Conformal group
Encyclopedia
In mathematics
, the conformal group is the group
of transformations from a space to itself that preserve all angles within the space. More formally, it is the group of transformations that preserve the conformal geometry
of the space. Several specific conformal groups are particularly important:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the conformal group is the group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of transformations from a space to itself that preserve all angles within the space. More formally, it is the group of transformations that preserve the conformal geometry
Conformal geometry
In mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces...
of the space. Several specific conformal groups are particularly important:
- The conformal orthogonal groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
. If V is a vector space with a quadratic formQuadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
Q, then the conformal orthogonal group CO(V,Q) is the group of linear transformations T of V such that for all x in V there exists a scalar λ such that
-
- The conformal orthogonal group is equal to the orthogonal groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
times the group of dilationDilationDilation refers to an enlargement or expansion in bulk or extent, the opposite of contraction. It derives from the Latin dilatare, "to spread wide".In physiology:* Pupillary dilation, dilation of the pupil of the eye...
s.
- The conformal group of the sphere. The group of conformal transformations of the n-sphere is generated by the inversions in circles. This group is also known as the Möbius group.