Hyperbola
A hyperbola is a curve formed by the intersection of a right circular cone and a plane. When the plane cuts both nappes of the cone, the intersection is a hyperbola. Because the plane is cutting two nappes, the curve it forms has two U-shaped branches opening in opposite directions.
A hyperbola can be defined in several other ways, all of them mathematically equivalent:
1. A hyperbola is a set of points P such that PF1 - PF2 = ± C, where C is a constant and F1 and F2 are fixed points called the "foci." That is, a hyperbola is the set of points the difference of whose distances from two fixed points is constant.
The positive value of ± C gives one branch of the hyperbola; the negative value, the other branch.
2. A hyperbola is a set of points whose distances from a fixed point (the "focus") and a fixed line (the "directrix") are in a constant ratio (the "eccentricity"). That is, PF/PD = e.
For this set of points to be a hyperbola, e has to be greater than 1. This definition gives only one branch of the hyperbola.
3. A hyperbola is a set of points (x,y) on a Cartesian coordinate plane satisfying an equation of the form x2/A2 - y2/B2 = ± 1. The equation xy = k also represents a hyperbola, but of eccentricity =not equal2 only. Other second-degree equations can represent hyperbolas, but these two forms are the simplest. When the positive value in ± 1 is used, the hyperbola opens to the left and right. When the negative value is used, the hyperbola opens up and down.
When a hyperbola is drawn as in Figure 4, the line through the foci, F1 and F2, is the "transverse axis."
V1 and V2 are the "vertices," and C the "center." The transverse axis also refers to the distance, V1V2, between the vertices.
The ratio CF1/CV1 (or CF2/CV2) is the "eccentricity" and is numerically equal to the eccentricity e in the focus-directrix definition.
The lines CR and CQ are asymptotes. An asymptote is a straight line which the hyperbola approaches more and more closely as it extends farther and farther from the center. The point Q has been located so that it is the vertex of a right triangle, one of whose legs is CV2, and whose hypotenuse CQ equals CF2. The point R is similarly located.
The line ST, perpendicular to the transverse axis at C, is called the "conjugate axis." The conjugate axis also refers to the distance ST, where SC = CT = QV2.
A hyperbola is symmetric about both its transverse and its conjugate axes.
When a hyperbola is represented by the equation x2/A2 - y2/B2 = 1, the x-axis is the transverse axis and the y-axis is the conjugate axis. These axes, when thought of as distances rather than lines, have lengths 2A and 2B respectively. The foci are at A2 + B2,0 and A2 + B2,0; the eccentricity is A2 + B2 ÷ A.
The equations of the asymptotes are y = Bx/A and y = -Bx/A. (Notice that the constant 1 in the equation above is positive. If it were -1, the y-axis would be the transverse axis and the other points would change accordingly. The asymptotes would be the same, however. In fact, the hyperbolas x2/A2 - y2/B2 = 1 and x2/A2 - y2/B2 = -1 are called "conjugate hyperbolas.") Hyperbolas whose asymptotes are perpendicular to each other are called "rectangular" hyperbolas. The hyperbolas xy = k and x2 - y2 = ± C2 are rectangular hyperbolas. Their eccentricity is =not equal2. Such hyperbolas are geometrically similar, as are all hyperbolas of a given eccentricity.
If one draws the angle F1PF2 the tangent to the hyperbola at point P will bisect that angle.
Hyperbolas can be sketched quite accurately by first locating the vertices, the foci, and the asymptotes. Starting with the axes, locate the vertices and foci.
Draw a circle with its center at C, passing through the two foci. Draw lines through the vertices perpendicular to the transverse axis. This determines four points, which are corners of a rectangle. These diagonals are the asymptotes.
Using the vertices and asymptotes as guides, sketch in the hyperbola as shown in Figure 4. The hyperbola approaches the asymptotes, but never quite reaches them. Its curvature, therefore, approaches, but never quite reaches, that of a straight line.
If the lengths of the transverse and conjugate axes are known, the rectangle in Figure 5 can be drawn without using the foci, since the rectangle 's length and width are equal to these axes.
One can also draw hyperbolas by plotting points on a coordinate plane. In doing this, it helps to draw the asymptotes, whose equations are given above.
Hyperbolas have many uses both mathematical and practical. The hyperbola y = 1/x is sometimes used in the definition of the natural logarithm. In Figure 6 the logarithm of a number n is represented by the shaded area, that is, by the area bounded by the x-axis, the line x = 1, the line x = n, and the hyperbola.
Of course one needs calculus to compute this area, but there are techniques for doing so.
The coordinates of the point (x,y) on the hyperbola x2- y2 = 1 represent the hyperbolic cosine and hyperbolic sine functions. These functions bear the same relationship to this particular hyperbola that the ordinary cosine and sine functions bear to a unit circle:
x = cosh u = (eU + e-u)2
y = sinh u = (eU - e-u) 2
Unlike ordinary sines and cosines, the values of the hyperbolic functions can be represented with simple exponential functions, as shown above. That these representations work can be checked by substituting them in the equation of the hyperbola. The parameter u is also related to the hyperbolas. It is twice the shaded area in Figure 7.
The definition PF1 - PF2 = ± C, of a hyperbola is used directly in the LORAN navigational system. A ship at P receives simultaneous pulsed radio signals from stations at A and B. It can't measure the time it takes for the signals to arrive from each of these stations, but it can measure how much longer it takes for the signal to arrive from one station than from the other. It can therefore compute the difference PA - PB in the distances. This locates the ship somewhere along a hyperbola with foci at A and B, specifically the hyperbola with that constant difference. In the same way, by timing the difference in the time it takes to receive simultaneous signals from stations B and C, it can measure the difference in the distances PB and PC. This puts it somewhere on a second hyperbola with B and C as foci and PC - PB as the constant difference. The ship's position is where these two hyperbolas cross.
Maps with grids of crossing hyperbolas are available to the ship's navigator for use in areas served by these stations.
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