{"id":404,"date":"2013-08-21T16:00:00","date_gmt":"2013-08-21T20:00:00","guid":{"rendered":"https:\/\/jasoncantarella.com\/wordpress\/?p=404"},"modified":"2014-06-04T14:17:21","modified_gmt":"2014-06-04T18:17:21","slug":"radiosity-in-math-4510-applied-math-course","status":"publish","type":"post","link":"https:\/\/jasoncantarella.com\/wordpress\/radiosity-in-math-4510-applied-math-course\/","title":{"rendered":"Radiosity in Math 4510 applied math course"},"content":{"rendered":"<figure id=\"attachment_405\" aria-describedby=\"caption-attachment-405\" style=\"width: 420px\" class=\"wp-caption alignleft\"><a href=\"https:\/\/jasoncantarella.com\/wordpress\/wp-content\/uploads\/2013\/08\/64_sphere_radiosity1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-405 \" src=\"https:\/\/jasoncantarella.com\/wordpress\/wp-content\/uploads\/2013\/08\/64_sphere_radiosity1.png\" alt=\"64_sphere_radiosity\" width=\"420\" height=\"419\" srcset=\"https:\/\/jasoncantarella.com\/wordpress\/wp-content\/uploads\/2013\/08\/64_sphere_radiosity1.png 420w, https:\/\/jasoncantarella.com\/wordpress\/wp-content\/uploads\/2013\/08\/64_sphere_radiosity1-150x150.png 150w, https:\/\/jasoncantarella.com\/wordpress\/wp-content\/uploads\/2013\/08\/64_sphere_radiosity1-300x300.png 300w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><\/a><figcaption id=\"caption-attachment-405\" class=\"wp-caption-text\">A 2048 polygon radiosity scene. You can see light being absorbed and reradiated among the polyhedra in the scene; only two of the polyhedra are light sources.<\/figcaption><\/figure>\n<p>When ambient light is reflected and reradiated in an image, finding the final distribution of light in the image requires the solution of a very large linear algebra problem. The total light <i>R_i\u00a0<\/i>radiating from a face\u00a0<i>i<\/i>\u00a0is given by R_i = E_i +\u00a0\u03a3 F_ij R_j, where\u00a0<sub><i>i<\/i><\/sub>\u00a0is the light emitted by face\u00a0<i>i<\/i>\u00a0and the\u00a0<i>F<\/i><sub><i>i<\/i><i>j<\/i><\/sub>\u00a0are &#8220;view factors&#8221; describing the relative geometry of faces\u00a0<i>i\u00a0<\/i>and\u00a0<i>j<\/i>. In the scene on the left, there are 64 polyhedra, each with 32 faces. The resulting system of 2048 linear equations in 2048 variables is solved in several tenths of a second using an iterative method implemented in Mathematica, but would take much longer with a standard solver. In a real application, like a scene from an animated movie, there would be a few million polygons in the scene and the resulting solution would require solving a system\u00a0<i>A<\/i><i>x<\/i>\u00a0=\u00a0<i>b<\/i>\u00a0where the matrix\u00a0<i>A<\/i>\u00a0was (say) 2,000,000 by 2,000,000. Even storing such a matrix would take\u00a0on the order of 4 terabytes of memory! Luckily, such matrices are very sparse, so they are (barely) tractable with good computing hardware.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>When ambient light is reflected and reradiated in an image, finding the final distribution of light in the image requires the solution of a very large linear algebra problem. The total light R_i\u00a0radiating from a face\u00a0i\u00a0is given by R_i = E_i +\u00a0\u03a3 F_ij R_j, where\u00a0i\u00a0is the light emitted by face\u00a0i\u00a0and the\u00a0Fij\u00a0are &#8220;view factors&#8221; describing the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,6],"tags":[],"class_list":["post-404","post","type-post","status-publish","format-standard","hentry","category-art","category-math"],"_links":{"self":[{"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/posts\/404","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/comments?post=404"}],"version-history":[{"count":2,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/posts\/404\/revisions"}],"predecessor-version":[{"id":674,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/posts\/404\/revisions\/674"}],"wp:attachment":[{"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/media?parent=404"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/categories?post=404"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jasoncantarella.com\/wordpress\/wp-json\/wp\/v2\/tags?post=404"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}