{"id":114,"date":"2017-09-12T19:16:39","date_gmt":"2017-09-12T22:16:39","guid":{"rendered":"http:\/\/www.professores.uff.br\/ralphteixeira\/?page_id=114"},"modified":"2017-09-12T19:35:03","modified_gmt":"2017-09-12T22:35:03","slug":"equidistants","status":"publish","type":"page","link":"https:\/\/www.professores.uff.br\/ralphteixeira\/equidistants\/","title":{"rendered":"Equidistants"},"content":{"rendered":"<p>&nbsp;<\/p>\n<h4><b>Definition<\/b><\/h4>\n<p>Given a CPOS polygon <em>P<sub>1<\/sub>P<sub>2<\/sub>\u2026P<sub>2n<\/sub><\/em> and a real number <em>\u03bb<\/em>, we define its <strong><em>\u03bb-equidistant polygon<\/em><\/strong> <strong><em>P(<\/em><\/strong><strong><em>\u03bb)<\/em><\/strong> by the vertices which split the great diagonals in the ratio <em>\u03bb:(1- \u03bb)<\/em>. More explicitly, the vertices of <em>P(<\/em><em>\u03bb)<\/em> are<\/p>\n<p style=\"text-align: center;\"><em>P<sub>i<\/sub>(<\/em><em>\u03bb)=P<sub>i<\/sub> + \u03bb(P<sub>n+i<\/sub>-P<sub>i<\/sub>),\u00a0\u00a0 i=1,2,\u2026,2n<\/em><\/p>\n<p>It is easy to see that:<\/p>\n<ul>\n<li>The <em>0<\/em>-equidistant is the original polygon (and so is the <em>1<\/em>-equidistant);<\/li>\n<li>Actually, the <em>\u03bb<\/em>-equidistant coincides with the <em>(1- \u03bb)<\/em>-equidistant, just reordered;<\/li>\n<li>The <em>5<\/em>-equidistant is the Area Evolute (AE), traveled twice;<\/li>\n<li>All equidistants are POSPs. Not all of them are convex, but there must be a neighborhood of <em>\u03bb=0<\/em> which produces convex equidistants;<\/li>\n<li>All (convex) equidistants have the same Central Symmetry Set (CSS) and the same Area Evolute (AE).<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h4><b>Cusps of the equidistants and the CSS<\/b><\/h4>\n<p>When the <em>\u03bb<\/em>-triangle <em>P<sub>i-1<\/sub>P<sub>i<\/sub>P<sub>i+1<\/sub><\/em> has a different orientation than the original <em>P<sub>i-1<\/sub>P<sub>i<\/sub>P<sub>i+1<\/sub><\/em>, we say that <em>P<sub>i<\/sub>(<\/em><em>\u03bb)<\/em> is a <em>cusp of the \u03bb-equidistant<\/em>.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><em><strong><span style=\"color: #539494;\">The 0.2-equidistant of a CPOS polygon (pink). Labeled green vertices are cusps.<\/span><\/strong><\/em><\/p>\n<p><\/BR>In this next <a href=\"http:\/\/www.geogebra.org\/\">Geogebra<\/a> Applet, you can construct your 10-sided CPOSP as before (moving P<sub>1<\/sub>-P<sub>9<\/sub>) and play with the <em>\u03bb<\/em>-equidistants (drag the <em>\u03bb<\/em>-slider or press the \u201cplay\u201d button on the lower left corner; you can use the mouse wheel to zoom in or out AND erase previous cusp traces, if you wish). Whenever a vertex of the equidistant turns into a cusp, we mark it green. As you move <em>\u03bb<\/em>, what do they trace?<\/p>\n<p style=\"text-align: left;\">\u00a0\u00a0 \u00a0\u00a0This 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>&nbsp;<\/p>\n<p>In fact (see [1] for proofs):<\/p>\n<p>\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Consecutive edges of the equidistant form a cusp if, and only if, such edges <em>do not cross<\/em> the corresponding great diagonal;<\/p>\n<p>\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <strong>The cusps of the equidistants form the Central Symmetry Set of the original CPOSP.<\/strong><br \/>\n<\/BR><\/p>\n<h4><b>Reference<\/b><\/h4>\n<p>[1] M. Craizer, R. Teixeira and M. Horta, \u201cParallel Opposite Sides Polygons\u201d, preprint.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>&nbsp; Definition Given a CPOS polygon P1P2\u2026P2n and a real number \u03bb, we define its \u03bb-equidistant polygon P(\u03bb) by the vertices which split the great diagonals in the ratio \u03bb:(1- \u03bb). More explicitly, the vertices of P(\u03bb) are Pi(\u03bb)=Pi + \u03bb(Pn+i-Pi),\u00a0\u00a0 i=1,2,\u2026,2n It is easy to see that: The 0-equidistant is the original polygon (and [&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-114","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/114","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=114"}],"version-history":[{"count":14,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/114\/revisions"}],"predecessor-version":[{"id":129,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/114\/revisions\/129"}],"wp:attachment":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/media?parent=114"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/categories?post=114"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/tags?post=114"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}