Download e-book for kindle: Computational Geometry on Surfaces: Performing Computational by Clara I. Grima

By Clara I. Grima

ISBN-10: 9048159083

ISBN-13: 9789048159086

ISBN-10: 9401598096

ISBN-13: 9789401598095

In the final thirty years Computational Geometry has emerged as a brand new self-discipline from the sector of layout and research of algorithms. That dis­ cipline reviews geometric difficulties from a computational standpoint, and it has attracted huge, immense learn curiosity. yet that curiosity is usually enthusiastic about Euclidean Geometry (mainly the airplane or ecu­ clidean three-d space). after all, there are a few vital rea­ sons for this prevalence because the first applieations and the bases of all advancements are within the airplane or in third-dimensional area. yet, we will be able to locate additionally a few exceptions, and so Voronoi diagrams at the sphere, cylin­ der, the cone, and the torus were thought of formerly, and there are lots of works on triangulations at the sphere and different surfaces. The exceptions pointed out within the final paragraph have seemed to attempt to resolution a few quest ions which come up within the growing to be record of components during which the result of Computational Geometry are acceptable, considering, in practiee, many occasions in these parts result in difficulties of Com­ putational Geometry on surfaces (probably the sector and the cylinder are the commonest examples). we will be able to point out the following a few particular components within which those occasions take place as engineering, machine aided layout, production, geographie info structures, operations re­ seek, roboties, special effects, strong modeling, etc.

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Extra resources for Computational Geometry on Surfaces: Performing Computational Geometry on the Cylinder, the Sphere, the Torus, and the Cone

Example text

These sections are devoted to summarizing the results presented in the chapter and so as to present some open problems. Also we include some bibliographical references which could be useful for extending the study of the topics covered. Chapter 2 EUCLIDEAN POSITION If we try to compute the convex hull or a triangulation of a set of sites in a surface and all those sites are very elose to each other, we can intuitively think that planar methods will be valid in this situation. This intuition has been used on several occasions by many authors, but sometimes it is not elear what 'very elose to each other' means.

3. CYLINDRICAL POSITION IN THE TORUS In the case of the torus, in addition to Euclidean position we have another situation in which we can apply directIy some methods obtained in this book for the cylinder. So we will say that a set of sites on this surface is in cylindrical position if the set of them is contained between two opposite parallels or two opposite meridians. Obviously, if a set is in Euclidean position it is in cylindrical position as weIl, but, of course, the converse is not true in general.

20. 53 The shaded region must be contained in Cm(P) ALGORITHM CH-CONE(P) INPUT: P = {Vl, V2, ... , VN} set OUTPUT: Metrically convex hull 0/ points on the cone. 0/ P on the cone. 1/ P is in Euclidean position go to Step 3. STEP 2 Compute the convex hull 0/ ifJ-l(p) in X adapting a planar convex hull algorithm. The m-convex hull 0/ P is STEP 1 CH(ifJ-l(p)) n C2 • (END) 3 Compute the convex hull 0/ P in the cone using a planar STEP convex hUll algorithm. 4. Algorithm for computing the metrically convex hull of a set on the cone.

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Computational Geometry on Surfaces: Performing Computational Geometry on the Cylinder, the Sphere, the Torus, and the Cone by Clara I. Grima

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