1911 Encyclopædia Britannica/Lemniscate

LEMNISCATE (from Gr. λημνίσκος, ribbon), a quartic curve invented by Jacques Bernoulli (Acta Eruditorum, 1694) and afterwards investigated by Giulio Carlo Fagnano, who gave its principal properties and applied it to effect the division of a quadrant into 2⋅2^m, 3⋅2^m and 5⋅2^m equal parts. Following Archimedes, Fagnano desired the curve to be engraved on his tombstone. The complete analytical treatment was first given by Leonhard Euler. The lemniscate of Bernoulli may be defined as the locus of a point which moves so that the product of its distances from two fixed points is constant and is equal to the square of half the distance between these points. It is therefore a particular form of Cassini’s oval (see Oval). Its cartesian equation, when the line joining the two fixed points is the axis of x and the middle point of this line is the origin, is (x² + y ²)² = 2a²(x² − y ²) and the polar equation is r ² = 2a² cos 2θ. The curve (fig. 1) consists of two loops symmetrically placed about the coordinate axes. The pedal equation is r ³ = a²p, which shows that it is the first positive pedal of a rectangular hyperbola with regard to the centre. It is also the inverse of the same curve for the same point. It is the envelope of circles described on the central radii of an ellipse as diameters. The area of the complete curve is 2a², and the length of any arc may be expressed in the form ∫(1−x⁴)^−1/2dx, an elliptic integral sometimes termed the lemniscatic integral.

Fig. 1.		Fig. 2.		Fig. 3.

The name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis. Such curves are given by the equation x² − y ² = ax⁴ + bx²y ² + cy ⁴. If a be greater than b the curve resembles fig. 2 and is sometimes termed the fishtail-lemniscate; if a be less than b, the curve resembles fig. 3. The same name is also given to the first positive pedal of any central conic. When the conic is a rectangular hyperbola, the curve is the lemniscate of Bernoulli previously described. The elliptic lemniscate has for its equation (x² + y ²)² = a²x² + b²y ² or r ² = a² cos²θ + b² sin²θ (a > b). The centre is a conjugate point (or acnode) and the curve resembles fig. 4. The hyperbolic lemniscate has for its equation (x² + y ²)² = a²x² − b²y ² or r ² = a² cos²θ − b² sin²θ. In this case the centre is a crunode and the curve resembles fig. 5. These curves are instances of unicursal bicircular quartics.

Fig. 4.		Fig. 5.