{"id":485,"date":"2019-06-29T21:59:57","date_gmt":"2019-06-29T21:59:57","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=485"},"modified":"2019-11-19T00:18:53","modified_gmt":"2019-11-19T00:18:53","slug":"ellipsoidal-paths","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/ellipsoidal-paths\/","title":{"rendered":"Ellipsoidal Paths"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Given an\u00a0ellipse, and a smaller ellipse strictly inside it, start at a point on the outer ellipse, and in a counterclockwise fashion (say), follow a line tangent to the inner ellipse until you hit the outer ellipse again. Repeat. Figure 1 shows an example. <\/p>\n\n\n\n

Now it is quite possible that this path will never hit the same points on the outer ellipse twice. But if it does “close up” in a certain number of steps, then something amazing is true: all<\/em> such paths, starting at any point<\/em> on the outer ellipse, close up in the same number of steps!<\/p>\n\n\n\n

This fact is known as Poncelet’s Theorem<\/em>.<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Intuition may be gained by presenting special cases, such as where the ellipses are concentric circles.<\/p>\n\n\n\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
This process that produces this path may be thought of as a\u00a0dynamical system<\/em>\u00a0on the outer ellipse, and is related to the study of\u00a0circle maps<\/em>\u00a0and\u00a0rotation numbers<\/em>\u00a0in\u00a0dynamical systems. You can learn more about Poncelet’s theorem in any classical text on algebraic geometry.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Ellipsoidal Paths.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

Fun Fact suggested by<\/strong>:
Jorge Aarao, Johannes Huisman <\/p>\n","protected":false},"excerpt":{"rendered":"

Given an\u00a0ellipse, and a smaller ellipse strictly inside it, start at a point on the outer ellipse, and in a...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[96,20,7,22,4],"class_list":["post-485","page","type-page","status-publish","hentry","tag-algebraic-geometry","tag-analysis","tag-calculus","tag-ellipse","tag-medium"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/485","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=485"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/485\/revisions"}],"predecessor-version":[{"id":1418,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/485\/revisions\/1418"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=485"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}