Convex Polyhedra by A.D. Alexandrov, N.S. Dairbekov, S.S. Kutateladze, A.B.

By A.D. Alexandrov, N.S. Dairbekov, S.S. Kutateladze, A.B. Sossinsky

Convex Polyhedra belongs to the classics in geometry. There easily isn't any different e-book that offers with the various points of the idea of three-dimensional convex polyhedra in a related method, and in at any place close to its aspect and completeness. it's a definitive resource of the classical box of convex polyhedra and comprises the to be had solutions to the query of the information which can uniquely be sure a convex polyhedron. this query matters all information pertinent to a polyhedron, e.g. the lengths of edges, components of faces, etc.

This important and obviously written ebook contains the fundamentals of convex polyhedra and collects the main normal life theorems for convex polyhedra which are proved through a brand new and unified strategy. it's a extraordinary resource of principles for college kids.

The English version comprises a number of reviews in addition to extra fabric and a finished bibliography by way of V.A. Zalgaller to convey the paintings modern. in addition, similar papers via L.A.Shor and Yu.A.Volkov were extra as supplementations to this booklet.

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Sample text

If we now move the plane Q to infinity, then in the limit the pyramid P transforms into a convex solid polyhedral angle V which is the convex hull of the collection of the half-lines a1 , . . , am . 3, a point Ak is a vertex of the pyramid P if and only if it does not belong to the convex hull of the remaining points O, A1 , . . , Ak−1 , Ak+1 , . . , Am . Therefore, a segment OAk is an edge of P if 38 1 Basic Concepts and Simplest Properties of Convex Polyhedra and only if it does not lie in the convex hull of the remaining segments OAi .

We have thus proved that to every edge of V corresponds a parallel unbounded edge of P . Hence, V is the limit angle of P , which completes the proof of the theorem. Theorem 5a. Under the hypotheses of Theorem 5, the polyhedron P may have vertices only among the given points A1 , A2 , . . , Am . Each of them is a vertex if and only if it is not contained in the convex hull of the figure would include an entire straight line, which is excluded by assumption. Therefore, all preceding arguments may be repeated verbatim.

We choose the vertex of V inside the polyhedron. 1, the angle V lies in the polyhedron. Given a support plane Q of the polyhedron, we can obviously find a support plane of V parallel to Q by shifting Q inside the polyhedron so that it begins to touch the angle V . Hence, the spherical image of the polyhedron lies in the spherical image of its limit angle. It remains to show that the spherical image of the angle V is contained in the spherical image of the polyhedron P . In accordance with Theorem 1, both are spherical polygons; hence, its suffices to prove that every interior point N of the spherical image of V belongs to the spherical image of P .

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