{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22398"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22398","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Shapes and implementations in three-dimensional geometry","abstract":"\"Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is often useful or required to compute what one might call the \"\"shape\"\" of the set. For that purpose, this thesis deals with the formal notion of the family of alpha shapes of a finite point set in three-dimensional space. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a real parameter controlling the desired level of detail. Algorithms and data structures are presented that construct and store the entire family of shapes, with a quadratic time and space complexity, in the worst case.\"","abstract_html":""Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is often useful or required to compute what one might call the ""shape"" of the set. For that purpose, this thesis deals with the formal notion of the family of alpha shapes of a finite point set in three-dimensional space. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a real parameter controlling the desired level of detail. Algorithms and data structures are presented that construct and store the entire family of shapes, with a quadratic time and space complexity, in the worst case."","abstract_has_math":false,"creators":["Mucke, Ernst Peter"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":["Edelsbrunner, Herbert"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:38:33Z","date_published":"2011-05-07T13:38:33Z","updated_at":"2026-07-22T22:25:19Z","subjects":["Computer Science"],"languages":["eng"],"rights":["Copyright 1994 Mucke, Ernst Peter"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9416411","(UMI)AAI9416411"],"render_values":[{"text":"AAI9416411","href":null,"code":true},{"text":"(UMI)AAI9416411","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22398","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Edelsbrunner, Herbert"]},{"key":"dc:creator","label":"Author","values":["Mucke, Ernst Peter"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:38:33Z","10000-01-01","1994"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Computer Science"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1994 Mucke, Ernst Peter"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9416411","(UMI)AAI9416411","http://hdl.handle.net/2142/22398"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is often useful or required to compute what one might call the \"\"shape\"\" of the set. 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The programs that use the technique tend to be considerably smaller and more robust than others.","Made available in DSpace on 2011-05-07T13:38:33Z (GMT). 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For that purpose, this thesis deals with the formal notion of the family of alpha shapes of a finite point set in three-dimensional space. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a real parameter controlling the desired level of detail. Algorithms and data structures are presented that construct and store the entire family of shapes, with a quadratic time and space complexity, in the worst case.\"","Implementations of the algorithms are discussed, with an emphasis on the robust construction of three-dimensional Delaunay triangulations. A general-purpose programming technique, called Simulation of Simplicity, is used to cope with degenerate input data. This method relieves the programmer from the task of providing a consistent treatment for every single special case that can occur. 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