Search results
Results From The WOW.Com Content Network
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 ...
The four bitangents of two disjoint convex polygons may be found efficiently by an algorithm based on binary search in which one maintains a binary search pointer into the lists of edges of each polygon and moves one of the pointers left or right at each steps depending on where the tangent lines to the edges at the two pointers cross each other.
Vertical and horizontal subspaces for the Möbius strip. The Möbius strip is a line bundle over the circle, and the circle can be pictured as the middle ring of the strip. . At each point on the strip, the projection map projects it towards the middle ring, and the fiber is perpendicular to the middle ri
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.
In linear algebra, given a vector space with a basis of vectors indexed by an index set (the cardinality of is the dimension of ), the dual set of is a set of vectors in the dual space with the same index set such that and form a biorthogonal system.
In mathematics, a biorthogonal system is a pair of indexed families of vectors ~ ~ such that ~, ~ =,, where and form a pair of topological vector spaces that are in duality, , is a bilinear mapping and , is the Kronecker delta.
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.
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