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If we recursively substitute the formula for the first term on the right then the following summation emerges: NR(n,d)=\u2211k=0d(nk)\u223cO(nd)."},{"key":"10.1016\/S0734-189X(89)80018-0_bib25","unstructured":"There are O(n2) unbounded regions because each plane containing a face of the polyhedra intersects a sphere through the unbounded regions in a circle. Each such circle intersects every other non-parallel circle at two points. The circles on the sphere may be thought of as an arrangement of pseudo lines. Because of the isomorphism of an arrangement of pseudo lines with an arrangement of ordinary lines we know the number of pseudo regions to be O(n2) and thereby the number of unbounded regions for the polyhedra is O(n2)."},{"key":"10.1016\/S0734-189X(89)80018-0_bib26","unstructured":"The following mapping provides a way to extend the plane R2 to a space in which the point at infinity is naturally included. Place a sphere of finite radius so that it osculates the plane R2 at the origin (some point inside the polygon) defining the south pole. We then define the mapping of any point in R2 to a point on the sphere by the point on the sphere lying on a line between the point in the plane and the north pole of the sphere. The plane is thus mapped to the surface of the sphere and the extension is to associate the point at infinity with north pole of the sphere."}],"container-title":["Computer Vision, Graphics, and Image Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0734189X89800180?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0734189X89800180?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2019,1,14]],"date-time":"2019-01-14T20:52:57Z","timestamp":1547499177000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0734189X89800180"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,4]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1989,4]]}},"alternative-id":["S0734189X89800180"],"URL":"https:\/\/doi.org\/10.1016\/s0734-189x(89)80018-0","relation":{},"ISSN":["0734-189X"],"issn-type":[{"value":"0734-189X","type":"print"}],"subject":[],"published":{"date-parts":[[1989,4]]}}}