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Cassini Oval
In geometry, a Cassini oval is a quartic plane curve defined as the locus (mathematics), locus of points in the plane (geometry), plane such that the Product_(mathematics), product of the distances to two fixed points (Focus (geometry), foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree of a polynomial, degree 2. Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in the late 17th century. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval. Other names include Cassinian ovals, Cassinian curves and ovals of Cassini. Formal definition A Cassini oval is a set of points, such that for any point P of the set, the ''product'' of the distances , PP_1, ,\, , PP_2, to two fixed points P_1, P_2 is a consta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemniscate Of Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from , 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. This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisector o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Hyperbolas arise in many ways: * as the curve representing the reciprocal function y(x) = 1/x in the Cartesian plane, * as the path followed by the shadow of the tip of a sundial, * as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Hyperbolas arise in many ways: * as the curve representing the reciprocal function y(x) = 1/x in the Cartesian plane, * as the path followed by the shadow of the tip of a sundial, * as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Confocal Conic Sections
In geometry, two conic sections are called confocal, if they have the same foci. Because ellipses and hyperbolas possess two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture of confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally (at right angles). Parabolas possess only one focus, so, by convention, confocal parabolas have the same focus ''and'' the same axis of symmetry. Consequently, any point not on the axis of symmetry lies on two confocal parabolas which intersect orthogonally (see below). The formal extension of the concept of confocal conics to surfaces leads to confocal quadrics. Confocal ellipses An ellipse which is not a circle is uniquely determined by its foci F_1,\; F_2 and a point not on the major axis (see the definition of an ellipse as a locus of points). The pencil of confocal ellipses with the foci F_1=(c,0),\; F_2=(-c,0) can be described by the equation * \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pencil (mathematics)
In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane. Although the definition of a pencil is rather vague, the common characteristic is that the pencil is completely determined by any two of its members. Analogously, a set of geometric objects that are determined by any three of its members is called a bundle. Thus, the set of all lines through a point in three-space is a bundle of lines, any two of which determine a pencil of lines. To emphasize the two dimensional nature of such a pencil, it is sometimes referred to as a ''flat pencil''. Any geometric object can be used in a pencil. The common ones are lines, planes, circles, conics, spheres, and general curves. Even points can be used. A pencil of points is the set of all points on a given line. A more common term for this set is a ''range'' of points. Penci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Trajectories
In mathematics an orthogonal trajectory is a curve, which intersects any curve of a given pencil of (planar) curves ''orthogonally''. For example, the orthogonal trajectories of a pencil of ''concentric circles'' are the lines through their common center (see diagram). Suitable methods for the determination of orthogonal trajectories are provided by solving differential equations. The standard method establishes a first order ordinary differential equation and solves it by separation of variables. Both steps may be difficult or even impossible. In such cases one has to apply numerical methods. Orthogonal trajectories are used in mathematics for example as curved coordinate systems (i.e. elliptic coordinates) or appear in physics as electric fields and their equipotential curves. If the trajectory intersects the given curves by an arbitrary (but fixed) angle, one gets an isogonal trajectory. Determination of the orthogonal trajectory In cartesian coordinates Generally one as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plücker Formula
In mathematics, a Plücker formula, named after Julius Plücker, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of algebraic curves to corresponding invariants of their dual curves. The invariant called the genus, common to both the curve and its dual, is connected to the other invariants by similar formulae. These formulae, and the fact that each of the invariants must be a positive integer, place quite strict limitations on their possible values. Plücker invariants and basic equations A curve in this context is defined by a non-degenerate algebraic equation in the complex projective plane. Lines in this plane correspond to points in the dual projective plane and the lines tangent to a given algebraic curve ''C'' correspond to points in an algebraic curve ''C''* called the dual curve. In the correspondence between the projective plane and its dual, points on ''C'' correspond to lines tangent ''C''*, so the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circular Algebraic Curve
In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation ''F''(''x'', ''y'') = 0, where ''F'' is a polynomial with real coefficients and the highest-order terms of ''F'' form a polynomial divisible by ''x''2 + ''y''2. More precisely, if ''F'' = ''F''''n'' + ''F''''n''−1 + ... + ''F''1 + ''F''0, where each ''F''''i'' is homogeneous of degree ''i'', then the curve ''F''(''x'', ''y'') = 0 is circular if and only if ''F''''n'' is divisible by ''x''2 + ''y''2. Equivalently, if the curve is determined in homogeneous coordinates by ''G''(''x'', ''y'', ''z'') = 0, where ''G'' is a homogeneous polynomial, then the curve is circular if and only if ''G''(1, ''i'', 0) = ''G''(1, −''i'', 0) = 0. In other words, the curve is circular if it contains the circular points at infinity, (1, ''i'', 0) and (1, −''i'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circular Points At Infinity
In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle. Coordinates A point of the complex projective plane may be described in terms of homogeneous coordinates, being a triple of complex numbers , where two triples describe the same point of the plane when the coordinates of one triple are the same as those of the other aside from being multiplied by the same nonzero factor. In this system, the points at infinity may be chosen as those whose ''z''-coordinate is zero. The two circular points at infinity are two of these, usually taken to be those with homogeneous coordinates : and . Trilinear coordinates Let ''A''. ''B''. ''C'' be the measures of the vertex angles of the reference triangle ABC. Then the trilinear coordinates of the circular points at infinity in the plane of the reference triangle are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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