21 examples
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- In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers.
- An affine n-manifold is then a manifold which looks infinitesimally like n-dimensional affine space.
- It is also possible to provide a system of axioms for the higher-dimensional affine spaces which does not refer to the corresponding projective space.
- Many 2-dimensional affine buildings have been constructed using hyperbolic reflection groups or other more exotic constructions connected with orbifolds.
- Shifting it by a vector external to it, one obtains an n -dimensional affine space.
- Two-dimensional affine transformations can produce any combination of translation, scaling, reflection, rotation and shearing -- and nothing else.
- In geometry, an affine plane is a two-dimensional affine space.
- Choosing an n -dimensional affine space as before one observes that the affine space is embedded as a proper subset into the projective space.
- This reflects the space along an -dimensional affine subspace.
- In geometric language, the Beauville-Laszlo theorem allows one to glue sheaves on a one-dimensional affine scheme over an infinitesimal neighborhood of a point.
- Synthetically, affine planes are 2-dimensional affine geometries defined in terms of the relations between points and lines (or sometimes, in higher dimensions, hyperplanes).
- In a further development, he gave an alternative formulation of this idea that did not use the imaginary time coordinate, but represented the four variables (x, y, z, t) of space and time in coordinate form in a four dimensional affine space.
- Let V be a \kappa -dimensional affine space over a field F, Given a set define its closure to be its affine hull (i.e. the smallest affine subspace containing it).
- The Theorem of Eisenbud-Evans-Storch states that every algebraic variety in n-dimensional affine space is given geometrically (i.e. up to radical) by n polynomials.
- It is also possible to generalize the Radon transform still further by integrating instead over k-dimensional affine subspaces of R n, The X-ray transform is the most widely used special case of this construction, and is obtained by integrating over straight lines.
- For instance, if are homogeneous coordinates for n-dimensional projective space, then the equation defines a hyperplane at infinity for the n-dimensional affine space with coordinates, H is also called the ideal hyperplane.
- The k-dimensional affine subspaces of R n are in one-to-one correspondence with the linear subspaces of R n+1 that are in general position with respect to the plane x n+1 = 1.
- In one it is the manifold of all k-dimensional affine subspaces of R n (described on this page), while in the other the affine Grassmannian is a quotient of a group-ring based on formal Laurent series.
- Geometrically, the polynomial ring in two variables represents the two-dimensional affine space A 2, so the existence of this one-parameter family says that affine space admits non-commutative deformations to the space determined by the Weyl algebra.
- The Birkhoff polytope lies within an dimensional affine subspace of the n 2 -dimensional space of all matrices: this subspace is determined by the linear equality constraints that the sum of each row and of each column be one.