{"id":178,"date":"2017-09-13T17:53:23","date_gmt":"2017-09-13T20:53:23","guid":{"rendered":"http:\/\/www.professores.uff.br\/ralphteixeira\/?page_id=178"},"modified":"2017-09-13T18:35:21","modified_gmt":"2017-09-13T21:35:21","slug":"cusps-and-the-affine-evolute","status":"publish","type":"page","link":"https:\/\/www.professores.uff.br\/ralphteixeira\/cusps-and-the-affine-evolute\/","title":{"rendered":"Cusps and the Affine Evolute"},"content":{"rendered":"

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

Definitions<\/B><\/h4>\n

Given a CEA polygon P1<\/sub>P2<\/sub>\u2026PN<\/sub>, we define:<\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The affine normal<\/em><\/strong> at Pi<\/sub> is the vector ni<\/sub>=Pi-1<\/sub>+Pi+1<\/sub>-2Pi<\/sub> (which is \u201ctwice the median\u201d of Pi-1<\/sub>Pi<\/sub>Pi+1<\/sub>);<\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The \u03bb-parallel polygon<\/em><\/strong> has vertices Pi<\/sub>(\u03bb)=Pi<\/sub>+\u03bb.ni<\/sub>. Its sides are indeed parallel to the sides of the original CEA polygon. As \u03bb varies, the \u03bb-parallel polygons perform an affine evolution<\/em><\/strong> of the original CEA polygon.<\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 When the \u03bb-triangle Pi-1<\/sub>Pi<\/sub>Pi+1<\/sub> has a different orientation than the original Pi-1<\/sub>Pi<\/sub>Pi+1<\/sub>, we say that Pi<\/sub>(\u03bb) is a cusp of the \u03bb-parallel polygon<\/em><\/strong>.<\/p>\n

\n

\"\"\n<\/div>\n

 <\/p>\n

Affine normals and the 0.6-parallel polygon. No cusps yet\u2026<\/span><\/p>\n

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

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The affine evolute <\/em><\/strong>is a polygonal curve with vertices qi<\/sub>= Pi<\/sub>(1\/\u03bci<\/sub>)=Pi<\/sub>+ni<\/sub>\/\u03bci<\/sub>. More precisely, if \u03bci<\/sub>\u03bci+1<\/sub>>0, the segment qi<\/sub>qi+1<\/sub> is part of the evolute; but, if \u03bci<\/sub>\u03bci+1<\/sub><0, the affine evolute includes the portion of the line qi<\/sub>qi+1<\/sub>outside<\/em><\/strong> the segment qi<\/sub>qi+1.<\/sub><\/p>\n

\n\"\"<\/p>\n

Affine evolute. Since \u03bc6<\/sub><0, \u201csides\u201d q5<\/sub>q6 <\/sub>and q6<\/sub>q7<\/sub> go through \u201cimproper points at infinity\u201d.<\/span><\/div>\n

 <\/p>\n

Cusps and the Affine Evolute<\/B><\/h4>\n

In the Geogebra<\/a> Applet below, you can construct your own CEA octagon (moving P1<\/sub>-P6<\/sub>) and play with the \u03bb-parallel polygons (dragging \u03bb \u2013 if your mouse hand is tired, just press the \u201cplay\u201d button on the lower left corner and we will move \u03bb around for you J). Whenever a vertex turns into a cusp, we mark it red.<\/p>\n

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now<\/a>)<\/p>\n

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

Things to note:<\/B><\/h4>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 In a CEAP, Pi<\/sub>(1\/\u03bci<\/sub>)= Pi+1<\/sub>(1\/\u03bci<\/sub>), so the vertices qi<\/sub> correspond to \u201cshocks\u201d of the Affine Evolution.<\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 In fact, the affine evolute turns out to be exactly the set of all cusps!<\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The affine evolute reduces to a single point if, and only if, the original polygon is regular (modulo an affine transformation).<\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 When a curvature associated to an edge of the original polygon is negative, that edge becomes red. For example, in the original position, P6<\/sub>P7<\/sub> is red because length(P5<\/sub>P8<\/sub>) > 3 .length(P6<\/sub>P7<\/sub>) and thus \u03bc6<\/sub><0. Note how P6<\/sub>(\u03bb) determines two disconnected branches of the Affine Evolute (and so does P7<\/sub>(\u03bb)).<\/p>\n

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

Reference<\/B><\/h4>\n

[2] M. Craizer, R. Teixeira and M. Horta, \u201cAffine properties of convex equal-area polygons\u201d; Discrete & Computational Geometry, October 2012, Volume 48, Issue 3, pp 580-595.<\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Definitions Given a CEA polygon P1P2\u2026PN, we define: \u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The affine normal at Pi is the vector ni=Pi-1+Pi+1-2Pi (which is \u201ctwice the median\u201d of Pi-1PiPi+1); \u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The \u03bb-parallel polygon has vertices Pi(\u03bb)=Pi+\u03bb.ni. Its sides are indeed parallel to the sides of the original CEA polygon. As \u03bb varies, the \u03bb-parallel polygons perform an affine evolution […]<\/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-178","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/178","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=178"}],"version-history":[{"count":16,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/178\/revisions"}],"predecessor-version":[{"id":209,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/pages\/178\/revisions\/209"}],"wp:attachment":[{"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/media?parent=178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/categories?post=178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.professores.uff.br\/ralphteixeira\/wp-json\/wp\/v2\/tags?post=178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}