{"id":164,"date":"2017-09-12T20:41:19","date_gmt":"2017-09-12T23:41:19","guid":{"rendered":"http:\/\/www.professores.uff.br\/ralphteixeira\/?page_id=164"},"modified":"2017-09-13T18:32:10","modified_gmt":"2017-09-13T21:32:10","slug":"convex-equal-area-polygons-ceap","status":"publish","type":"page","link":"https:\/\/www.professores.uff.br\/ralphteixeira\/convex-equal-area-polygons-ceap\/","title":{"rendered":"Convex Equal-Area Polygons (CEAP)"},"content":{"rendered":"<p><\/BR><\/p>\n<h4><B>Definition<\/B><\/h4>\n<p>A plane polygon is called <strong><em>Equal-Area<\/em><\/strong> when all triangles formed by 3 consecutive vertices have the same area (see <a href=\"#anc01\">[1]<\/a>). This property is invariant by Affine Transformations \u2013 when applying an Affine Transformation to a polygon, all areas are multiplied by a common factor, so they remain equal to each other.<\/p>\n<div id=\"01\" style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-168\" src=\"http:\/\/www.professores.uff.br\/ralphteixeira\/wp-content\/uploads\/sites\/129\/2017\/09\/Equal-Area-300x251.png\" alt=\"\" width=\"388\" height=\"324\" srcset=\"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-content\/uploads\/sites\/129\/2017\/09\/Equal-Area-300x251.png 300w, https:\/\/www.professores.uff.br\/ralphteixeira\/wp-content\/uploads\/sites\/129\/2017\/09\/Equal-Area.png 388w\" sizes=\"auto, (max-width: 388px) 100vw, 388px\" \/><\/div>\n<p style=\"text-align: center;\"><span style=\"color: #609e9e;\">Highlighted areas are equal to each other<\/span><\/p>\n<p>We are interested in <strong><em>Convex<\/em><\/strong> Equal-Area polygons with <em>N<\/em> sides. What do they look like?<\/p>\n<ul>\n<li><em>N=3<\/em>: every triangle is CEA. Unremarkable. L<\/li>\n<li><em>N=4<\/em>: a CEA quadrilateral must be a parallelogram.<\/li>\n<li><em>N=5<\/em>: every CEA pentagon is affinely equivalent to the regular one!<\/li>\n<li><em>N<\/em><em>\u2265<\/em><em>5<\/em>: discounting affine transformations, there are <em>N-5<\/em> degrees of freedom in your choice of a CEAP (<a href=\"#anc01\">[1]<\/a> or <a href=\"#anc02\">[2]<\/a>.<\/li>\n<\/ul>\n<p><\/BR><\/p>\n<h4><B>Construct your own CEA decagon<\/B><\/h4>\n<p>You can use the Applet below (created with <a href=\"http:\/\/www.geogebra.org\/\">Geogebra<\/a>) to construct your own 10-sided CEA polygon!<\/p>\n<p>This is a Java Applet created using GeoGebra from www.geogebra.org &#8211; it looks like you don&#8217;t have Java installed, please go to <a href=\"http:\/\/www.java.com\">www.java.com<\/a><\/p>\n<p>You can move P<sub>1<\/sub>, P<sub>2<\/sub> and P<sub>3<\/sub> at will (but they just apply an affine transformation to the whole figure).<br \/>\nYou can move P<sub>4<\/sub>, P<sub>5<\/sub>, P<sub>6<\/sub>, P<sub>7<\/sub> and P<sub>8<\/sub> along the dotted lines (5 degrees of freedom).<br \/>\nFrom P<sub>4<\/sub> on, each vertex only affects the ones with higher indices.<br \/>\nThe applet will determine P<sub>9<\/sub> and P<sub>10<\/sub> in order to close a convex equal-area polygon.<br \/>\n<\/BR><\/p>\n<h4><B>How does it work?<\/B><\/h4>\n<p>There are two keys to the construction of P<sub>9<\/sub> and P<sub>10<\/sub>:<\/p>\n<ul>\n<li>The hyperbola <em>h<\/em> through P<sub>7<\/sub>, with asymptotes r<sub>1<\/sub>=P<sub>1<\/sub>P<sub>8<\/sub> and r<sub>2<\/sub>=P<sub>6<\/sub>P<sub>9<\/sub>, must have a second branch going through P<sub>10<\/sub>.<\/li>\n<li>If P<sub>3<\/sub>, I and K are lined up in this order, the correct branch of <em>h<\/em> intersects r<sub>3 <\/sub>(usually at two points) <strong>if and only if <em>area(<\/em><em>IJK)<\/em><\/strong><strong><em> \u22654 area(P<sub>1<\/sub>P<sub>2<\/sub>P<sub>3<\/sub>)<\/em><\/strong>.<\/li>\n<li>If I, P<sub>3<\/sub> and K are lined up in this order, the correct branch of <em>h<\/em> intersects r<sub>3<\/sub> at one point, which is P<sub>10<\/sub> (the alternative intersection does not give us a convex polygon).<\/li>\n<\/ul>\n<p><\/BR><\/p>\n<h4><B>Open Questions<\/B><\/h4>\n<p>\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Given a strictly convex curve C, a point P<sub>1<\/sub> on it, and a number N, we can <em>approximate<\/em> C by a CEAP passing through P<sub>1<\/sub> (essentially doing the construction above; details in <a href=\"#anc02\">[2]<\/a>). Now, can we always find an N-sided CEAP passing through P<sub>1<\/sub> <em>inscribed<\/em> in C?<br \/>\n[We believe the answer is \u201cyes\u201d if N is odd \u2013 but we think the answer is \u201cno\u201d for even N!]<\/p>\n<ul>\n<li>Starting from a regular polygon, and moving the vertices continuously, can we get to any CEAP \u2013 keeping it CEA throughout the whole process? In other words, is the space of all N-sided CEAPs connected?<br \/>\n[We believe the answer is \u201cyes\u201d.]<\/li>\n<\/ul>\n<p><\/BR><\/p>\n<h4><B>References<\/B><\/h4>\n<p><a name=\"anc01\">[1]<\/a> G.Harel and J.M.Rabin, \u201cPolygons whose vertex triangles have equal area\u201d; Amer. Math. Monthly 110 (2003), 606-619.<\/p>\n<p><a name=\"anc02\">[2]<\/a> M. Craizer, R. Teixeira and M. Horta, \u201cAffine properties of convex equal-area polygons\u201d; Discrete &amp; Computational Geometry, October 2012, Volume 48, Issue 3, pp 580-595.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Definition A plane polygon is called Equal-Area when all triangles formed by 3 consecutive vertices have the same area (see [1]). This property is invariant by Affine Transformations \u2013 when applying an Affine Transformation to a polygon, all areas are multiplied by a common factor, so they remain equal to each other. Highlighted areas are [&hellip;]<\/p>\n","protected":false},"author":46,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"categories":[],"tags":[],"class_list":["post-164","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/164","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/users\/46"}],"replies":[{"embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/comments?post=164"}],"version-history":[{"count":8,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/164\/revisions"}],"predecessor-version":[{"id":207,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/164\/revisions\/207"}],"wp:attachment":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/media?parent=164"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/categories?post=164"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/tags?post=164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}