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The definition of flat excludes non-straight curves and non-planar surfaces, which are subspaces having different notions of distance: arc length and geodesic length, respectively. Flats occur in linear algebra, as geometric realizations of solution sets of systems of linear equations.
Flat (geometry), the generalization of lines and planes in an n-dimensional Euclidean space; Flat (matroids), a further generalization of flats from linear algebra to the context of matroids; Flat module in ring theory; Flat morphism in algebraic geometry; Flat space, a space with zero curvature
Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric, which determines how distances are measured near each point) or extrinsic (where the object under study is a part of some ambient flat Euclidean space).
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense. In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2 ...
The definition of a flat morphism of schemes results ... (see also below for the geometric meaning). Such flat extensions can be used to yield examples of flat ...
Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed. [2] Given P and Q in σ, the elliptic distance between them is the measure of the angle POQ, usually taken in radians. Arthur Cayley initiated the study of elliptic geometry when he wrote "On the definition of distance".
Let and be flats of dimensions and in the -dimensional Euclidean space .By definition, a translation of or does not alter their mutual angles. If and do not intersect, they will do so upon any translation of which maps some point in to some point in .
In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.