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188 lines
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HTML
188 lines
9.4 KiB
HTML
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<h1 class="center">Hyperbola</h1>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/hyperbola1.svg" alt="hyperbola path of spacecraft" /></p>
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<p>Did you know that the orbit of a spacecraft can sometimes be a hyperbola?</p>
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<p>A spacecraft can use the gravity of a planet to alter its path and propel it at high speed away from the planet and back out into space using a technique called "gravitational slingshot".</p>
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<p>If this happens, then the path of the spacecraft is a <b>hyperbola</b>.</p>
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<p>(Play with this at <a href="../numbers/gravity-freeplay.html">Gravity Freeplay</a>)</p>
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<h2>Definition</h2>
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<p>A hyperbola is two curves that are like infinite bows.</p>
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<p>Looking at just one of the curves:</p>
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<p class="center larger">any point <b>P</b> is closer to <b>F</b> than to <b>G</b> by some constant amount</p>
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<p>The other curve is a mirror image, and is closer to G than to F.</p>
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<p class="center"><img src="images/hyperbola-distance.svg" alt="hyperbola distances" /></p>
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<p>In other words, the distance from <b>P to F</b> is always less than the distance <b>P to G</b> by some constant amount. (And for the other curve <b>P to G</b> is always less than <b>P to F</b> by that constant amount.)</p>
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<div class="def">
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<p>As a formula:</p>
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<p class="center larger">|PF − PG| = constant</p>
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<ul>
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<li>PF is the distance P to F</li>
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<li>PG is the distance P to G</li>
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<li>|| is the <a href="../numbers/absolute-value.html">absolute value</a> function (makes any negative a positive)</li>
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</ul>
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</div>
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<p>Each bow is called a <b>branch</b> and F and G are each called a <b>focus</b>.</p>
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<p>Have a try yourself:</p>
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<iframe src="ihyperbola.html" scrolling="no" style="width:362px; height:362px; overflow:hidden; margin:auto; display:block; border:none;"></iframe>
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<p>Try moving point <b>P</b>: what do you notice about the lengths <b>PF</b> and <b>PG</b> ?</p>
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<p>Also try putting point <b>P</b> on the other branch.</p>
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<p>There are some other interesting things, too:</p>
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<p class="center"><img src="images/hyperbola-foci.svg" alt="hyperbola foci etc" /></p>
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<p>On the diagram you can see:</p>
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<ul>
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<li>an <b>axis of symmetry</b> (that goes through each focus)</li>
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<li>two <b>vertices</b> (where each curve makes its sharpest turn)</li>
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<li>the distance between the vertices (2a on the diagram) is the <b>constant difference</b> between the lengths <b>PF</b> and <b>PG</b></li>
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<li>two <b>asymptotes</b> which are not part of the hyperbola but show where the curve would go if continued indefinitely in each of the four directions</li>
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</ul>
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<p>And, strictly speaking, there is also <b>another axis of symmetry</b> that goes down the middle and separates the two branches of the hyperbola.</p>
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<table style="border: 0; margin:auto;">
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<tr>
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<td><h2>Conic Section</h2>
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<p>You can also get a hyperbola when you slice through a double cone.</p>
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<p>The slice must be steeper than that for a parabola, but does not<br>
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have to be parallel to the cone's axis for the hyperbola to be symmetrical.</p>
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<p>So the hyperbola is a <a href="conic-sections.html">conic section</a> (a section of a cone).</p></td>
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<td> </td>
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<td><img src="images/conic-hyperbola2.jpg" alt="conic section hyperbola" width="116" height="200" /></td>
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</tr>
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</table>
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<h2><a name="equation" id="equation"></a>Equation</h2>
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<p>By placing a hyperbola on an <a href="../data/cartesian-coordinates.html">x-y graph</a> (centered over the x-axis and 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 style="float:right; margin: 0 0 5px 10px;"><img src="images/hyperbola6.svg" alt="hyperbola on xy graph" /></p>
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<p>Also:</p>
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<p>One <b>vertex</b> is at (a, 0), and the other is at (−a, 0)</p>
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<p>The <b>asymptotes</b> are the straight lines:</p>
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<ul>
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<li>y = (b/a)x</li>
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<li>y = −(b/a)x</li>
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</ul>
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<p>(Note: the equation is similar to the equation of the <a href="ellipse.html#equation">ellipse</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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<h2>Eccentricity</h2>
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<p>Any branch of a hyperbola can also be defined as a curve where the distances of any point from:</p>
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<ul>
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<li>a fixed point (the <b>focus</b>), and</li>
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<li>a fixed straight line (the <b>directrix</b>)
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are always in the same ratio.</li>
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</ul>
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<p class="center"><img src="images/hyperbola2.svg" alt="hyperbola focus directrix" /></p>
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<p>This ratio is called the <a href="eccentricity.html">eccentricity</a>, and for a hyperbola it is always greater than 1.</p>
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<p>The eccentricity (usually shown as the letter <span class="large">e</span>) shows how "uncurvy" (varying from being a circle) the hyperbola
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is.</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/hyperbola7.svg" alt="hyperbola with points N P F" /></p>
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<p>On this diagram:</p>
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<ul>
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<li>P is a point on the curve,</li>
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<li>F is the focus and</li>
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<li>N is the point on the directrix so that PN is perpendicular to the directrix.</li>
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</ul>
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<p>The eccentricity is the ratio PF/PN, and has the formula:</p>
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<p class="center large">e = <span class="intbl"><em>√(a<sup>2</sup>+b<sup>2</sup>)</em><strong>a</strong></span></p>
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<p>Using "a" and "b" from the diagram above.</p>
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<h2>Latus Rectum</h2>
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<table style="border: 0; margin:auto;">
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<tr>
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<td><img src="images/hyperbola8.svg" alt="hyperbola latus rectum" /></td>
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<td><p>The Latus Rectum is the line through the focus and parallel to the directrix.</p>
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<p>The length of the Latus Rectum is 2b<sup>2</sup>/a.</p></td>
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</tr>
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</table>
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<h2>1/x</h2>
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<p class="center"><img src="../sets/images/function-reciprocal.svg" alt="Reciprocal function" /><br>
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The <a href="../sets/function-reciprocal.html">reciprocal</a> function y = 1/x is a hyperbola!</p>
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<p> </p>
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<script>getQ(835, 3336, 836, 3337, 837, 3338, 838, 3339, 9068, 9069);</script>
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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="index.html">Geometry Index</a>
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