In geometry, a flat is an affine subspace, i.e. a subset of an affine space that is itself an affine space. Particularly, in the case the parent space is Euclidean, a flat is a Euclidean subspace which inherits the notion of distance from its parent space.
In an n-dimensional space, there are k-flats of every dimension k from 0 to n; flats one dimension lower than the parent space, (n − 1)-flats, are called hyperplanes.
The flats in a plane (two-dimensional space) are points, lines, and the plane itself; the flats in three-dimensional space are points, lines, planes, and the space itself. 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.
A flat is a manifold and an algebraic variety, and is sometimes called a linear manifold or linear variety to distinguish it from other manifolds or varieties.
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:
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:
while the description of a plane would require two parameters:
In general, a parameterization of a flat of dimension k would require k parameters, e.g. t1, …, tk.
Operations and relations on flats
Intersecting, parallel, and skew flats
An intersection of flats is either a flat or the empty set.
If each line from one flat is parallel to some line from another flat, then these two 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 k1 and k2 there exists the minimal flat which contains them, of dimension at most k1 + k2 + 1. If two flats intersect, then the dimension of the containing flat equals to k1 + k2 minus the 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.
However, the lattice of all flats is not a distributive lattice. If two lines ℓ1 and ℓ2 intersect, then ℓ1 ∩ ℓ2 is a point. If p is a point not lying on the same plane, then (ℓ1 ∩ ℓ2) + p = (ℓ1 + p) ∩ (ℓ2 + p), both representing a line. But when ℓ1 and ℓ2 are parallel, this distributivity fails, giving p on the left-hand side and a third parallel line on the right-hand side.
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.)
- 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). See also Angles between flats.)
See also
- N-dimensional space
- Matroid
- Coplanarity
- Isometry
Notes
- Gallier, J. (2011). "Basics of Affine Geometry". Geometric Methods and Applications. New York: Springer. doi:10.1007/978-1-4419-9961-0_2. p. 21:
An affine subspace is also called a flat by some authors.
References
- Heinrich Guggenheimer (1977), Applicable Geometry, Krieger, New York, page 7.
- 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 Archived 2021-10-17 at the Wayback Machine.
External links
- Weisstein, Eric W. "Hyperplane". MathWorld.
- Weisstein, Eric W. "Flat". MathWorld.
In geometry a flat is an affine subspace i e a subset of an affine space that is itself an affine space Particularly in the case the parent space is Euclidean a flat is a Euclidean subspace which inherits the notion of distance from its parent space In an n dimensional space there are k flats of every dimension k from 0 to n flats one dimension lower than the parent space n 1 flats are called hyperplanes The flats in a plane two dimensional space are points lines and the plane itself the flats in three dimensional space are points lines planes and the space itself 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 A flat is a manifold and an algebraic variety and is sometimes called a linear manifold or linear variety to distinguish it from other manifolds or varieties DescriptionsBy 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 displaystyle 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 tz 32 4t displaystyle x 2 3t y 1 t z frac 3 2 4t while the description of a plane would require two parameters x 5 2t1 3t2 y 4 t1 2t2z 5t1 3t2 displaystyle 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 k parameters e g t1 tk Operations and relations on flatsIntersecting parallel and skew flats An intersection of flats is either a flat or the empty set If each line from one flat is parallel to some line from another flat then these two 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 k1 and k2 there exists the minimal flat which contains them of dimension at most k1 k2 1 If two flats intersect then the dimension of the containing flat equals to k1 k2 minus the 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 However the lattice of all flats is not a distributive lattice If two lines ℓ1 and ℓ2 intersect then ℓ1 ℓ2 is a point If p is a point not lying on the same plane then ℓ1 ℓ2 p ℓ1 p ℓ2 p both representing a line But when ℓ1 and ℓ2 are parallel this distributivity fails giving p on the left hand side and a third parallel line on the right hand side Euclidean geometryThe 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 There is the angle between two flats which belongs to the interval 0 p 2 between 0 and the right angle See for example Dihedral angle between two planes See also Angles between flats See alsoN dimensional space Matroid Coplanarity IsometryNotesGallier J 2011 Basics of Affine Geometry Geometric Methods and Applications New York Springer doi 10 1007 978 1 4419 9961 0 2 p 21 An affine subspace is also called a flat by some authors ReferencesHeinrich Guggenheimer 1977 Applicable Geometry Krieger New York page 7 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 Archived 2021 10 17 at the Wayback Machine External linksWeisstein Eric W Hyperplane MathWorld Weisstein Eric W Flat MathWorld
