Inversive geometry - Inversive geometry In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. This is an angle-preserving geometry, which is why it is called conformal. For greater than two dimensions, this is also the same as conformal geometry. For two dimensions, however, conformal geometry is simply the Riemann sphere. Basically, in the spirit of the Erlanger program, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversions, which in coordinate form, basically are conjugate to where r is the radius of the inversion. Note that in inversive geometry, there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is just nothing more and nothing less.
Integral geometry - Integral geometry In mathematics, the term integral geometry in is used in two ways, which, although related, imply different views of the content of the subject. The more traditional usage is that of Santalo and Blaschke. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the affine group of the plane acts. A probability measure is sought on this space, invariant under the symmetry group. If, as in this case, we can find a unique such invariant measure, that solves the problem of formulating accurately what.
Incidence (geometry) - Incidence (geometry) In geometry, the relations of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L'), and 'intersects' (as in 'line L1 intersects line L2', in three-dimensional space). That is, they are the binary relations describing how subsets meet. The propositions of incidence stated in terms of them are statements such as 'any two lines in a plane meet'. This is true in a projective plane, though not true in Euclidean space of two dimensions where lines may be parallel. Historically, projective geometry was introduced in order to make the propositions of incidence true (without exceptions such as are caused by parallels). From the point of view of synthetic geometry it was considered that projective geometry should.
Italian school of algebraic geometry - Italian school of algebraic geometry In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major contributions; about half of those being in fact Italian. There is no question that the leadership fell to the group in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi; who were involved in some of the deepest discoveries, as well as setting the style. The fashion and foundational attitude changed in algebraic geometry from 1950 onwards, leading to an axiomatisation and some acrimony as to the status of some results. For a while it.
Hyperbolic geometry - Hyperbolic geometry Hyperbolic geometry, also called saddle or Lobachevskian geometry, is the non-Euclidean geometry obtained by replacing the parallel postulate with the hyperbolic postulate, which states: "Given a line L and any point A not on L, at least two distinct lines exist which pass through A and are parallel to L." In this case parallel means that the lines do not intersect L, even when extended, rather than that they are a constant distance from L. Hyperbolic geometry was explored by Saccheri in the 1700s, who nevertheless believed it was inconsistent, and later by Bolyai, Gauss, and Lobachevsky, after whom it is sometimes named. (See article on non-Euclidean geometry for more history.) There are three models commonly used for hyperbolic geometry. The Klein model uses the.
Geometry - Geometry simple:Geometry Geometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible of proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. Because of its immediate practical applications, geometry was one of the first branches of mathematics to be developed. Likewise, it was the first field to be put on an axiomatic basis, by Euclid. The Greeks were interested in many questions about ruler-and-compass constructions. The next most significant development had to wait until a millennium later, and that was analytic geometry, in which coordinate systems are introduced and points are.
Geometry of numbers - Geometry of numbers In number theory, the geometry of numbers refers to a topic and method arising from the work of Hermann Minkowski, on the relationship between convex sets and lattices in n-dimensional space. It has frequently been used in an auxiliary role in proofs, particularly in diophantine approximation. The subject was given a great deal of attention in the period 1930-1960 by some leading number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel). To begin with, Minkowski's theorem establishes a relation between symmetric convex sets and integer points; we might as well say, between any lattice and any Banach space norm in n dimensions. The topic therefore belongs properly to a sort of affine geometry simplification of the theory of quadratic forms (Hilbert.
Finite geometry - Finite geometry Finite geometry describes any geometric system that has only a finite number of points. Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact precisely the same number of points as there are real numbers. There are two main kinds of finite geometry: affine and projective. In an affine geometry, the parallel postulate holds, meaning that the normal sense of parallel lines applies. In a projective geometry, by contrast, any two lines intersect at a unique point, and so parallel lines do not exist. Both finite affine geometry and finite projective geometry may be described by fairly simple axioms. For affine geometry, the axioms are as follows: Given any two distinct points, there is exactly one line that.
Finsler geometry - Finsler geometry A Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property: For each point x of M, and for every vector v in the tangent space TxM, the second derivative of the function L:TxM->R given by at v is positive definite. Riemannian manifolds (but not pseudo Riemannian manifolds) are special cases of Finsler manifolds. The length of γ, a differentiable curve in M is given by . Note that it's reparametrization invariant. Geodesics are curves in M whose length is extremal under functional derivatives. This article is a stub. You can help Wikipedia by fixing it..
Elliptic geometry - Elliptic geometry Elliptic geometry is a non-Euclidean geometry developed by German geometer Bernhard Riemann. It is based on a change in Euclid's fifth postulate, that through a point outside a line, there is one line parallel to the other line. In elliptic geometry, there are no parallel lines at all. It is very much like spherical geometry. Elliptic geometry has the property that a triangle will have more than 180 degrees. This is best understood on a globe. Two right angles on the equator, a quarter of the circumference away from each other, make the base, and the angle on the pole is also 90 degrees. This creates a 270 degree triangle, impossible in Euclidean geometry..
Elliptic or Riemannian geometry - Elliptic or Riemannian geometry Elliptic geometry or Riemannian geometry is a non-Euclidean geometry, in which, given a line A and a point B outside A, there exists no line parallel to A passing through B. Its simplest model is that of the Spherical geometry..
Vector (spatial) - object with a "magnitude" (size) and "direction", a vector is more formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry. These definitions are discussed in more detail below. Such a vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry). Vectors are the building blocks of vector fields and vector calculus. Table of contents showTocToggle("show","hide") 1 Definitions 1..1 Generalizations 2 Representation of a vector 3 Vector Equality 4 Vector Addition and Subtraction.
Karl Pearson - the German department at Cambridge University. His next career move was to Lincoln's Inn, where he read law until 1881 (although he never practised). After this, he returned to mathematics, deputising for the mathematics professor at King's College London in 1881 and for the professor at University College London in 1883. In 1884, he was appointed to the Goldshmid Chair of Applied Mathematics and Mechanics at University College London. 1891 saw him also appointed to the professorship of Geometry at Gresham College; here he met W.F.R. Weldon, a zoologist who had some interesting problems requiring quantitative solutions. The collaboration, in biometry and evolutionary theory, was a fruitful one and lasted until Weldon died in 1906. Weldon introduced Pearson to Francis Galton, who was interested in aspects of evolution such as heredity.
Vector calculus - and direction of change in a scalar field; the gradient of a scalar field is a vector field. curl: measures a vector field's tendency to rotate about a point; the curl of a vector field is another vector field. divergence: measures a vector field's tendency to originate from or converge upon certain points; the divergence of a vector field is a scalar field. Most of the analytic results are more easily understood using the machinery of differential geometry, for which vector calculus forms a subset..
Kähler manifold - gives rise to complex-linear mappings on the tangent spaces). In terms of local coordinates it is specified in this way: if is the metric, then the associated Kähler form ω defined up to a factor of i/2 by is closed: that is, dω = 0. If M carries such a metric it is called a Kähler manifold. Such metrics are common because there are simple examples: on Cn the usual metric from 2n-dimensional Euclidean space is one. Important for algebraic geometry is the Fubini-Study metric on complex projective space. It is essentially determined by the condition that it is invariant under the action of the unitary group (of dimension one larger, acting on the complex vector space giving rise to the projective space). The restriction properties of the Fubini-Study metric mean.
Vertex - vertex (Latin: whirl, whirlpool; plural vertices), in geometry, is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). In 3D computer graphics, a vertex is a point in 3D space with a particular location, usually given in terms of its x, y, and z coordinates. It is one of the fundamental structures in polygonal modelling: two vertices, taken together, can be used to define the endpoints of a line; three vertices can be used to define a planar triangle. In graph theory, a graph describes a set of connections between nodes. Each node or vertex can map to an object. The connections between the nodes are called edges or arcs. See also parabola..
Kissing number problem - Kissing number problem In geometry, the kissing number problem is to find the maximal number of spheres of radius 1 that can simultaneously touch the unit sphere in n-dimensional Euclidean space (or, with the restriction for their centres to be in a particular lattice). See also: sphere packing.
Vehicle dynamics - Vehicle dynamics Definitions Ackermann steering geometry Camber angle Castor angle Circle of forces Live axle Oversteer Understeer Unsprung weight Performance Driving Techniques Double declutching Handbrake turn Heel-and-Toe Left-foot braking Opposite lock.
J. H. C. Whitehead - He was brought up in Oxford, as was educated at Eton College and Balliol College of Oxford University, reading mathematics. After a year working as a stockbroker, he started a Ph.D. in Princeton in differential geometry under Veblen in 1929. He worked also with Lefschetz. He became a fellow of Balliol in 1933. During the Second World War he worked on operations research for submarine warfare. He became a professor at Oxford in 1947. His definition of CW complexes gave a setting for homotopy theory that became standard. He introduced the idea of simple homotopy theory, which was later much developed in connection with algebraic K-theory. The Whitehead product is an operation in homotopy theory. The Whitehead problem on abelian groups was solved (as an independence proof) by Saharon Shelah. He.
James Clerk Maxwell - fundamental models of nature. In 1931, on the centennial anniversary of Maxwell's birth, Einstein described Maxwell's work as the "most profound and the most fruitful that physics has experienced since the time of Newton." Algebraic mathematics with elements of geometry are a feature of much of Maxwell's work. Maxwell demonstrated that electric and magnetic forces are two complementary aspects of electromagnetism. He showed that electric and magnetic fields travel through space, in the form of waves, at a constant velocity of 3.0 × 108 m/s. He also proposed that light was a form of electromagnetic radiation. The scientific compound derived CGS unit measuring magnetic flux (commonly abbreviated as f ), the maxwell (Mx), was named in his honor. There is a mountain range on Venus, Maxwell Montes, named after James Clerk.
