Linear System Of Conics
In algebraic geometry, the conic sections in the projective plane form a linear system of dimension five, as one sees by counting the constants in the degree two equations. The condition to pass through a given point ''P'' imposes a single linear condition, so that conics ''C'' through ''P'' form a linear system of dimension 4. Other types of condition that are of interest include tangency to a given line ''L''. In the most elementary treatments a linear system appears in the form of equations :\lambda C + \mu C' = 0\ with λ and μ unknown scalars, not both zero. Here ''C'' and ''C′'' are given conics. Abstractly we can say that this is a projective line in the space of all conics, on which we take : lambda : \mu as homogeneous coordinates. Geometrically we notice that any point ''Q'' common to ''C'' and ''C′'' is also on each of the conics of the linear system. According to Bézout's theorem ''C'' and ''C′'' will intersect in four points (if counted co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, 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 point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conic Section
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes considered a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a '' focus'', and some particular line, called a ''directrix'', are in a fixed ratio, called the ''eccentricity''. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear System Of Divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the form of a ''linear system'' of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisor (algebraic geometry), divisors ''D'' on a general Scheme (mathematics), scheme or even a ringed space (X, \mathcal_X). Linear systems of dimension 1, 2, or 3 are called a Pencil (mathematics), pencil, a net, or a web, respectively. A map determined by a linear system is sometimes called the Kodaira map. Definitions Given a general variety X, two divisors D,E \in \text(X) are linearly equivalent if :E = D + (f)\ for some non-zero rational function f on X, or in other words a non-zero element f of the Function field of an algebraic variety, func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Equation
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. The " =" symbol, which appears in every equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Conic
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes considered a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a ''focus'', and some particular line, called a ''directrix'', are in a fixed ratio, called the ''eccentricity''. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Line
In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case). There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field ''K'', commonly denoted P1(''K''), as the set of one-dimensional subspaces of a two-dimensional ''K''-vector space. This definition is a special instance of the general definition of a projective space. The projective line over the reals is a manifold; see '' Real projective line'' for details. Homogeneous coordinates An arbitrary point in the projective line P1(''K'') may be represented by an equivalence class of '' homogeneous coordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Homogeneous Coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. They are also used in fundamental elliptic curve cryptography algorithms. If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bézout's Theorem
In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the degrees of the polynomials. It is named after Étienne Bézout. In some elementary texts, Bézout's theorem refers only to the case of two variables, and asserts that, if two plane algebraic curves of degrees d_1 and d_2 have no component in common, they have d_1d_2 intersection points, counted with their multiplicity, and including points at infinity and points with complex coordinates. In its modern formulation, the theorem states that, if is the number of common points over an algebraically closed field of projective hypersurfaces defined by homogeneous polynomials in indeterminates, then is either infinite, or equals the product of the degrees of the polynomials. Moreover, the finite case occurs almost always. In the case of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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General Position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings. For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a ''generic point''. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see '' generic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension. Definition Codimension is a ''relative'' concept: it is only defined for one object ''inside'' another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector ''sub''space. If ''W'' is a linear subspace of a finite-dimensional vector space ''V'', then the codimension of ''W'' in ''V'' is the difference between the dimensions: :\operatorname(W) = \dim(V) - \dim(W). It is the complement of the dimension of ''W,'' in that, with the dimension of ''W,'' it adds up to the dimension of the ambient space ''V:'' :\dim(W) + \operatorname(W) = \dim(V). Simi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Degenerate Conic
In geometry, a degenerate conic is a conic (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers (or more generally over an algebraically closed field) as the product of two linear polynomials. Using the alternative definition of the conic as the intersection in three-dimensional space of a plane and a double cone, a conic is degenerate if the plane goes through the vertex of the cones. In the real plane, a degenerate conic can be two lines that may or may not be parallel, a single line (either two coinciding lines or the union of a line and the line at infinity), a single point (in fact, two complex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines). All these degenerate conics may occur in pencils of conics. That is, if two real non-degenerated conics are defined by quadratic polynomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Multinomial Coefficient
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. Theorem For any positive integer and any non-negative integer , the multinomial theorem describes how a sum with terms expands when raised to the th power: (x_1 + x_2 + \cdots + x_m)^n = \sum_ x_1^ \cdot x_2^ \cdots x_m^ where = \frac is a multinomial coefficient. The sum is taken over all combinations of nonnegative integer indices through such that the sum of all is . That is, for each term in the expansion, the exponents of the must add up to . In the case , this statement reduces to that of the binomial theorem. Example The third power of the trinomial is given by (a+b+c)^3 = a^3 + b^3 + c^3 + 3 a^2 b + 3 a^2 c + 3 b^2 a + 3 b^2 c + 3 c^2 a + 3 c^2 b + 6 a b c. This can be computed by hand using the distributive property of multiplication over additi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |