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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 ...
The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography. In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical ...
An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue, if the eigenvalues are distinct. [ 1 ] A biorthogonal system in which E = F {\displaystyle E=F} and v ~ i = u ~ i {\displaystyle {\tilde {v}}_{i}={\tilde {u}}_{i}} is an orthonormal system .
The edge bipartization problem is the algorithmic problem of deleting as few edges as possible to make a graph bipartite and is also an important problem in graph modification algorithmics. This problem is also fixed-parameter tractable , and can be solved in time O ( 2 k m 2 ) {\textstyle O\left(2^{k}m^{2}\right)} , [ 33 ] where k is the ...
Here the vectors N, B and the torsion are not well defined. If the torsion is always zero then the curve will lie in a plane. A curve may have nonzero curvature and zero torsion. For example, the circle of radius R given by r(t) = (R cos t, R sin t, 0) in the z = 0 plane has zero torsion and curvature equal to 1/R. The converse, however, is false.
Sylvester's theorem or Sylvester's formula describes a particular interpretation of the sum of three pairwise distinct vectors of equal length in the context of triangle geometry. It is also referred to as Sylvester's (triangle) problem in literature, when it is given as a problem rather than a theorem.
Contravariant vectors are often just called vectors and covariant vectors are called covectors or dual vectors. The terms covariant and contravariant were introduced by James Joseph Sylvester in 1851. [3] [4] Curvilinear coordinate systems, such as cylindrical or spherical coordinates, are often used in physical and geometric problems ...
In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R 3 , the Gauss map is a map N : X → S 2 (where S 2 is the unit sphere ) such that for each p in X , the function value N ( p ) is ...