Flat (geometry)

In geometry, a flat is a subset of $n$ -dimensional space that is congruent to a Euclidean space of lower dimension. The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes. In $n$ -dimensional space, there are flats of every dimension from 0 to $n - 1$ .^[1] Flats of dimension $n - 1$ are called hyperplanes.

Flats are similar to linear subspaces, except that they need not pass through the origin. If Euclidean space is considered as an affine space, the flats are precisely the affine subspaces. Flats are important in linear algebra, where they provide a geometric realization of the solution set for a system of linear equations.

A flat is also called a linear manifold or linear variety.

Descriptions

By equations

A flat can be described by a system of linear equations. For example, a line in two-dimensional space can be described by a single linear equation involving $x$ and $y$ :

3x + 5y = 8.

In three-dimensional space, a single linear equation involving $x$ , $y$ , and $z$ defines a plane, while a pair of linear equations can be used to describe a line. In general, a linear equation in $n$ variables describes a hyperplane, and a system of linear equations describes the intersection of those hyperplanes. Assuming the equations are consistent and linearly independent, a system of $k$ equations describes a flat of dimension $n - k$ .

Parametric

A flat can also be described by a system of linear parametric equations. A line can be described by equations involving one parameter:

x=2+3t,\;\;\;\;y=-1+t\;\;\;\;z=\frac{3}{2}-4t

while the description of a plane would require two parameters:

x=5+2t_1-3t_2,\;\;\;\; y=-4+t_1+2t_2\;\;\;\;z=5t_1-3t_2.\,\!

In general, a parameterization of a flat of dimension $k$ would require parameters $t 1, \dots , t k$ .

Operations and relations on flats

Intersecting, parallel, and skew flats

An intersection of flats is either a flat or the empty set.^[2]

If every line from the first flat is parallel to some line from the second flat, then these flats are parallel. 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 are skew flats. It is possible only if sum of their dimensions is less than dimension of the ambient space.

Join

For two flats of dimensions $k 1$ and $k 2$ there exists the minimal flat which contains them, of dimension at most $k 1 + k 2 + 1$ . If two flats intersect, then the dimension of the containing flat equals to $k 1 + k 2 -$ dimension of the intersection.

Properties of operations

These two operations (referred to as meet and join) make the set of all flats in the Euclidean $n$ -space a lattice and can build systematic coordinates for flats in any dimension, leading to Grassmann coordinates or dual Grassmann coordinates. For example, a line in three-dimensional space is determined by two distinct points or by two distinct planes.

Though, the lattice of all flats is not a distributive lattice. If two lines $ℓ 1$ and $ℓ 2$ intersect, then $ℓ 1 \cap ℓ 2$ is a point. If $p$ is a point not lying on the same plane, then $(ℓ 1 \cap ℓ 2) + p = (ℓ 1 + p) \cap (ℓ 2 + p)$ , both representing a line. But when $ℓ 1$ and $ℓ 2$ are parallel, this distributivity fails, giving $p$ on the left-hand side and a third parallel line on the right-hand side. The ambient space would be a projective space to accommodate intersections of parallel flats, which lead to objects "at infinity".

Euclidean geometry

The aforementioned facts do not depend on the structure being that of Euclidean space (namely, involving Euclidean distance) and are correct in any affine space. In a Euclidean space:

There is the distance between a flat and a point. (See for example Distance from a point to a plane and Distance from a point to a line.)
There is the distance between two flats, equal to 0 if they intersect. (See for example Distance between two lines (in the same plane) and Skew lines#Distance.)
If two flats intersect, then there is the angle between two flats, which belongs to the interval $[0, π /2]$ between 0 and the right angle. (See for example Dihedral angle (between two planes).)

Notes

↑ In addition, all of $n$ -dimensional space is sometimes considered an $n$ -dimensional flat as a subset of itself.
↑ Can be considered as −1-flat.

References

Heinrich Guggenheimer (1977) Applicable Geometry,page 7, Krieger, New York.
Stolfi, Jorge (1991), Oriented Projective Geometry, Academic Press, ISBN 978-0-12-672025-9
From original Stanford Ph.D. dissertation, Primitives for Computational Geometry, available as DEC SRC Research Report 36.
PlanetMath: linear manifold

External links

Weisstein, Eric W., "Hyperplane", MathWorld.
Weisstein, Eric W., "Flat", MathWorld.

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