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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 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.
The local (non-unit) basis vector is b 1 (notated h 1 above, with b reserved for unit vectors) and it is built on the q 1 axis which is a tangent to that coordinate line at the point P. The axis q 1 and thus the vector b 1 form an angle with the Cartesian x axis and the Cartesian basis vector e 1. It can be seen from triangle PAB that
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 ...
Generalizing this to an n-dimensional vector space, the Hodge star is a one-to-one mapping of k-vectors to (n – k)-vectors; the dimensions of these spaces are the binomial coefficients = (). The naturalness of the star operator means it can play a role in differential geometry, when applied to the cotangent bundle of a pseudo-Riemannian ...
A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.
Some r-vectors are scalars (r = 0), vectors (r = 1) and bivectors (r = 2). One may generate a finite-dimensional GA by choosing a unit pseudoscalar (I). The set of all vectors that satisfy = is a vector space. The geometric product of the vectors in this vector space then defines the GA, of which I is a member.