First examined by Jacob Bernoulli (1654 - 16705) in 1694, the lemniscate of Bernoulli has the general form of a figure eight, or the mathematical symbol for infinity. Bernoulli's conceptualization of the lemniscate followed his earlier research on parabolas, logarithmic spirals, and epicycloids. Bernoulli originally termed the curve lemniscus, which is Latin for "a pendant ribbon," to describe the unique shape of the curve. Roughly defined, the lemniscate of Bernoulli is the infinite set of all points (called a locus) satisfying the property that the product of the distances between the point and two foci separated by 2a is exactly a2. In other words, if (a,0) and (-a,0) are the two foci of the curve, each point (x,y) along the curve satisfies the condition that sqrt[(x - a)2 + (y - 0)2)] * sqrt[(x - (-a))2 + (y - 0)2) = a2. This stipulation is expressed here in Cartesian coordinates, but can be translated to other coordinate systems (such as polar or cylindrical) as well.
Interestingly enough, Bernoulli's lemniscate is actually a special case of a family of curves known as Cassini's ovals (and alternatively known as Cassini's curve). Named for the astronomer Gian Domenico Cassini (1625 - 1712), Cassini first described his ovals in 1680 while researching the relative motion of the earth and the sun. Cassini's curves are more general than Bernoulli's lemniscate; instead of describing just a single curve, Cassini described a family of curves in which the locus of each curve satisfies the condition that the product of the distances to each foci is exactly a constant, generally denoted c2. In other words, if (a,0) and (-a,0) are the two foci of the curve, each point (x,y) along the curve satisfies the condition that sqrt[(x - a)2 + (y - 0)2)] * sqrt[(x - (-a))2 + (y - 0)2) = c2. Bernoulli's lemniscate is the special case in which c = a. If c > a, the curve has the form of two distinct loops; if c < a, the curve appears almost peanut-shaped, and when a = 0, the curve forms an oval. Cassini's ovals were apparently unknown to Bernoulli at the time of Bernoulli's investigation.
In Cartesian coordinates, the lemniscate is commonly expressed by the equation (x2 + y2)2 = 2a2(x2 - y2, where 2a is the distance between the two foci. The expression, however, is much simpler in polar coordinates. In that system, two common forms of Bernoulli's lemniscate are r2 = c2sin( and r2 = c2cos(. In this case, c represents the radius of each petal of the lemniscate, and c is related to a by c = sqrt(2) * a. Of course, Bernoulli's lemniscate can also be written in parametrically. One common form is x = ccost / (1 + sin2t) and y = csintcost / (1 + sin2t).
Both Gauss and Euler studied the arc length of the lemniscate curve. Much like the relationship between pi and the diametercircle, Bernoulli's lemniscate has a unique constant that relates its arc length to the distance between the foci. Specifically, Gauss determined for a leminscate in which the distance between the two foci is 2 (that is, a = 1), the arc length can be represented by L = 2c times the integral over the interval [0,1] of the quantity (1 - t4)-(1/2)dt = 1 / sqrt(2 * pi) * [Gamma(1 / 4)] 2 " 5.24411.... Then, the lemniscate constant is L/2, which is related to Gauss' constant M by L = 2 * pi / M.
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