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232 lines
11 KiB
HTML
232 lines
11 KiB
HTML
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<title>Ellipse</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Ellipse</h1>
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<p class="center">An ellipse usually looks like a <b>squashed circle</b>:</p>
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<div class="script" style="height: 330px;">
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images/geom-ellipse.js?mode=foci
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</div>
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<p class="center">"F" is a <b>focus</b>, "G" is a <b>focus</b>,<br>
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and together they are called <b>foci</b>.<br>
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(pronounced "fo-sigh")</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/ellipse.svg" alt="ellipse" height="210" width="220"></p>
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<p> </p>
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<p class="center"><span class="larger">The total distance from <b>F to P to G</b> stays the same</span></p>
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<p>In other words, we always travel <b>the same distance</b> when going from:</p>
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<ul>
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<li>point "F" to</li>
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<li>to <b>any point on the ellipse</b></li>
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<li>and then on to point "G"</li>
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</ul>
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<div style="clear:both"></div>
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<h2>You Can Draw It Yourself</h2>
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<p>Put two pins in a board, and then ...</p>
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<div class="boxa"><img src="images/ellipse-draw-1.jpg" alt="ellipse drawing pins and string" height="117" width="180"><br>
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put a loop of string around them,</div>
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<div class="boxa"><img src="images/ellipse-draw-2.jpg" alt="ellipse drawing stretch" height="120" width="180"><br>
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insert a pencil into the loop,</div>
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<div class="boxa"><img src="images/ellipse-draw-3.jpg" alt="ellipse draw " height="139" width="180"><br>
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stretch the string so it forms a triangle,</div>
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<div class="boxa"><img src="images/elllipse-draw-4.jpg" alt="elllipse drawing pins" height="127" width="180"><br>
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and draw a curve.<br>
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It is an ellipse!</div>
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<p> </p>
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<p>It works because the string naturally forces the <b>same distance</b> from <b>pin-to-pencil-to-other-pin</b>.</p>
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<h2>A Circle is an Ellipse</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/circle-draw.jpg" alt="circle draw" height="180" width="180"></p>
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<p>In fact a Circle <b>is</b> an Ellipse, where both foci are at the same point (the center). So to draw a circle we only need one pin!</p>
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<p><br></p>
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<p class="center"><img src="images/ellipse-stretch.svg" alt="ellipse as stretched circle" height="133" width="207"><br>
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A circle is a "special case" of an ellipse. Ellipses Rule!</p>
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<h2 align="left">Definition</h2>
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<div class="def">
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<p>An ellipse is the <b><a href="../sets/set-of-points.html">set of all points</a></b> on a plane whose distance from two fixed points F and G add up to a constant.</p>
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</div>
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<h2>Major and Minor Axes</h2>
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<p>The <b>Major Axis</b> is the longest diameter. It goes from one side of the ellipse, through the center, to the other side, at the widest part of the ellipse. And the <b>Minor Axis</b> is the shortest diameter (at the narrowest part of the ellipse).</p>
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<div class="script" style="height: 330px;">
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images/geom-ellipse.js?mode=axes
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</div>
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<p>The <b>Semi-major Axis</b> is half of the Major Axis, and the <b>Semi-minor Axis</b> is half of the Minor Axis.</p>
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<div class="script" style="height: 330px;">
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images/geom-ellipse.js?mode=semiaxes
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</div>
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<h3>Major Axis Equals f+g</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/ellipse-major-axis.svg" alt="ellipse major axis" height="159" width="221"></p>
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<p>Remember from the top how the distance "f+g" stays the same for an ellipse?</p>
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<p>Well f+g is equal to the <b>length of the major axis</b>.</p>
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<p>Can you think why? (Try moving the point P at the top.)</p><br>
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<h2>Calculations</h2>
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<p>Area is easy, perimeter is not!</p>
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<h2>Area</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/ellipse-axes.svg" alt="ellipse axes" height="143" width="287"></p>
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<p>The <a href="../area.html">area</a> of an ellipse is:</p>
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<p class="larger center"><span class="times">π</span><b> × a × b</b></p>
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<p>where <b>a</b> is the length of the Semi-major Axis, and <b>b</b> is the length of the Semi-minor Axis.</p>
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<div style="clear:both"></div>
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<p class="center larger">Be careful: <b>a</b> and <b>b</b> are <b>from the center</b> outwards (not all the way across).</p>
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<p>(Note: for a circle, <b>a</b> and <b>b</b> are equal to the radius, and you get <span class="times">π</span><b> × r × r = <span class="times">π</span><b>r<sup>2</sup></b></b>, which is right!)</p>
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<h2>Perimeter Approximation</h2>
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<p>Rather strangely, the perimeter of an ellipse is <b>very difficult to calculate</b>, so I created a special page for the subject: read <a href="ellipse-perimeter.html">Perimeter of an Ellipse</a> for more details.</p>
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<p>But a <b>simple approximation</b> that is within about 5% of the true value (so long as <b>a</b> is not more than 3 times longer than <b>b</b>) is as follows:</p>
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<div class="center large">p ≈ 2<span class="times">π</span> <span class="tall">√</span><span class="intbl"><span class="overline"><em>a<sup>2</sup>+b<sup>2</sup></em><strong>2</strong></span></span></div>
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<p>Remember this is only an approximation! (That is why the "equals sign" is squiggly.)</p>
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<h2>Tangent</h2>
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<p>A <a href="tangent-secant-lines.html">tangent line</a> just touches a curve at one point, without cutting across it.
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Here is a tangent to an ellipse:</p>
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<div class="script" style="height: 330px;">
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images/geom-ellipse.js?mode=tangent
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</div>
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<p>Here is a cool thing: the tangent line has equal angles with the two lines going to each focus! Try bringing the two focus points together (so the ellipse is a circle) ... what do you notice?</p>
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<h2>Reflection</h2>
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<p>Light or sound starting at one focus point reflects to the other focus point (because angle in matches angle out):</p>
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<p class="center"><img src="images/ellipse-focus.svg" alt="ellipse focus points" height="143" width="220"></p>
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<p class="center larger">Have a play with a simple <a href="ellipse-reflect-model.html">computer model of reflection inside an ellipse</a>.</p>
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<h2>Eccentricity</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/ellipse-eccentricity.svg" alt="ellipse eccentricity" height="127" width="193"></p>
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<p>The <a href="eccentricity.html">eccentricity</a> is a measure of how "un-round" the ellipse is.</p>
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<p>The formula (using semi-major and semi-minor axis) is:</p>
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<p class="center large"><span class="intbl"><em>√(a<sup>2</sup>−b<sup>2</sup>)</em><strong>a</strong></span></p><br>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/conic-ellipse.jpg" alt="conic section parabola" height="200" width="133"></p>
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<h2>Section of a Cone</h2>
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<p>We also get an ellipse when we <b>slice through a cone</b> (but not too steep a slice, or we get a <a href="parabola.html">parabola</a> or <a href="hyperbola.html">hyperbola</a>).</p>
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<p>In fact the ellipse is a <a href="conic-sections.html">conic section</a> (a section of a cone) with an <a href="eccentricity.html">eccentricity</a> between 0 and 1.</p>
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<p> </p>
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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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<h2><a name="equation" id="equation"></a>Equation</h2>
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<p>By placing an ellipse on an <a href="../data/cartesian-coordinates.html">x-y graph</a> (with its major axis on the x-axis and minor axis on the y-axis), the equation of the curve 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>(similar to the <a href="hyperbola.html">equation of the hyperbola</a>: <b>x<sup>2</sup>/a<sup>2</sup> − y<sup>2</sup>/b<sup>2</sup> = 1</b>, except for a "+" instead of a "−")</p>
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<p><b>Or</b> we can use "parametric equations", where we have another variable "t" and we calculate x and y from it, like this:</p>
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<ul>
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<li>x = a cos(t)</li>
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<li>y = b sin(t)</li>
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</ul>
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<p>(Just imagine "t" going from 0° to 360°, what x and y values would we get?)</p>
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<p> </p>
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<div class="questions">635, 3330, 3332, 3333, 636, 3331, 7417, 7418, 7419, 7420</div>
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<div class="related">
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<a href="../area-calculation-tool.html">Area Calculation Tool</a>
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<a href="index.html">Geometry Index</a>
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</div>
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<footer id="footer" class="centerfull noprint"></footer>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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