{"id":359,"date":"2011-09-08T20:05:37","date_gmt":"2011-09-08T20:05:37","guid":{"rendered":"http:\/\/devnot.wordpress.com\/?p=359"},"modified":"2022-03-24T19:19:51","modified_gmt":"2022-03-24T18:19:51","slug":"check-whether-two-2d-vectors-are-coincident","status":"publish","type":"post","link":"https:\/\/thomas-jansen.eu\/?p=359","title":{"rendered":"Check whether two 2D lines are coincident."},"content":{"rendered":"\n\n\n

To check whether two 2D lines are coincident, the distance of \"C\" and \"D\" to \"AB\" need to be calculated. For a coincident line both need to be (almost) zero.<\/p>\n\n\n\n

\"\"<\/a><\/figure>\n\n\n\n

The z component of the cross vector can be used for this quite elegantly.<\/p>\n\n\n\n

Note:<\/strong> The cross vector is noted as an x e.g. \"A\"x\"B\" <\/p>\n\n\n\n

To check whether the two lines \"AB\" and \"CD\" are coincident, the z-component of two cross vectors need to be calculated. Both calculate the distance of the points on line \"CD\" to \"AB\".<\/p>\n\n\n\n

\"AB\" x \"AC\" is the area of the parallelogram \"AB AC\", dividing it with the length of \"AB\" (\"|AB|\") gives the shortest distance of point \"C\" to line \"AB\".<\/p>\n\n\n\n

\"AB\" x \"AD\" gives the area of the parallelogram \"AB AD\" <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

After dividing with \"|AB|\" this is also the shortest distance of \"D\" to line \"AB\". If both distances are small enough the lines are coincident. <\/p>\n\n\n\n

\"dist_{abc} = \overline{AB}_x·\overline{AC}_y - \overline{AB}_y·\overline{AC}_x\"
\"dist_{abd} = \overline{AB}_x·\overline{AD}_y - \overline{AB}_y·\overline{AD}_x\" <\/p>\n","protected":false},"excerpt":{"rendered":"

To check whether two 2D lines are coincident, the distance of $C$ and $D$ to $AB$ need to be calculated. For a coincident line both need to be (almost) zero. The z component of the cross vector can be used for this quite elegantly. Note: The cross vector is noted as an x e.g. $A$x$B$ […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-359","post","type-post","status-publish","format-standard","hentry","category-geometry"],"_links":{"self":[{"href":"https:\/\/thomas-jansen.eu\/index.php?rest_route=\/wp\/v2\/posts\/359","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/thomas-jansen.eu\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/thomas-jansen.eu\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/thomas-jansen.eu\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/thomas-jansen.eu\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=359"}],"version-history":[{"count":5,"href":"https:\/\/thomas-jansen.eu\/index.php?rest_route=\/wp\/v2\/posts\/359\/revisions"}],"predecessor-version":[{"id":683,"href":"https:\/\/thomas-jansen.eu\/index.php?rest_route=\/wp\/v2\/posts\/359\/revisions\/683"}],"wp:attachment":[{"href":"https:\/\/thomas-jansen.eu\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=359"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/thomas-jansen.eu\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=359"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/thomas-jansen.eu\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=359"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}