameterized surfaces are performed on these geometric models. This intersection problem, as a typical example linking together geom-etry, algebra and numeric computation, received a lot of attention in the past literature. See for instance [158, 280, 233]. It requires robust methods, for solv-ing (semi)-algebraic problems. Different techniques (subdivision, lattice eval-uation, marching methods) have been developed [278, 176, 14, 191, 190, 280].
A critical question is to certify or to control the topology of the result.
From a theoretical point of view, the study of algebraic surfaces is also a fascinating area where important developments of mathematics such as singu-larity theory interact with visualization problems and the rendering of math-ematical objects. The classification of singularities [29] provides simple alge-braic formulas for complicated shapes, which geometrically may be difficult to
Chapter coordinator
handle. Such models can be visualized through techniques such as ray-tracing1 in order to produce beautiful pictures of these singularities. Many open ques-tions, related for instance to the topological types of real algebraic curves or surfaces, remain to be solved in this area. Computation tools, which allow to treat such algebraic models are thus important to understand their geometric properties.
In this chapter, we will describe methods for the treatment of algebraic models. We focus on the problem of computing the topology of implicit curves or surfaces. Our objective is to devise certified and output-sensitive methods, in order to combine control and efficiency. We distinguish two types of sub-problems:
• the construction of new geometric objects such as points of intersection,
• predicates such as the comparison of coordinates of intersection points.
In the first case, a good approximation of the exact algebraic object, which usually cannot be described explicitly by an analytic formula, may be enough.
On the contrary for the second subproblem, the result has to be exact in order to avoid incoherence problems, which might be dangerous from an implemen-tation point of view, leading to well known non-robustness issues.
These two types of geometric problems, which appear for instance in arrangement computations (see Chapter 1) lead to the solution of algebraic questions. In particular, the construction or the comparison of coordinates of points of intersections of two curves or three surfaces involve computations with algebraic numbers. In the next section, we will describe exact methods for their treatment. Then we show how to apply these tools to compute the topology of implicit curves. This presentation includes effective aspects and pointers to software. It does not include proofs, which can be found in the cited literature.