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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.
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.
where s and t range over all real numbers, v and w are given linearly independent vectors defining the plane, and r 0 is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors v and w can be visualized as vectors starting at r 0 and pointing in different directions along the plane.
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.
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 ...
The signature of a metric tensor is defined as the signature of the corresponding quadratic form. [2] It is the number (v, p, r) of positive, negative and zero eigenvalues of any matrix (i.e. in any basis for the underlying vector space) representing the form, counted with their algebraic multiplicities.
Consider the point 1 ∈ R +, and x ∈ R an element of the tangent space at 1. The usual straight line emanating from 1, namely y ( t ) = 1 + xt covers the same path as a geodesic, of course, except we have to reparametrize so as to get a curve with constant speed ("constant speed", remember, is not going to be the ordinary constant speed ...