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A description of the projective geometry can be constructed in the geometric algebra using basic operations. For example, given two distinct points in RP n−1 represented by vectors a and b the line containing them is given by a ∧ b (or b ∧ a). Two lines intersect in a point if A ∧ B = 0 for their bivectors A and B. This point is given ...
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.
In the study of geometric algebras, a k-blade or a simple k-vector is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a k-blade is a k-vector that can be expressed as the exterior product (informally wedge product) of 1-vectors, and is of grade k. In detail: [1] A 0-blade is a ...
In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors.Geometric algebra is built out of two fundamental operations, addition and the geometric product.
Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. [1] There is also a third system, based on two poles ( biangular coordinates ). The term "bipolar" is further used on occasion to describe other curves having two singular points (foci), such as ellipses , hyperbolas , and Cassini ovals .
Plane-based GA usually handles the (3D version of) the middle case here. But we instead choose to have a basis element squaring to 1 or −1 instead of 0, euclidean geometry can be changed to spherical or hyperbolic geometry. In these systems, the points, planes, and lines have the same coordinates that they have in plane-based GA.
The point x = 0 in R p,q maps to n o in R p+1,q+1, so n o is identified as the (representation) vector of the point at the origin. A vector in R p+1,q+1 with a nonzero n ∞ coefficient, but a zero n o coefficient, must (considering the inverse map) be the image of an infinite vector in R p,q.
The tangent vectors e 1 and e 2 of a frame on M define smooth functions from E with values in R 3, so each gives a 3-vector of functions and in particular de 1 is a 3-vector of 1-forms on E. The connection form is given by