In geometry, a Cartesian oval is a plane curve consisting of points that have the same linear combination of distances from two fixed points (foci). These curves are named after French mathematician René Descartes, who used them in optics.
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Cartesian oval in the context of Focus (geometry)
In geometry, focuses or foci (/ˈfoʊsaɪ/ or /ˈfoʊkaɪ/; sg.: focus) are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse.
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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:
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