{"id":80,"date":"2017-09-12T18:40:56","date_gmt":"2017-09-12T21:40:56","guid":{"rendered":"http:\/\/www.professores.uff.br\/ralphteixeira\/?page_id=80"},"modified":"2017-09-13T18:37:28","modified_gmt":"2017-09-13T21:37:28","slug":"parallel-opposite-sides-polygons-posp","status":"publish","type":"page","link":"https:\/\/www.professores.uff.br\/ralphteixeira\/parallel-opposite-sides-polygons-posp\/","title":{"rendered":"Parallel Opposite Sides Polygons (POSP)"},"content":{"rendered":"

<\/BR><\/p>\n

Definition<\/B><\/h4>\n

As the name indicates, a plane polygon with 2n<\/em> vertices P1<\/sub>P2<\/sub>\u2026P2n<\/sub><\/em> is called Parallel Opposite Sides<\/em><\/strong> when Pi<\/sub>Pi+1<\/sub><\/em> is parallel to Pi+n<\/sub>Pi+n+1<\/sub><\/em> for i<\/em>=1,2,\u2026,n<\/em> (indices are taken modulo 2n<\/em>, so P2n+1<\/sub>=P1<\/sub><\/em>). Note that such property is invariant by affine transformations! In these pages, we are interested in Convex<\/em><\/strong> POS polygons.<\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 n=2<\/em>: all 4-sided POSPs are parallelograms; they are all equivalent via affine transformations of the plane.<\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 n\u22653<\/em>: there are 3n degrees of freedom when defining a 2n<\/em>-sided POSP (of which, 6 are spent in affine transformations).<\/p>\n

 <\/p>\n

Construct your own POS decagon<\/B><\/h4>\n

You can use the Applet below (created with Geogebra<\/a>) to construct your own 10-sided POS polygon!<\/p>\n

This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com<\/p>\n

You can move P1<\/sub>, P2<\/sub>, \u2026, P6<\/sub><\/em> at will, defining the 5 directions of the sides (12 degrees of freedom);
\nYou can move P7<\/sub><\/em>, P8<\/sub><\/em> and P9<\/sub><\/em> along the dotted lines (3 more degrees);
\nFinally, the Applet will determine P10<\/sub><\/em> in order to close a POS. Can you see how this is done?<\/p>\n

In the next pages<\/a> we shall explore some properties of convex POS polygons.<\/p>\n

 <\/p>\n

References<\/B><\/h4>\n

[1] M. Craizer, R. Teixeira and M. Horta, \u201cParallel Oposite Side Polygons\u201d, preprint.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Definition As the name indicates, a plane polygon with 2n vertices P1P2\u2026P2n is called Parallel Opposite Sides when PiPi+1 is parallel to Pi+nPi+n+1 for i=1,2,\u2026,n (indices are taken modulo 2n, so P2n+1=P1). Note that such property is invariant by affine transformations! In these pages, we are interested in Convex POS polygons. \u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 n=2: all 4-sided […]<\/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-80","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/80","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=80"}],"version-history":[{"count":7,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/80\/revisions"}],"predecessor-version":[{"id":212,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/80\/revisions\/212"}],"wp:attachment":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/media?parent=80"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/categories?post=80"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/tags?post=80"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}