Clifford algebraIn mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and .
Screw axisA screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw axis, and the displacement can be decomposed into a rotation about and a slide along this screw axis. Plücker coordinates are used to locate a screw axis in space, and consist of a pair of three-dimensional vectors. The first vector identifies the direction of the axis, and the second locates its position.
Hyperbolic manifoldIn mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.
Quotient space (linear algebra)In linear algebra, the quotient of a vector space by a subspace is a vector space obtained by "collapsing" to zero. The space obtained is called a quotient space and is denoted (read " mod " or " by "). Formally, the construction is as follows. Let be a vector space over a field , and let be a subspace of . We define an equivalence relation on by stating that if . That is, is related to if one can be obtained from the other by adding an element of .
Hyperbolic sectorA hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a) and (b, 1/b) on the rectangular hyperbola xy = 1, or the corresponding region when this hyperbola is re-scaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola. A hyperbolic sector in standard position has a = 1 and b > 1. Hyperbolic sectors are the basis for the hyperbolic functions.
