Cassini oval in the context of "Focus (geometry)"

In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. For example:

• In statistics, the limiting case of the binomial distribution is the Poisson distribution. As the number of events tends to infinity in the binomial distribution, the random variable changes from the binomial to the Poisson distribution.
• A circle is a limiting case of various other figures, including the Cartesian oval, the ellipse, the superellipse, and the Cassini oval. Each type of figure is a circle for certain values of the defining parameters, and the generic figure appears more like a circle as the limiting values are approached.
• Archimedes calculated an approximate value of π by treating the circle as the limiting case of a regular polygon with 3 × 2 sides, as n gets large.
• In electricity and magnetism, the long wavelength limit is the limiting case when the wavelength is much larger than the system size.
• In economics, two limiting cases of a demand curve or supply curve are those in which the elasticity is zero (the totally inelastic case) or infinity (the infinitely elastic case).
• In finance, continuous compounding is the limiting case of compound interest in which the compounding period becomes infinitesimally small, achieved by taking the limit as the number of compounding periods per year goes to infinity.

A limiting case is sometimes a degenerate case in which some qualitative properties differ from the corresponding properties of the generic case. For example:

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.

The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular points, inflection points and points at infinity. More advanced questions involve the topology of the curve and the relationship between curves defined by different equations.

In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F₁ and F₂, known as foci, at distance 2c from each other as the locus of points P so that ${\displaystyle |PF_{1}||PF_{2}|=c^{2}}$. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscatus, which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4.

This lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed focal points is a constant. A Cassini oval, by contrast, is the locus of points for which the product of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli.