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CGA has particularly been applied in connection with the projective mapping of the everyday Euclidean space R 3 into a five-dimensional vector space R 4,1, which has been investigated for applications in robotics and computer vision. It can be applied generally to any pseudo-Euclidean space - for example, Minkowski space R 3,1 to the space R 4,2.
In the late 1990s, plane-based geometric algebra and conformal geometric algebra (CGA) respectively provided a framework for euclidean geometry and classical geometries. [2] In practice, these and several derived operations allow a correspondence of elements, subspaces and operations of the algebra with geometric interpretations.
Inversive geometry itself can be performed with the larger system known as Conformal Geometric Algebra(CGA), of which Plane-based GA is a subalgebra. CGA is also usually applied to 3D space, and is able to model general spheres, circles, and conformal (angle-preserving) transformations, which include the transformations seen on the Poincare ...
In two dimensions, every conformal metric is locally conformally flat. In dimension n > 3 a conformal metric is locally conformally flat if and only if its Weyl tensor vanishes; in dimension n = 3, if and only if the Cotton tensor vanishes. Conformal geometry has a number of features which distinguish it from (pseudo-)Riemannian geometry.
Poincaré disk with hyperbolic parallel lines Poincaré disk model of the truncated triheptagonal tiling.. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or ...
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
Geometric constraint solving is constraint satisfaction in a computational geometry setting, which has primary applications in computer aided design. [1] A problem to be solved consists of a given set of geometric elements and a description of geometric constraints between the elements, which could be non-parametric (tangency, horizontality, coaxiality, etc) or parametric (like distance, angle ...
The geometry and topology of three-manifolds is a set of widely circulated notes for a graduate course taught at Princeton University by William Thurston from 1978 to 1980 describing his work on 3-manifolds. They were written by Thurston, assisted by students William Floyd and Steven Kerchoff. [1]