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288 lines
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HTML
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HTML
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<article id="content" role="main">
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<h1 class="center">Parabola</h1>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><img src="images/parabola-soccer.svg" alt="parabola" height="274" width="344"></td>
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<td>
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<p>When you kick a soccer ball (or shoot an arrow, fire a missile or throw a stone) it arcs up into the air and comes down again ...</p>
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<p class="large">... following the path of a parabola!</p>
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<p><i>(Except for how the air affects it.)</i></p></td>
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</tr>
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</tbody></table>
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<p>Try kicking the ball:</p>
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<div class="script" style="height: 360px;">
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images/parabola-ball.js?mode=ball
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</div>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/parabola-distances.svg" alt="parabola equal distances" height="225" width="296"></p>
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<h2>Definition</h2>
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<p>A parabola is a curve where any point is at an <b>equal distance</b> from:</p>
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<ul>
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<li>a fixed point (the
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<strong>focus</strong>
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), and</li>
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<li>a fixed straight line (the
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<strong>directrix</strong>
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)</li>
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</ul>
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<div style="clear:both"></div>
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<h2>On Paper</h2>
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<p>Get a piece of paper, draw a straight line on it, then make a big dot for the focus (not on the line!).</p>
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<p>Now play around with some measurements until you have another dot that is exactly the same distance from the focus and the straight line.</p>
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<p>Keep going until you have lots of little dots, then join the little dots and you will have a parabola!</p>
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<p>Just like in this interactive (try moving point P):</p>
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<div class="script" style="height: 320px;">
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../sets/images/geom-locus.js?mode=parabola
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</div>
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<p> </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/parabola-names.svg" alt="parabola directrix vertex focus and axis of symmetry" height="212" width="288"></p>
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<h2>Names</h2>
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<p>Here are the important names:</p>
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<ul>
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<li>the
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<strong>directrix</strong>
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and <b>focus</b> (explained above)</li>
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<li>the <b>axis of symmetry</b> (goes through the focus, at right angles to the directrix)</li>
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<li>the
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<strong>vertex</strong>
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(where the parabola makes its sharpest turn) is halfway between the focus and directrix.</li>
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</ul>
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<p> </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/parabola-reflector.svg" alt="parabola rays go to focus" height="235" width="299"></p>
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<h2>Reflector</h2>
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<p>And a parabola has this amazing property:</p>
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<p class="center larger">Any ray parallel to the axis of symmetry gets <b>reflected</b> off the surface straight <b>to the focus</b>.</p>
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<div style="clear:both"></div>
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<p class="center">And that explains why that dot is called the <b>focus</b> ...</p>
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<p class="center">... because that's where all the rays get focused!</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/parabolic-dish.jpg" alt="parabolic dish" height="109" width="116"></p>
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<p>So the parabola can be used for:</p>
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<ul>
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<li>satellite dishes,</li>
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<li>radar dishes,</li>
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<li>concentrating the sun's rays to make a hot spot,</li>
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<li>the reflector on spotlights and torches,</li>
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<li>etc</li>
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</ul>
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<p> </p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/conic-parabola.jpg" alt="conic section parabola" height="200" width="122"></td>
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<td>
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<p>We also get a parabola when we <b>slice through a cone</b> (the slice must be parallel to the side of the cone).</p>
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<p>So the parabola is a <a href="conic-sections.html">conic section</a> (a section of a cone).</p></td>
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</tr>
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</tbody></table>
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<h2>Equations</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/parabola-x2.svg" alt="x-squared is a parabola" height="227" width="148"></p>
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<p> </p>
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<p>The simplest equation for a parabola is <b>y = x<sup>2</sup></b></p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/parabola-y2.svg" alt="x-squared is a parabola" height="160" width="208"></p>
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<p> </p>
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<p class="center">Turned on its side it becomes <b> y<sup>2</sup> = x</b></p>
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<p class="center">(or <b>y = √x</b> for just the top half)</p>
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<div style="clear:both"></div>
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<p> </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/parabola-formula.svg" alt="parabola on coordinates" height="267" width="322"></p>
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<p>A little more generally:</p>
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<p class="center large">y<sup>2</sup> = 4ax</p>
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<p>where <b>a</b> is the distance from the origin to the focus (and also from the origin to directrix)</p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example:
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Find the focus for the equation y<sup>2</sup>=5x</h3>
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<p><br>
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Converting <b>y<sup>2</sup> = 5x</b> to <b>y<sup>2</sup> = 4ax</b> form, we get <b>y<sup>2</sup> = 4 (5/4) x</b>,</p>
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<p>so <b>a = 5/4</b>, and the focus of y<sup>2</sup>=5x is:</p>
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<div class="larger">
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<p class="center">F = (a, 0) = (5/4, 0)</p>
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</div>
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</div>
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<p>The equations of parabolas in different orientations are as follows:</p>
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<div class="centerfull">
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<div class="boxa">
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<p><img src="images/parabola-rt.svg" alt="parabola orientation right" height="123" width="112"><br>
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<span class="large">y<sup>2</sup> = 4ax</span></p>
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</div>
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<div class="boxa">
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<p><img src="images/parabola-lt.svg" alt="parabola orientation left" height="122" width="148"><br>
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<span class="large">y<sup>2</sup> = −4ax</span></p>
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</div>
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<div class="boxa">
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<p><img src="images/parabola-up.svg" alt="parabola orientation up" height="127" width="153"><br>
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<span class="large">x<sup>2</sup> = 4ay</span></p>
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</div>
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<div class="boxa">
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<p><img src="images/parabola-dn.svg" alt="parabola orientation down" height="134" width="153"><br>
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<span class="large">x<sup>2</sup> = −4ay</span></p>
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</div>
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</div>
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<h2>Measurements for a Parabolic Dish</h2>
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<p>If you want to build a parabolic dish where the focus is 200 mm above the surface, what measurements do you need?</p>
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<p>To make it easy to build, let's have it pointing upwards, and so we choose the <span class="large">x<sup>2</sup> = 4ay</span> equation.</p>
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<p>And we want "a" to be 200, so the equation becomes:</p>
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<p class="center"><span class="large">x<sup>2</sup> = 4ay = 4 × 200 × y = 800y</span></p>
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<p>Rearranging so we can calculate heights:</p>
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<p class="center"><span class="large">y</span> <span class="large">= x<sup>2</sup>/800</span></p>
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<p>And here are some height measurements as you run along:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td rowspan="9">
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<img src="images/parabola-dish-graph.svg" alt="parabolic dish graph" height="227" width="287">
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</td>
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<td style="text-align:right;">Distance Along ("x")</td>
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<td style="text-align:right; width:100px;">Height ("y")</td>
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</tr>
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<tr>
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<td style="text-align:right;">0 mm</td>
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<td style="text-align:right; width:100px;">0.0 mm</td>
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</tr>
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<tr>
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<td style="text-align:right;">100 mm</td>
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<td style="text-align:right; width:100px;">12.5 mm</td>
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</tr>
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<tr>
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<td style="text-align:right;">200 mm</td>
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<td style="text-align:right; width:100px;">50.0 mm</td>
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</tr>
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<tr>
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<td style="text-align:right;">300 mm</td>
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<td style="text-align:right; width:100px;">112.5 mm</td>
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</tr>
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<tr>
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<td style="text-align:right;">400 mm</td>
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<td style="text-align:right; width:100px;">200.0 mm</td>
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</tr>
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<tr>
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<td style="text-align:right;">500 mm</td>
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<td style="text-align:right; width:100px;">312.5 mm</td>
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</tr>
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<tr>
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<td style="text-align:right;">600 mm</td>
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<td style="text-align:right; width:100px;">450.0 mm</td>
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</tr>
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<tr>
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<td> </td>
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<td style="width:100px;"> </td>
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</tr>
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</tbody></table>
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<p>Try to build one yourself, it could be fun! Just be careful, a reflective surface can concentrate a lot of heat at the focus.</p>
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<p> </p>
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<div class="questions">567,568,833,834, 2088, 2089, 2086, 2087, 3334, 3335</div>
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<div class="related">
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<a href="conic-sections.html">Conic Sections</a>
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<a href="projectile-animation.html">Projectile Animation</a>
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<a href="index.html">Geometry Index</a>
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</div>
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<!-- #EndEditable -->
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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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