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:
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 cannot 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 (Figure 8). 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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J. Paul Moulton