By Clara I. Grima
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 generally interested by Euclidean Geometry (mainly the airplane or european clidean three-dimensional space). in fact, there are a few very important rea sons for this prevalence because the first applieations and the bases of all advancements are within the airplane or in three-dimensional house. 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 turning out to be checklist of parts during which the result of Computational Geometry are appropriate, on the grounds that, in practiee, many occasions in these components bring about difficulties of Com putational Geometry on surfaces (probably the sector and the cylinder are the commonest examples). we will point out the following a few particular components within which those events ensue as engineering, computing device aided layout, production, geographie details structures, operations re seek, roboties, special effects, good 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
VN} be a set of points on this surfacej in order to compute its m-convex hull the main idea will be to determine whether P is in Euclidean position or not. We know that if the set of points in the cylinder is in Euclidean position we must use a planar convex hull algorithm [Edelsbrunner, 1987, o'Rourke , 1994, Preparata and Shamos, 1985, Seidel, 1997]. 4 we have that the convex hull will be the union of the open strip defined by the points with the greatest and the smallest ordinate and the m-top and the m-bottom.
Thus we can conclude that any significant collection of points in the cylinder will be in non-Euclidean position; in this way planar results cannot be applied for most point sets in the cylinder. 3 It is possible to decide whether a collection of points in the cone is in Euclidean position in linear time. 2 EUCLIDEAN POSITION ON THE TORUS Regarding the torus, using considerations similar to those in the case of the cylinder it is easy to design a linear algorithm to decide if a set of points is in Euclidean position or not.
If the ans wer is YES the points in the torus are in cylindrical position and we can use algorithm CH-CYLINDER(P). If the answer is NO go to next step. 2. Now cut the torus by a parallel and test whether P is in Euclidean position in the obtained cylinder. If the answer is YES P is contained between two opposite meridians, therefore it is in cylindrical position and one can use algorithm CH-CYLINDER(P) in order to compute its m-convex hull. 6. 2. 15. , , I I Cutting the torus by a meridian we obtain a cylinder.