In
Euclidean geometry, a plane is a
flat
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), ...
two-
dimensional
surface that extends indefinitely.
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
s often arise as
subspaces of
three-dimensional space .
A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.
While a pair of real numbers
suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their
embedding in the
ambient space .
Derived concepts
A or (or simply "plane", in lay use) is a planar surface
region; it is analogous to a
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
.
A ''
bivector'' is an
oriented
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
plane segment, analogous to
directed line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between it ...
s.
A ''
face'' is a plane segment bounding a
solid object
In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
.
A ''
slab'' is a region bounded by two parallel planes.
A ''
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
'' is a region bounded by three pairs of parallel planes.
Background
Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry. He selected a small core of undefined terms (called ''common notions'') and postulates (or
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s) which he then used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the ''
Elements
Element or elements may refer to:
Science
* Chemical element, a pure substance of one type of atom
* Heating element, a device that generates heat by electrical resistance
* Orbital elements, parameters required to identify a specific orbit of ...
'', it may be thought of as part of the common notions. Euclid never used numbers to measure length, angle, or area. The Euclidean plane equipped with a chosen
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
is called a ''Cartesian plane''; a non-Cartesian Euclidean plane equipped with a
polar coordinate system would be called a ''polar plane''.
A plane is a
ruled surface.
Euclidean plane
Representation
This section is solely concerned with planes embedded in three dimensions: specifically, in
.
Determination by contained points and lines
In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following:
* Three non-
collinear points (points not on a single line).
* A line and a point not on that line.
* Two distinct but intersecting lines.
* Two distinct but
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of IBM ...
lines.
Properties
The following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues:
* Two distinct planes are either parallel or they intersect in a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
.
* A line is either parallel to a plane, intersects it at a single point, or is contained in the plane.
* Two distinct lines
perpendicular to the same plane must be parallel to each other.
* Two distinct planes perpendicular to the same line must be parallel to each other.
Point–normal form and general form of the equation of a plane
In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
) to indicate its "inclination".
Specifically, let be the position vector of some point , and let be a nonzero vector. The plane determined by the point and the vector consists of those points , with position vector , such that the vector drawn from to is perpendicular to . Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points such that
The dot here means a
dot (scalar) product.
Expanded this becomes
which is the ''point–normal'' form of the equation of a plane. This is just a
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
where
which is the expanded form of
In mathematics it is a common convention to express the normal as a
unit vector, but the above argument holds for a normal vector of any non-zero length.
Conversely, it is easily shown that if , , , and are constants and , , and are not all zero, then the graph of the equation
is a plane having the vector as a normal.
This familiar equation for a plane is called the ''general form'' of the equation of the plane or just the ''plane equation''.
Thus for example a
regression equation of the form (with ) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.
Describing a plane with a point and two vectors lying on it
Alternatively, a plane may be described parametrically as the set of all points of the form
where and range over all real numbers, and are given
linearly independent vectors defining the plane, and is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors and can be visualized as vectors starting at and pointing in different
directions
Direction may refer to:
*Relative direction, for instance left, right, forward, backwards, up, and down
** Anatomical terms of location for those used in anatomy
** List of ship directions
*Cardinal direction
Mathematics and science
*Direction ...
along the plane. The vectors and can be
perpendicular, but cannot be parallel.
Describing a plane through three points
Let , , and be non-collinear points.
Method 1
The plane passing through , , and can be described as the set of all points (''x'',''y'',''z'') that satisfy the following
determinant equations:
Method 2
To describe the plane by an equation of the form
, solve the following system of equations:
This system can be solved using
Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...
and basic matrix manipulations. Let
If is non-zero (so for planes not through the origin) the values for , and can be calculated as follows:
These equations are parametric in ''d''. Setting ''d'' equal to any non-zero number and substituting it into these equations will yield one solution set.
Method 3
This plane can also be described by the "#Point-normal form and general form of the equation of a plane, point and a normal vector" prescription above. A suitable normal vector is given by the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
and the point can be taken to be any of the given points , or
(or any other point in the plane).
Operations
Distance from a point to a plane
Line–plane intersection
Line of intersection between two planes
Sphere–plane intersection
Occurrence in nature
A plane serves as a mathematical model for many physical phenomena, such as
specular reflection in a
plane mirror
A plane mirror is a mirror with a flat (planar) reflective surface. For light rays striking a plane mirror, the angle of reflection equals the angle of incidence. The angle of the incidence is the angle between the incident ray and the surface n ...
or
wavefronts in a
traveling plane wave In mathematics and physics, a traveling plane wave is a special case of plane wave, namely a field (physics), field whose evolution in time can be described as simple translation of its values at a constant wave speed c, along a fixed direction of p ...
.
The
free surface of undisturbed liquids tends to be nearly flat (see
flatness).
The flattest surface ever manufactured is a quantum-stabilized atom mirror.
In astronomy, various ''
reference planes'' are used to define positions in orbit.
''
Anatomical planes'' may be lateral ("sagittal"), frontal ("coronal") or transversal.
In geology, ''
beds'' (layers of sediments) often are planar.
Planes are involved in different forms of
imaging, such as the ''
focal plane
In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the '' focal points'', the principal points, and the nodal points. For ''ideal'' ...
'', ''
picture plane'', and ''
image plane
In 3D computer graphics, the image plane is that plane in the world which is identified with the plane of the display monitor used to view the image that is being rendered. It is also referred to as screen space. If one makes the analogy of taking ...
''.
Miller indices
The attitude of a
lattice plane In crystallography, a lattice plane of a given Bravais lattice is any plane containing at least three noncollinear Bravais lattice points. Equivalently, a lattice plane is a plane whose intersections with the lattice (or any crystalline structure of ...
is the orientation of the line normal to the plane,
[
] and is described by the plane's
Miller indices
Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices.
In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''â ...
. In three-space a family of planes (a series of parallel planes) can be denoted by its
Miller indices
Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices.
In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''â ...
(''hkl''),
[
][
] so the family of planes has an attitude common to all its constituent planes.
Strike and dip
Many features observed in geology are planes or lines, and their orientation is commonly referred to as their ''attitude''. These attitudes are specified with two angles.
For a line, these angles are called the ''trend'' and the ''plunge''. The trend is the compass direction of the line, and the plunge is the downward angle it makes with a horizontal plane.
[
]
For a plane, the two angles are called its ''strike (angle)'' and its ''dip (angle)''. A ''strike line'' is the intersection of a horizontal plane with the observed planar feature (and therefore a horizontal line), and the strike angle is the ''bearing'' of this line (that is, relative to
geographic north
True north (also called geodetic north or geographic north) is the direction along Earth's surface towards the geographic North Pole or True North Pole.
Geodetic north differs from ''magnetic'' north (the direction a compass points toward the ...
or from
magnetic north). The dip is the angle between a horizontal plane and the observed planar feature as observed in a third vertical plane perpendicular to the strike line.
See also
*
Flat (geometry)
*
Half-plane
*
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
*
Plane coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
*
Plane of incidence
*
Plane of rotation
*
Plane orientation
*
Polygon
Notes
Explanatory notes
Citations
References
*
*
*
External links
*
*
"Easing the Difficulty of Arithmetic and Planar Geometry"is an Arabic manuscript, from the 15th century, that serves as a tutorial about plane geometry and arithmetic.
{{DEFAULTSORT:Plane (Geometry)
*