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Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space R p,q to null vectors in R p+1,q+1.
Conformal geometry has a number of features which distinguish it from (pseudo-)Riemannian geometry. The first is that although in (pseudo-)Riemannian geometry one has a well-defined metric at each point, in conformal geometry one only has a class of metrics. Thus the length of a tangent vector cannot be defined, but the angle between two ...
In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important: The conformal orthogonal group.
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Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space) and the conformal geometric algebra. Geometric calculus , an extension of GA that incorporates differentiation and integration , can be used to formulate other theories such as complex analysis and differential ...
In plane geometry there are three types of angles that may be preserved in a conformal map. [3] Each is hosted by its own real algebra, ordinary complex numbers, split-complex numbers, and dual numbers. The conformal maps are described by linear fractional transformations in each case. [4]
Many Dirac type operators have a covariance under conformal change in metric. This is true for the Dirac operator in euclidean space, and the Dirac operator on the sphere under Möbius transformations. Consequently, this holds true for Dirac operators on conformally flat manifolds and conformal manifolds which are simultaneously spin manifolds.
QGA is a super-algebra over , conformal geometric algebra (CGA) and , spacetime algebra (STA), which can each be defined within sub-algebras of QGA. CGA provides representations of spherical entities (points, spheres, planes, and lines) and a complete set of operations ( translation , rotation , dilation , and intersection ) that apply to them.