{"id":130,"date":"2017-09-12T19:37:48","date_gmt":"2017-09-12T22:37:48","guid":{"rendered":"http:\/\/www.professores.uff.br\/ralphteixeira\/?page_id=130"},"modified":"2017-09-13T18:26:24","modified_gmt":"2017-09-13T21:26:24","slug":"the-dual-family","status":"publish","type":"page","link":"https:\/\/www.professores.uff.br\/ralphteixeira\/the-dual-family\/","title":{"rendered":"The Dual Family"},"content":{"rendered":"

<\/h4>\n

 <\/p>\n

Definition<\/strong><\/h4>\n

The midlines<\/strong> of a CPOS polygon are the lines which are equidistant from opposite sides (so, they connect successive midpoints Mi and support the AE). Now, given a CPOS polygon P=P1P2\u2026P2n<\/em>, it is possible (see [1] for proof) to construct another CPOS polygon Q=Q1Q2\u2026Q2n<\/em> such that:<\/p>\n

\u00b7 The great diagonals of Q are the midlines of P;<\/p>\n

\u00b7 The sides of Q are parallel to the great diagonals of P.<\/p>\n

In fact, given the polygon P, there is a 1-parameter family of such polygons Q, which we call the Dual Family of P<\/strong>.<\/p>\n

 <\/p>\n

\n

\"\"<\/p>\n

The orange polygon Q is a member of the dual family of P.<\/span><\/p>\n

 <\/p>\n

\n<\/div>\n

Next applet allows you to play with a CPOS polygon P, and generates a dual polygon Q. You can move the slider \u03b1 to see other members of the dual family.<\/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

Things to note:<\/p>\n

\u00b7 Whenever Q is convex, the CSS of Q is the AE of P;<\/p>\n

\u00b7 The members of the dual family are equidistants of each other, and all have the same AE;<\/p>\n

\u00b7 Hence, there is a member of the dual family which is this very AE, traveled twice \u2013 set \u03b1 to zero to see it!<\/p>\n

The AE of the Dual Family<\/strong><\/h4>\n

Now, call N=N1+\u00bdN2+\u00bd\u2026Nn+\u00bd<\/em> the AE of Q (where Ni+\u00bd is on the midline parallel to PiPi+1). Here are some properties relating P directly to N:<\/p>\n

\u00b7 If the line joining Pi<\/em> and Ni+\u00bd<\/em> intersects the segment Pi+nPi+n+1<\/em><\/strong>, then it divides the polygon P in two regions of the same area (proof in [1]);<\/p>\n

\u00b7 Moreover, in this case, since Ni+\u00bd<\/em> is on the midline, it will be the midpoint of the chord that such line determines inside P.<\/p>\n

 <\/p>\n

\n

\"\"<\/p>\n

The chord through P1 and N1+\u00bd splits areas equally!<\/span><\/p>\n

 <\/p>\n

Reference<\/strong><\/h4>\n<\/div>\n

[1] M. Craizer, R. Teixeira and M. Horta, \u201cParallel Opposite Sides Polygons\u201d, preprint.<\/p>\n","protected":false},"excerpt":{"rendered":"

  Definition The midlines of a CPOS polygon are the lines which are equidistant from opposite sides (so, they connect successive midpoints Mi and support the AE). Now, given a CPOS polygon P=P1P2\u2026P2n, it is possible (see [1] for proof) to construct another CPOS polygon Q=Q1Q2\u2026Q2n such that: \u00b7 The great diagonals of Q are […]<\/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-130","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/130","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=130"}],"version-history":[{"count":11,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/130\/revisions"}],"predecessor-version":[{"id":203,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/130\/revisions\/203"}],"wp:attachment":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/media?parent=130"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/categories?post=130"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/tags?post=130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}