{"id":154313,"date":"1977-07-01T00:00:00","date_gmt":"1977-07-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/a-homogeneous-formulation-for-lines-in-2-space\/"},"modified":"2018-10-16T20:34:28","modified_gmt":"2018-10-17T03:34:28","slug":"a-homogeneous-formulation-for-lines-in-2-space","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/a-homogeneous-formulation-for-lines-in-2-space\/","title":{"rendered":"A Homogeneous Formulation for Lines in 2 Space"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Homogeneous coordinates have long been a standard tool of computer graphics. They afford a convenient representation for various geometric quantities in two and three dimensions. The representation of lines in three dimensions has, however, never been fully described. This paper presents a homogeneous formulation for lines in 3 dimensions as an anti-symmetric 4&#215;4 matrix which transforms as a tensor. This tensor actually exists in both covariant and contravariant forms, both of which are useful in different situations. The derivation of these forms and their use in solving various geometric problems is described.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Homogeneous coordinates have long been a standard tool of computer graphics. They afford a convenient representation for various geometric quantities in two and three dimensions. The representation of lines in three dimensions has, however, never been fully described. This paper presents a homogeneous formulation for lines in 3 dimensions as an anti-symmetric 4&#215;4 matrix which [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13551],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-154313","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-graphics-and-multimedia","msr-locale-en_us"],"msr_publishername":"Association for Computing Machinery, Inc.","msr_edition":"Siggraph 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