Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighbourhood of the point. The existence of a complex derivative in a neighbourhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0. The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε where a and b are uniquely determined real numbers. The dual numbers can also be thought of as the exterior algebra of a one-dimensional vector space; the general case of n dimensions leads to the Grassmann numbers.
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.
In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relationship between points and lines of the plane determined by the conic. The technique does not require putting the equation of a conic section into a standard form, thus making it easier to investigate those conic sections whose axes are not parallel to the coordinate system.
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, 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.
In algebra, a cubic function is a function of the form
In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables. More generally an algebraic curve is similar but may be embedded in a higher dimensional space or defined over some more general field.
In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation
In mathematics, a real-valued function defined on a connected open set is said to have a conjugate (function) if and only if they are respectively the real and imaginary parts of a holomorphic function of the complex variable That is, is conjugate to if is holomorphic on As a first consequence of the definition, they are both harmonic real-valued functions on . Moreover, the conjugate of if it exists, is unique up to an additive constant. Also, is conjugate to if and only if is conjugate to .
In the mathematical field of algebraic geometry, a singular point of an algebraic varietyV is a point P that is 'special', in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.
In projective geometry, a collineation is a one-to-one and onto map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. The set of all collineations of a space to itself form a group, called the collineation group.
In complex geometry, an imaginary line is a straight line that only contains one real point. It can be proven that this point is the intersection point with the conjugated line.
A generalized circle, also referred to as a "cline" or "circline", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and are best treated together.
In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with 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, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties.
In geometry, trilinear polarity is a certain correspondence between the points in the plane of a triangle not lying on the sides of the triangle and lines in the plane of the triangle not passing through the vertices of the triangle. "Although it is called a polarity, it is not really a polarity at all, for poles of concurrent lines are not collinear lines." It was Poncelet (1788–1867), a French engineer and mathematician, who introduced the idea of the trilinear polar of a point in 1865.
