Ingeometry, aflat is anaffine subspace, i.e. a subset of anaffine space that is itself an affine space.[1] Particularly, in the case the parent space isEuclidean, a flat is aEuclidean subspace which inherits the notion ofdistance from its parent space.
In ann-dimensional space, there arek-flats of everydimensionk from 0 ton; flats one dimension lower than the parent space,(n − 1)-flats, are calledhyperplanes.
The flats in aplane (two-dimensional space) arepoints,lines, and the plane itself; the flats inthree-dimensional space are points, lines, planes, and the space itself. The definition of flat excludes non-straightcurves and non-planarsurfaces, which aresubspaces having different notions of distance:arc length andgeodesic length, respectively.
Flats occur inlinear algebra, as geometric realizations of solution sets ofsystems of linear equations.
A flat is amanifold and analgebraic variety, and is sometimes called alinear manifold orlinear variety to distinguish it from other manifolds or varieties.
A flat can be described by asystem of linear equations. For example, a line in two-dimensional space can be described by a single linear equation involvingx andy:
In three-dimensional space, a single linear equation involvingx,y, andz defines a plane, while a pair of linear equations can be used to describe a line. In general, a linear equation inn variables describes a hyperplane, and a system of linear equations describes theintersection of those hyperplanes. Assuming the equations are consistent andlinearly independent, a system ofk equations describes a flat of dimensionn −k.
A flat can also be described by a system of linearparametric equations. A line can be described by equations involving oneparameter:
while the description of a plane would require two parameters:
In general, a parameterization of a flat of dimensionk would requirek parameters, e.g.t1, …, tk.
Anintersection of flats is either a flat or theempty set.
If each line from one flat is parallel to some line from another flat, then these two flats areparallel. Two parallel flats of the same dimension either coincide or do not intersect; they can be described by two systems of linear equations which differ only in their right-hand sides.
If flats do not intersect, and no line from the first flat is parallel to a line from the second flat, then these areskew flats. It is possible only if sum of their dimensions is less than dimension of the ambient space.
For two flats of dimensionsk1 andk2 there exists the minimal flat which contains them, of dimension at mostk1 +k2 + 1. If two flats intersect, then the dimension of the containing flat equals tok1 +k2 minus the dimension of the intersection.
These two operations (referred to asmeet andjoin) make the set of all flats in the Euclideann-space alattice and can build systematic coordinates for flats in any dimension, leading toGrassmann coordinates or dual Grassmann coordinates. For example, a line in three-dimensional space is determined by two distinct points or by two distinct planes.
However, the lattice of all flats is not adistributive lattice.If two linesℓ1 andℓ2 intersect, thenℓ1 ∩ ℓ2 is a point. Ifp is a point not lying on the same plane, then(ℓ1 ∩ ℓ2) +p = (ℓ1 +p) ∩ (ℓ2 +p), both representing a line. But whenℓ1 andℓ2 are parallel, thisdistributivity fails, givingp on the left-hand side and a third parallel line on the right-hand side.
The aforementioned facts do not depend on the structure being that of Euclidean space (namely, involvingEuclidean distance) and are correct in anyaffine space. In a Euclidean space:
An affine subspace is also called aflat by some authors.