{"id":1739,"date":"2024-02-26T11:29:16","date_gmt":"2024-02-26T16:29:16","guid":{"rendered":"http:\/\/brickisland.net\/DDGSpring2024\/?p=1739"},"modified":"2024-03-10T15:24:47","modified_gmt":"2024-03-10T19:24:47","slug":"lecture-11-discrete-curves","status":"publish","type":"post","link":"https:\/\/brickisland.net\/DDGSpring2024\/2024\/02\/26\/lecture-11-discrete-curves\/","title":{"rendered":"Lecture 11\u2014Discrete Curves"},"content":{"rendered":"<p>This lecture presents the discrete counterpart of the previous lecture on smooth curves. Here we also arrive at a discrete version of the fundamental theorem for plane curves: a discrete curve is completely determined by its discrete parameterization (a.k.a. edge lengths) and its discrete curvature (a.k.a. exterior angles). Can you come up with a discrete version of the fundamental theorem for space curves? If we think of torsion as the rate at which the binormal is changing, then a natural analogue might be to (i) associate a binormal \\(B_i\\) with each vertex, equal to the normal of the plane containing \\(f_{i-1}\\), \\(f_{i}\\), and \\(f_{i+1}\\), and (ii) associate a torsion \\(\\tau_{ij}\\) to each edge \\(ij\\), equal to the angle between \\(B_i\\) and \\(B_{i+1}\\). Using this data, can you recover a discrete space curve from edge lengths \\(\\ell_{ij}\\), exterior angles \\(\\kappa_i\\) at vertices, and torsions \\(\\tau_{ij}\\) associated with edges? What&#8217;s the actual algorithm? (If you find this problem intriguing, leave a comment in the notes! It&#8217;s not required for class credit.)<\/p>\n<p><a href=\"http:\/\/brickisland.net\/DDGSpring2024\/wp-content\/uploads\/2019\/02\/DDG_458_SP19_Lecture11_DiscreteCurves.pdf\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/brickisland.net\/DDGSpring2024\/wp-content\/uploads\/2019\/02\/DDG_458_SP19_Lecture11_DiscreteCurves_icon.jpg\" alt=\"\" width=\"419\" height=\"243\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This lecture presents the discrete counterpart of the previous lecture on smooth curves. Here we also arrive at a discrete version of the fundamental theorem for plane curves: a discrete curve is completely determined by its discrete parameterization (a.k.a. edge lengths) and its discrete curvature (a.k.a. exterior angles). Can you come up with a discrete &hellip; <a href=\"https:\/\/brickisland.net\/DDGSpring2024\/2024\/02\/26\/lecture-11-discrete-curves\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Lecture 11\u2014Discrete Curves&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":"","_links_to":"","_links_to_target":""},"categories":[3],"tags":[],"_links":{"self":[{"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/posts\/1739"}],"collection":[{"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/comments?post=1739"}],"version-history":[{"count":1,"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/posts\/1739\/revisions"}],"predecessor-version":[{"id":1742,"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/posts\/1739\/revisions\/1742"}],"wp:attachment":[{"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/media?parent=1739"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/categories?post=1739"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/tags?post=1739"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}