Graph Theory

Effective Computational Geometry for Curves and Surfaces by Jean-Daniel Boissonnat, Monique Teillaud

By Jean-Daniel Boissonnat, Monique Teillaud

The cause of this e-book is to settle the rules of non-linear computational geometry. It covers combinatorial info constructions and algorithms, algebraic matters in geometric computing, approximation of curves and surfaces, and computational topology.

Each bankruptcy presents a state-of-the-art, in addition to an instructional advent to big thoughts and effects. the point of interest is on equipment that are either good based mathematically and effective in practice.

References to open resource software program and dialogue of power functions of the offered concepts also are included.

This booklet can function a textbook on non-linear computational geometry. it's going to even be beneficial to engineers and researchers operating in computational geometry or different fields, like structural biology, third-dimensional clinical imaging, CAD/CAM, robotics, and graphics.

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Extra resources for Effective Computational Geometry for Curves and Surfaces

Sample text

There is, however, a subtlety regarding this combined approach since, as we will describe next, incremental construction of arrangements requires answering point-location queries. We discuss this issue after presenting various point-location strategies. The Insertion Process Fig. 5. The insertion process of a new curve (dotted) into an existing arrangement of planar curves. The zone of the new curve is lightly shaded and all the newly introduced vertices are marked k) log n), while the incremental construction may take O(n2 ) time.

The traits class exploits the functionality of the parameterized SegmentTraits type to handle the segments that comprise the polyline curves. , Arr segment traits 2 or Arr segment cached traits 2 ), and the number type used by the instantiated segment traits should satisfy the same conditions listed above. We next review the arrangement traits classes that handle non-linear curves. The first two traits classes are included in the public distribution of Cgal, while the others (Sect.

The first step consists of choosing a representation for the input data: - We will consider here implicit algebraic curves, and focus more specifically circles since this corresponds to the current Cgal release. Such a curve is represented by a bivariate polynomial of total degree two. A circular arc is represented by a supporting circle and its two delimiting endpoints, knowing that the arc is oriented counterclockwise. - Two kinds of points—intersection points and endpoints—can be considered in the same way, thus allowing us to have a unique representation.

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