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symbolized e (Don t confuse this with the exponential constant, which is also symbolized e) In the case of a parabola, these distances are both equal to u, so e = u /u = 1
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Specifications for an ellipse and a circle Suppose that we want to construct a curve in which the eccentricity is positive but less than 1 We can use a geometric arrangement similar to the one we used with the parabola, but the distance from the focus is eu instead of u, as shown in Fig 13-6 In this situation we get an ellipse The focus is at the origin (0,0) The ellipse has two vertices (points where the curvature is sharpest), both of which lie on the y axis, and the ellipse is taller than it is wide When we draw an ellipse this way, its variables and parameters are related according to the equations
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u = f + f /e + y and x2 + y2 = (eu)2
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Point (x, y) Focus eu
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f Vertex f/e u = f + f/e + y
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Directrix
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Figure 13-6
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Construction of an ellipse based on a defined focus and directrix The eccentricity e is an expression of the elongation of the ellipse
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As the eccentricity e approaches 0, the focus gets farther from the directrix, and the ellipse gets less elongated When e reaches 0, then f /e becomes meaningless, the directrix vanishes ( runs away to infinity ), and we have a circle where f is equal to the radius r A circle is actually an ellipse whose major and minor semi-axes are the same length Going the other way, as the eccentricity e approaches 1, the focus gets closer to the directrix, and the ellipse gets more elongated When e reaches 1, the ellipse breaks open at one end and becomes a parabola Summarizing the above we can say For a circle, e = 0 For an ellipse, 0 < e < 1 For a parabola, e = 1 The ellipse has another focus besides the one shown in Fig 13-6 It s located at the same distance from the upper vertex of the curve as the coordinate origin is from the lower vertex We can flip the ellipse in Fig 13-6 upside-down, putting the upper focus in place of the lower focus and vice versa, and we ll get a diagram that looks exactly the same The center of the ellipse is midway between the two foci
How the foci, directrix, and eccentricity relate Let s look at the circle, the ellipse, and the parabola in terms of the parameters we ve just described The circle has a single focus, which is at the center The directrix is at infinity The ellipse has two foci separated by a finite distance The curve is symmetrical with respect to a straight line that goes through the two foci The curve is also symmetrical with respect to a straight line equidistant from the foci The ellipse has two directrixes at finite distances from the vertices We can think of a parabola as having two foci: one within reach and the other at infinity Its single directrix is at a distance from the vertex equal to the focal length There s an alternative way to define the eccentricity of an ellipse Suppose we know the distance d between the foci, and we also know the length s of the major semi-axis The eccentricity can be found by taking the ratio
e = d /(2s)
Specifications for a hyperbola If we construct a conic section for which e > 1, we get a curve called a hyperbola Figure 13-7 shows an example The hyperbola looks like two parabolas back-to-back, but there s an important difference in the shape of a hyperbola compared with the shape of a double parabola The parameters that help define hyperbolas are straight lines called asymptotes Hyperbolas always have asymptotes, but parabolas never do In the scenario of Fig 13-7, the hyperbola has two asymptotes that happen to pass through the origin In this case, the equations of the asymptotes are
y = (b /a) x and y = (b /a) x The curve approaches the asymptotes as we move away from the center of the hyperbola, but the curves never quite reach the asymptotes, no matter how far from the center we go
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