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 sphere of layout and research of algorithms. That dis cipline reviews geometric difficulties from a computational viewpoint, and it has attracted huge, immense examine curiosity. yet that curiosity is usually all in favour of Euclidean Geometry (mainly the aircraft or european clidean third-dimensional space). after all, there are a few very important rea sons for this incidence because the first applieations and the bases of all advancements are within the aircraft 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 numerous works on triangulations at the sphere and different surfaces. The exceptions pointed out within the final paragraph have seemed to attempt to solution a few quest ions which come up within the turning out to be record of components within which the result of Computational Geometry are acceptable, considering that, in practiee, many events in these parts result in difficulties of Com putational Geometry on surfaces (probably the sector and the cylinder are the most typical examples). we will be able to point out right here a few particular parts during which those occasions occur as engineering, computing device aided layout, production, geographie info structures, operations re seek, roboties, special effects, sturdy modeling, etc.

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

**Example text**

Our purpose now is to characterize the m-convex huH of a set of points on the cylinder, the torus, the sphere, and the cone, and to give algorithms for computing it. 5. When the set of points in the cylinder is in Euclidean position the convex huH is similar to a set of points in the plane. 2 The m-convex hull 0/ three points in non-Euclidean position on the cylinder contains the open strip defined by those points. LEMMA 38 COMPUTATIONAL GEOMETRY ON SURFACES Proo/: Let P = {VI,V2,V3} be three points in non-Euclidean position on the cylinder, and let r be the curve defined by the segments joining those points pairwise.

LEMMA 38 COMPUTATIONAL GEOMETRY ON SURFACES Proo/: Let P = {VI,V2,V3} be three points in non-Euclidean position on the cylinder, and let r be the curve defined by the segments joining those points pairwise. Apriori we know that Sand rare contained in Cm{P). Let gi denote the generatrix containing Vi, and let g~ be its opposite generatrix. 6). r_. --_. -------,. _. --. -- -------------f -_. _ I --_. -_. ----_.!. -_. ---_. - :m~~L~~~,~' r~~Lt::::::::=i v ------------~-----------~-----_ J .. ~... 6.

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. ALGORITHM CH-TORUS(P) P = {VI, V2, ... , VN} set 0/ points on the torus. OUTPUT: Metrically convex hull 0/ P in the torus. INPUT: STEP STEP STEP 1 I/ P is in cylindrical position go to Step 3.

### 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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