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209 lines
9.8 KiB
HTML
209 lines
9.8 KiB
HTML
<!DOCTYPE html>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Conic Sections</h1>
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<p class="center"><i>Conic Section: a section (or slice) through a cone</i>.</p>
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<div class="large">
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<div class="center80">Did you know that by taking different slices through a cone you can create a circle, an ellipse, a parabola or a hyperbola?</div>
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</div>
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<p> </p>
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<div class="large" style="text-align:center;">
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<div style="float:left; margin: 0 10px 5px 0;"> <a href="cone.html"><img src="images/conic-solid.jpg" alt="cones" height="200" width="127"><br>
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Cones</a><br>
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</div>
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<div style="float:left; margin: 0 10px 5px 0;"> <a href="circle.html"><img src="images/conic-circle.jpg" alt="conic section circle" height="200" width="150"><br>
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Circle</a><br>
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straight through </div>
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<div style="float:left; margin: 0 10px 5px 0;"> <a href="ellipse.html"><img src="images/conic-ellipse.jpg" alt="conic section ellipse" height="200" width="133"><br>
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Ellipse</a><br>
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slight angle </div>
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<div style="float:left; margin: 0 10px 5px 0;"> <a href="parabola.html"><img src="images/conic-parabola.jpg" alt="conic section parabola" height="200" width="122"><br>
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Parabola</a><br>
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parallel to edge<br>
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of cone </div>
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<div style="float:left; margin: 0 10px 5px 0;"> <a href="hyperbola.html"><img src="images/conic-hyperbola2.jpg" alt="conic section hyperbola" height="200" width="116"><br>
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Hyperbola</a><br>
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steep angle </div>
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<div style="clear:both"></div>
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</div>
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<p class="center larger">So all those curves are related.</p>
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<h2>Focus!</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/focus-directrix-ratio.svg" alt="focus and directrix" height="248" width="315"></p>
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<p>The curves can also be defined using a straight line (the <b>directrix</b>) and a point (the <b>focus</b>).</p>
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<p>When we measure the distance:</p>
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<ul>
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<li>from the <b>focus</b> to any point on the curve, and</li>
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<li>perpendicularly from the <b>directrix</b> to that point</li>
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</ul>
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<p>the two distances will always have the same ratio.</p>
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<ul>
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<li>For an ellipse, the ratio is less than 1</li>
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<li>For a parabola, the ratio is 1 (so the two distances are <b>equal</b>)</li>
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<li>For a hyperbola, the ratio is greater than 1</li></ul>
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<p>That ratio is called the <a href="eccentricity.html">eccentricity</a>. Play with it here:</p>
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<div class="script" style="height: 400px;">
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images/eccentricity-graph.js
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</div>
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<h2>Eccentricity</h2>
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<p>We can say that any conic section is:</p>
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<div class="def">
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<p class="center larger">"all points whose distance to the <b>focus</b> is equal<br>
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to
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the <b>eccentricity</b> times the distance to the <b>directrix</b>"</p>
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</div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/eccentricity.svg" alt="Eccentricity" height="350" width="300"></p>
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<p>For:</p>
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<ul>
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<li>0 < <b>eccentricity</b> < 1 we get an ellipse,</li>
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<li><b>eccentricity</b> = 1 a parabola, and</li>
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<li><b>eccentricity</b> > 1 a hyperbola.</li>
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</ul>
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<p>A circle has an <b>eccentricity of zero</b>, so the eccentricity shows us how "un-circular" the curve is. The bigger the eccentricity, the less curved it is.</p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: Orbits have an eccentricity less than 1</h3>
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<p>An eccentricity above 1 is is not really an orbit as it does not loop back, but passes by.</p>
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<p class="center"><img src="images/oumuamua.jpg" alt="oumuamua" height="195" width="360"><br>
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Artist's Impression of <i>'Oumuamua</i><br>
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Credit: ESO/M. Kornmesser</p>
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<p>The interstellar asteroid <i>'Oumuamua</i> has an eccentricity of about <b>1.2</b> in it's path around the Sun, meaning it is not part of our solar system:</p>
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<p class="center"><img src="images/oumuamua-orbit.jpg" alt="oumuamua orbit" height="254" width="360"><br>
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Credit: Wikpedia authors nagualdesign and Tomruen</p>
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<p>The orbit of Earth has an eccentricity of about 0.0167 (nearly a circle)<br>
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The orbit of Mars has an eccentricity of about 0.0934 (a little less circular)</p>
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</div>
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<h2>Latus Rectum</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/latus-rectum.svg" alt="latus rectum" height="162" width="254"></p>
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<p>The <b>latus rectum</b> (no, it is not a rude word!) runs parallel to the directrix and passes through the focus. Its length:</p>
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<ul>
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<li>In a parabola, is four times the focal length</li>
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<li>In a circle, is the diameter</li>
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<li>In an ellipse, is 2b<sup>2</sup>/a (where a and b are one half of the major and minor diameter).</li>
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</ul>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/ellipse-directrix-focus.svg" alt="ellipse directrix, focus and latus rectum" height="176" width="303"></p>
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<p>Here is the <b>major axis</b> and <b>minor axis</b> of an ellipse.</p>
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<p>There is a focus and directrix <b>on each side</b> (ie a pair of them).</p>
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<div style="clear:both"></div>
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<h2>Equations</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/ellipse-cartesian.svg" alt="ellipse on xy graph" height="223" width="260"></p>
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<p>When placed like this on an x-y graph, the equation for an ellipse is:</p>
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<p class="center large"><span class="intbl"><em>x<sup>2</sup></em><strong>a<sup>2</sup></strong></span> + <span class="intbl"><em>y<sup>2</sup></em><strong>b<sup>2</sup></strong></span> = 1</p>
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<p>The special case of a circle (where radius=a=b) is:</p>
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<p class="center large"><span class="intbl"><em>x<sup>2</sup></em><strong>a<sup>2</sup></strong></span> + <span class="intbl"><em>y<sup>2</sup></em><strong>a<sup>2</sup></strong></span> = 1</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/hyperbola6.svg" alt="hyperbola on xy graph" height="220" width="260"></p>
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<p>And for a hyperbola it is:</p>
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<p class="center large"><span class="intbl"><em>x<sup>2</sup></em><strong>a<sup>2</sup></strong></span> − <span class="intbl"><em>y<sup>2</sup></em><strong>b<sup>2</sup></strong></span> = 1</p>
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<div style="clear:both">
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</div>
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<h2>General Equation</h2>
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<p>We can make an equation that covers all these curves.</p>
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<p>Because they are plane curves (even though cut out of the solid) we only have to deal with <a href="../data/cartesian-coordinates.html">Cartesian ("x" and "y") Coordinates</a>.</p>
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<p>But these are not straight lines, so just "x" and "y" will not do ... we need to go to the next level, and have:</p>
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<ul>
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<li><b>x<sup>2</sup></b> and <b>y<sup>2</sup></b>,</li>
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<li>and also <b>x</b> (without y), <b>y</b> (without x),</li>
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<li>x and y together (<b>xy</b>)</li>
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<li>and a constant term.</li>
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</ul>
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<p>There, that should do it!</p>
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<p>Give each one a factor (A,B,C etc) and we get a <b>general equation</b> that covers all conic sections:</p>
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<div class="center large">Ax<sup>2</sup> + Bxy + Cy<sup>2</sup> + Dx + Ey + F = 0</div>
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<!-- Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 -->
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<p>From that equation we can create equations for the circle, ellipse, parabola and hyperbola.</p>
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<p> </p>
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<div class="questions">9064, 9065, 9066, 9067, 637, 638, 3326, 3327, 3328, 3329</div>
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<div class="related">
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<a href="index.html">Geometry Index</a>
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</div>
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