{"id":395,"date":"2019-06-26T23:53:55","date_gmt":"2019-06-26T23:53:55","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=395"},"modified":"2019-11-18T23:58:02","modified_gmt":"2019-11-18T23:58:02","slug":"eccentricity-of-conics","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/eccentricity-of-conics\/","title":{"rendered":"Eccentricity of Conics"},"content":{"rendered":"\n<p>To each\u00a0conic section\u00a0(ellipse, parabola, hyperbola) there is a number called the\u00a0<em>eccentricity<\/em>\u00a0that uniquely characterizes the shape of the curve. A circle has eccentricity 0, an ellipse between 0 and 1, a parabola 1, and hyperbolae have eccentricity greater than 1.<\/p>\n\n\n\n<p>Although you might think that y=2x<sup>2<\/sup>&nbsp;and y=x<sup>2<\/sup>&nbsp;have different &#8220;shapes&#8221; because the former is skinnier, they really have the same &#8220;shape&#8221; (and thus same eccentricity) because the first curve is just the second curve viewed twice as far away (i.e., x and y are both increased by a factor of 2).<\/p>\n\n\n\n<p>One way to define a conic section is to specify a line in the plane, called the&nbsp;<em>directrix<\/em>, and a point in the plane off of the line, called the&nbsp;<em>focus<\/em>. The conic section is then the set of all points whose distance to the focus is a constant times the distance to the directrix. This constant is the eccentricity.<\/p>\n\n\n\n<p>It is easy to see that as the eccentricity of an ellipse grows, the ellipse becomes skinnier. The formula for the\u00a0ellipse\u00a0also shows that every ellipse can be obtained by taking a circle in a plane, lifting it up and out, tilting it, and projecting it back into the plane.<\/p>\n\n\n\n<p>Surprise: the eccentricity is equal to the sine of the angle of this tilt!<\/p>\n\n\n\n<p><strong>Presentation\u00a0Suggestions:<\/strong><br>If students are puzzled why the circle has eccentricity zero, you might explain that its directrix is the line &#8220;at infinity&#8221; in the\u00a0projective plane.<\/p>\n\n\n\n<p><strong>The&nbsp;Math&nbsp;Behind&nbsp;the&nbsp;Fact:<\/strong><br>Conic sections take their name from the fact that one can also obtain them by slicing a cone by a plane at various angles. Yet another way to obtain a conic section is by starting with a circle and performing a geometric transformation called reciprocation. The focus, directrix and eccentricity fall out as obvious parameters of this reciprocation operation. This approach to conic sections comes from the field of&nbsp;<em>projective geometry<\/em>.<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;Eccentricity of Conics.&#8221;&nbsp;<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts&gt;.<\/p>\n\n\n\n<p><strong>References:<\/strong><br> H.S.M. Coxeter and S.L. Greitzer, Geometry Revisited. <br> (for the material on projective geometry).<\/p>\n\n\n\n<p><strong>Fun Fact suggested by<\/strong>:  <br>Aaron Archer <\/p>\n","protected":false},"excerpt":{"rendered":"<p>To each\u00a0conic section\u00a0(ellipse, parabola, hyperbola) there is a number called the\u00a0eccentricity\u00a0that uniquely characterizes the shape of the curve. A circle&#46;&#46;&#46;<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[3,22,8,4,21,95],"class_list":["post-395","page","type-page","status-publish","hentry","tag-easy","tag-ellipse","tag-geometry","tag-medium","tag-parabola","tag-projective-geometry"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/395","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=395"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/395\/revisions"}],"predecessor-version":[{"id":1413,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/395\/revisions\/1413"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=395"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=395"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}