{"id":12662,"date":"2021-11-15T18:28:33","date_gmt":"2021-11-15T23:28:33","guid":{"rendered":"https:\/\/carleton.ca\/scs\/?page_id=12662"},"modified":"2026-06-02T14:59:26","modified_gmt":"2026-06-02T18:59:26","slug":"tr-134-separating-a-polyhedron-by-one-translation-from-a-set-of-obstacles","status":"publish","type":"page","link":"https:\/\/carleton.ca\/scs\/research\/scs-technical-reports\/technical-reports-1987\/tr-134-separating-a-polyhedron-by-one-translation-from-a-set-of-obstacles\/","title":{"rendered":"TR-134: Separating a Polyhedron by One Translation from a Set of Obstacles"},"content":{"rendered":"\n<section class=\"w-screen px-6 cu-section cu-section--white ml-offset-center md:px-8 lg:px-14\">\n    <div class=\"space-y-6 cu-max-w-child-5xl  md:space-y-10 cu-prose-first-last\">\n\n            <div class=\"cu-textmedia flex flex-col lg:flex-row mx-auto gap-6 md:gap-10 my-6 md:my-12 first:mt-0 max-w-5xl\">\n        <div class=\"justify-start cu-textmedia-content cu-prose-first-last\" style=\"flex: 0 0 100%;\">\n            <header class=\"font-light prose-xl cu-pageheader md:prose-2xl cu-component-updated cu-prose-first-last\">\n                                    <h1 class=\"cu-prose-first-last font-semibold !mt-2 mb-4 md:mb-6 relative after:absolute after:h-px after:bottom-0 after:bg-cu-red after:left-px text-3xl md:text-4xl lg:text-5xl lg:leading-[3.5rem] pb-5 after:w-10 text-cu-black-700 not-prose\">\n                        TR-134: Separating a Polyhedron by One Translation from a Set of Obstacles\n                    <\/h1>\n                \n                                \n                            <\/header>\n\n                    <\/div>\n\n            <\/div>\n\n    <\/div>\n<\/section>\n\n<p>Carleton University<br>\n<a href=\"https:\/\/carleton.ca\/scs\/research\/scs-technical-reports\/technical-reports-1987\/\">Technical Report<\/a> <strong>TR-134<\/strong><br>\nDecember&nbsp;1987<\/p>\n\n\n\n<h2 id=\"separating-a-polyhedron-by-one-translation-from-a-set-of-obstacles\" class=\"wp-block-heading tr_t1\">Separating a Polyhedron by One Translation from a Set of Obstacles<\/h2>\n\n\n\n<div class=\"tr_t3\">\n<div class=\"tr_t3\">\n<div class=\"tr_t3\">Otto Nurmi &amp; J\u00f6rg-R\u00fcdiger Sack<\/div>\n<\/div>\n<\/div>\n\n\n\n<div>\n<h3>Abstract<\/h3>\n<p>We present efficient algorithms for several problems of movable. separability in 3-dimensional apace. The first algorithm determines all directions in which a convex polyhedron can be translated through a planar convex polygon. The algorithm runs in O(n) time. This is a considerable improvement over the previous O(n2 log n) time algorithm of [17]. The second algorithm computes, in O(n) time, all directions in which a convex polyhedron can be translated to infinity without collisions with a convex obstacle (n is the number of vertices of the polyhedra). A generalization of the plane-sweep technique, called &#8220;sphere-sweep&#8221;, is given and provides an efficient algorithm for the last problem which is: determine all directions in which a convex polyhedron can be separated from a set of convex obstacles. Our results are obtained by avoiding the standard technique of motion planning problems, the O(n2 ) time computation of the Minkowski differences of the polyhedra.<\/p>\n<\/div>\n\n\n\n<p><a href=\"https:\/\/carleton.ca\/scs\/wp-content\/uploads\/sites\/260\/tr-134.pdf\">TR-134.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Carleton University Technical Report TR-134 December&nbsp;1987 Separating a Polyhedron by One Translation from a Set of Obstacles Otto Nurmi &amp; J\u00f6rg-R\u00fcdiger Sack Abstract We present efficient algorithms for several problems of movable. separability in 3-dimensional apace. The first algorithm determines all directions in which a convex polyhedron can be translated through a planar convex polygon. [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":11827,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"_cu_dining_location_slug":"","footnotes":"","_links_to":"","_links_to_target":""},"cu_page_type":[],"class_list":["post-12662","page","type-page","status-publish","hentry"],"acf":{"cu_post_thumbnail":false},"_links":{"self":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/12662","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/comments?post=12662"}],"version-history":[{"count":1,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/12662\/revisions"}],"predecessor-version":[{"id":12663,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/12662\/revisions\/12663"}],"up":[{"embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/11827"}],"wp:attachment":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/media?parent=12662"}],"wp:term":[{"taxonomy":"cu_page_type","embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/cu_page_type?post=12662"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}