Download Computational Geometry on Surfaces: Performing Computational by Clara I. Grima PDF

By Clara I. Grima

In the final thirty years Computational Geometry has emerged as a brand new self-discipline from the sphere of layout and research of algorithms. That dis­ cipline experiences geometric difficulties from a computational perspective, and it has attracted huge, immense study curiosity. yet that curiosity is usually eager about Euclidean Geometry (mainly the airplane or european­ clidean three-d space). in fact, 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 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 various works on triangulations at the sphere and different surfaces. The exceptions pointed out within the final paragraph have looked as if it would try and resolution a few quest ions which come up within the starting to be record of components within which the result of Computational Geometry are acceptable, in view that, in practiee, many events in these parts result in difficulties of Com­ putational Geometry on surfaces (probably the field and the cylinder are the commonest examples). we will point out the following a few particular parts during which those events take place as engineering, laptop aided layout, production, geographie info structures, operations re­ seek, roboties, special effects, reliable modeling, etc.

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

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Intuitively, if a set is in Euclidean position, all planar methods for solving problems in Computational Geometry are valid (or their adaptation will be very easy) for that set. In some sense this is a book devoted to Computational Geometry on sets that are not in Euclidean position. It is in those sets that there are situations more interesting to analyze, and where there is more work to do; but, from a practical point of view, there exist many situations 19 20 COMPUTATIONAL GEOMETRY ON SURFACES whieh ean be solved by adapting planar algorithms in a straightforward way and we need to know when we are facing one of those situations.

2, this then justifies the choice of the maximum gap problem. 2. Smallest arc covering a set of points on a circurnference be solved in optimal time O(N), as one can see in [Sacristan, 1997], using aseparability criterion and linear programming techniques [Megiddo, 1983]. However, we will give here a simpler proof of this property. 2 of N points time O(N). at the same of points in time. LEMMA The problem of deciding if the smallest are covering a set on a eireumferenee is less than 1r ean be solved in optimal Moreover, if that are is less than 1r then it can be found computational cost.

Proof: Let P = {pl,p2, ... ,PN} be a set of N points on the torus, and let (Xi, Vi) be the coordinates of each Pi. If we call C the event 'P is in cylindrical position' we will have that E = E 1 U E2, El being the event 'P is contained between two opposite paralleIs' and Eh the event 'P is contained between two opposite meridians'. 1, we can show that P{Et} = P{E2) = Nj2 N - 1 • So P{E) = P{Et} + P{E2) - P{E1 n Eh), and the result holds. 0 4. EUCLIDEAN POSITION IN ORBIFOLDS AND IN GENERAL SURFACES As has been said before, arguments or algorithms on general surfaces that are valid in the planar case have been used in many applications , but it is not dear whether those arguments are still valid out of the plane.

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