{"id":1959,"date":"2024-03-23T10:54:16","date_gmt":"2024-03-23T14:54:16","guid":{"rendered":"http:\/\/brickisland.net\/DDGSpring2024\/?p=1959"},"modified":"2024-04-04T14:34:33","modified_gmt":"2024-04-04T18:34:33","slug":"lectures-17-18-laplace-beltrami","status":"publish","type":"post","link":"https:\/\/brickisland.net\/DDGSpring2024\/2024\/03\/23\/lectures-17-18-laplace-beltrami\/","title":{"rendered":"Lectures 17 &#038; 18\u2014Laplace Beltrami"},"content":{"rendered":"<p>In the next two lectures we&#8217;ll take a deep dive into one of the most important objects not only in <em>discrete<\/em> differential geometry, but in differential geometry at large (not to mention physics!): the Laplace-Beltrami operator. This operator generalizes the familiar Laplacian you may have studied in vector calculus, which just gives the sum of second partial derivatives: \\(\\Delta \\phi = \\sum_i \\partial^2 \\phi_i \/ \\partial x_i^2\\). We&#8217;ll motivate Laplace-Beltrami from several points of view, talk about how to discretize it, and show how from a computational point of view it really is the \u2014Swiss army knife\u2014 of geometry processing algorithms, essentially replacing the discrete Fourier transform from classical signal processing.<\/p>\n<p><a href=\"http:\/\/brickisland.net\/DDGSpring2024\/wp-content\/uploads\/2019\/04\/SwissArmyLaplacian.pdf\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1960\" src=\"http:\/\/brickisland.net\/DDGSpring2024\/wp-content\/uploads\/2019\/04\/DDG_458_SP19_Lecture17_LaplaceBeltramiI_icon.jpg\" alt=\"\" width=\"838\" height=\"486\" srcset=\"https:\/\/brickisland.net\/DDGSpring2024\/wp-content\/uploads\/2019\/04\/DDG_458_SP19_Lecture17_LaplaceBeltramiI_icon.jpg 838w, https:\/\/brickisland.net\/DDGSpring2024\/wp-content\/uploads\/2019\/04\/DDG_458_SP19_Lecture17_LaplaceBeltramiI_icon-300x174.jpg 300w, https:\/\/brickisland.net\/DDGSpring2024\/wp-content\/uploads\/2019\/04\/DDG_458_SP19_Lecture17_LaplaceBeltramiI_icon-768x445.jpg 768w\" sizes=\"(max-width: 709px) 85vw, (max-width: 909px) 67vw, (max-width: 984px) 61vw, (max-width: 1362px) 45vw, 600px\" \/><\/a><a href=\"https:\/\/brickisland.net\/DDGSpring2021\/wp-content\/uploads\/2020\/04\/DDG_458_SP20_Lecture18_LaplaceOperator.pdf\">Here<\/a> are the slides used for the lecture.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the next two lectures we&#8217;ll take a deep dive into one of the most important objects not only in discrete differential geometry, but in differential geometry at large (not to mention physics!): the Laplace-Beltrami operator. This operator generalizes the familiar Laplacian you may have studied in vector calculus, which just gives the sum of &hellip; <a href=\"https:\/\/brickisland.net\/DDGSpring2024\/2024\/03\/23\/lectures-17-18-laplace-beltrami\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Lectures 17 &#038; 18\u2014Laplace Beltrami&#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\/1959"}],"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=1959"}],"version-history":[{"count":4,"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/posts\/1959\/revisions"}],"predecessor-version":[{"id":2388,"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/posts\/1959\/revisions\/2388"}],"wp:attachment":[{"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/media?parent=1959"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/categories?post=1959"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/brickisland.net\/DDGSpring2024\/wp-json\/wp\/v2\/tags?post=1959"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}