{"id":89,"date":"2017-09-12T18:48:00","date_gmt":"2017-09-12T21:48:00","guid":{"rendered":"http:\/\/www.professores.uff.br\/ralphteixeira\/?page_id=89"},"modified":"2017-09-12T19:15:25","modified_gmt":"2017-09-12T22:15:25","slug":"area-evolute-and-central-symmetry-set","status":"publish","type":"page","link":"https:\/\/www.professores.uff.br\/ralphteixeira\/area-evolute-and-central-symmetry-set\/","title":{"rendered":"Area Evolute and Central Symmetry Set"},"content":{"rendered":"<p>&nbsp;<\/p>\n<p>Given a convex parallel opposite sides polygon (CPOSP) <em>P<sub>1<\/sub>P<sub>2<\/sub>\u2026P<sub>2n<\/sub> <\/em>we can define:<\/p>\n<ul>\n<li>Its <strong><em>Great Diagonals:<\/em> <\/strong>segments <strong><em>d<sub>i<\/sub><\/em><\/strong> that join <em>P<sub>i<\/sub><\/em> and <em>P<sub>i+n<\/sub><\/em> (<em>i<\/em><em>=1,2,\u2026,n<\/em>).<\/li>\n<li>Its<strong><em>Area Evolute (AE):<\/em><\/strong> polygon whose vertices <strong><em>M<sub>i<\/sub><\/em><\/strong> are the midpoints of the Great Diagonals.<\/li>\n<li>Its <strong><em>Central Symmetry Set (CSS):<\/em><\/strong> polygon whose vertices <strong><em>D<sub>i+\u00bd<\/sub><\/em><\/strong> are intersections of successive great diagonals.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<div id=\"01\" style=\"text-align: center;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/www.professores.uff.br\/ralphteixeira\/wp-content\/uploads\/sites\/129\/2017\/09\/image002-300x173.png\" alt=\"\" width=\"300\" height=\"173\" \/><\/p>\n<p style=\"text-align: center;\"><em><strong><span style=\"color: #529e9e;\">Figure 1. AE (red) and CSS (green) of a CPOSP<\/span><\/strong><\/em><\/p>\n<\/div>\n<p>Note that:<\/p>\n<ul>\n<li>The AE reduces to a point if, and only if, the polygon is <em>symmetric<\/em> with relation to that point (by definition);<\/li>\n<li>The CSS of a CPOSP reduces to a point if, and only if, the polygon is symmetric with relation to that point.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3><\/h3>\n<h4><strong>Cusps of the AE and the CSS<\/strong><\/h4>\n<ul>\n<li>A cusp of the CSS is a vertex <em>D<sub>i+\u00bd <\/sub><\/em>such that the orientations of the triangles <em>D<sub>i-1+\u00bd<\/sub> D<sub>i+\u00bd<\/sub> D<sub>i+1+\u00bd<\/sub><\/em> and <em>P<sub>i-1<\/sub>P<sub>i<\/sub>P<sub>i+1<\/sub><\/em> are reversed.<\/li>\n<li>A cusp of the AE is a vertex <em>M<sub>i<\/sub><\/em> such that the orientations of the triangles <em>M<sub>i-1<\/sub>M<sub>i<\/sub>M<sub>i+1<\/sub><\/em> and <em>P<sub>i-1<\/sub>P<sub>i<\/sub>P<sub>i+1<\/sub><\/em> are reversed (equivalently: <em>M<sub>i<\/sub><\/em> is a cusp if <em>M<sub>i-1<\/sub><\/em> and <em>M<sub>i+1<\/sub><\/em> are on the same side of <em>d<sub>i<\/sub><\/em>).<\/li>\n<\/ul>\n<p>In this Geogebra applet, the vertices of the AE and of the CSS are marked red whenever they turn into cusps (as you play with the outer 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 www.java.com<\/p>\n<p>For non-degenerate positions (see [1] for proofs):<\/p>\n<ul>\n<li>The number of cusps in the CSS is odd and at least 3;<\/li>\n<li>The number of cusps in the AE is odd and at least 3;<\/li>\n<li>A vertex of the AE is a cusp if, and only if, it belongs to the CSS.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3><\/h3>\n<h4><strong>Alternate CSS cusp definition<\/strong><\/h4>\n<p>If we define <em>\u03bb<sub>i+\u00bd<\/sub><\/em> (<em>i<\/em><em>=1,2,\u2026,2n<\/em>) by the equations<\/p>\n<p style=\"text-align: center;\"><em>D<sub>i+\u00bd<\/sub> = P<sub>i<\/sub> + \u03bb<\/em><em><sub>i+\u00bd<\/sub><\/em><em>\u00b7(P<sub>i+n<\/sub> &#8211; P<sub>i<\/sub>) = P<sub>i+1<\/sub> + \u03bb<\/em><em><sub>i+\u00bd<\/sub><\/em><em>\u00b7(P<sub>i+1+n<\/sub> &#8211; P<sub>i+1<\/sub>)<\/em><\/p>\n<p>then we have<\/p>\n<p style=\"text-align: center;\"><em>D<sub>i+\u00bd<\/sub> &#8211; D<sub>i-1+\u00bd<\/sub> = (\u03bb<sub>i+\u00bd<\/sub> &#8211; \u03bb<sub>i-1+\u00bd<\/sub>)(P<sub>i+n<\/sub> &#8211; P<sub>i<\/sub>)<\/em><\/p>\n<p>Now, in a CPOSP, the great diagonals <em>P<sub>i+n<\/sub><\/em><em> &#8211; P<sub>i<\/sub><\/em> \u201cturn\u201d in the same orientation as the polygon itself. Therefore the triangles <em>D<sub>i-1+\u00bd<\/sub> D<sub>i+\u00bd<\/sub> D<sub>i+1+\u00bd<\/sub><\/em> and <em>P<sub>i-1<\/sub>P<sub>i<\/sub>P<sub>i+1<\/sub><\/em> have opposite orientations if and only if<\/p>\n<p style=\"text-align: center;\"><em>(\u03bb<sub>i+<\/sub><sub>\u00bd<\/sub> &#8211; \u03bb<sub>i-1+\u00bd<\/sub>)(\u03bb<sub>i+<\/sub><sub>1+\u00bd<\/sub> &#8211; \u03bb<sub>i+\u00bd<\/sub>)&lt;0<\/em><\/p>\n<p>In short<\/p>\n<p style=\"text-align: center;\"><strong><em>D<sub>i+\u00bd<\/sub><\/em> is a cusp of the CSS if, and only if, <em>\u03bb<sub>i+\u00bd<\/sub><\/em> is a local extremum of the cyclic sequence {<em>\u03bb<sub>i+\u00bd<\/sub><\/em>}<\/strong><\/p>\n<p>&nbsp;<\/p>\n<h3><\/h3>\n<h4><strong>References<\/strong><\/h4>\n<p>[1] M. Craizer, R. Teixeira and M. Horta, \u201cParallel Opposite Side Polygons\u201d, preprint.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>&nbsp; Given a convex parallel opposite sides polygon (CPOSP) P1P2\u2026P2n we can define: Its Great Diagonals: segments di that join Pi and Pi+n (i=1,2,\u2026,n). ItsArea Evolute (AE): polygon whose vertices Mi are the midpoints of the Great Diagonals. Its Central Symmetry Set (CSS): polygon whose vertices Di+\u00bd are intersections of successive great diagonals. &nbsp; Figure [&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-89","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/89","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=89"}],"version-history":[{"count":21,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/89\/revisions"}],"predecessor-version":[{"id":113,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/89\/revisions\/113"}],"wp:attachment":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/media?parent=89"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/categories?post=89"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/tags?post=89"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}