InEuclidean geometry, aplane is aflat two-dimensionalsurface that extends indefinitely.Euclidean planes often arise assubspaces ofthree-dimensional space${\displaystyle {\mathbb{R}}^3}$.A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.While a pair of real numbers${\displaystyle {\mathbb{R}}^2}$ suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for theirembedding in theambient space${\displaystyle {\mathbb{R}}^3}$.

Aplane segment orplanar region (or simply "plane", in lay use) is a planar surfaceregion; it is analogous to aline segment.Abivector is anoriented plane segment, analogous todirected line segments.^[a]Aface is a plane segment bounding asolid object.^[1]Aslab is a region bounded by two parallel planes.Aparallelepiped is a region bounded by three pairs of parallel planes.

Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry.^[2] He selected a small core of undefined terms (calledcommon notions) and postulates (oraxioms) which he then used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in theElements, it may be thought of as part of the common notions.^[3] Euclid never used numbers to measure length, angle, or area. The Euclidean plane equipped with a chosenCartesian coordinate system is called aCartesian plane; a non-Cartesian Euclidean plane equipped with apolar coordinate system would be called apolar plane.

A plane is aruled surface.

This section is an excerpt fromEuclidean plane.[edit]

Inmathematics, aEuclidean plane is aEuclidean space ofdimension two, denoted${\displaystyle {\mathbf{\text{E}}}^2}$ or${\displaystyle {\mathbb{E}}^2}$. It is ageometric space in which tworeal numbers are required to determine theposition of eachpoint. It is anaffine space, which includes in particular the concept ofparallel lines. It has alsometrical properties induced by adistance, which allows to definecircles, andangle measurement.

A Euclidean plane with a chosenCartesian coordinate system is called aCartesian plane.

The set${\displaystyle {\mathbb{R}}^2}$ of the ordered pairs of real numbers (thereal coordinate plane), equipped with thedot product, is often calledthe Euclidean plane orstandard Euclidean plane, since every Euclidean plane isisomorphic to it.

This section is solely concerned with planes embedded in three dimensions: specifically, inR³.

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 butparallel lines.

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 aline.
• A line is either parallel to a plane, intersects it at a single point, or is contained in the plane.
• Two distinct linesperpendicular to the same plane must be parallel to each other.
• Two distinct planes perpendicular to the same line must be parallel to each other.

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 (thenormal vector) to indicate its "inclination".

Specifically, letr₀ be the position vector of some pointP₀ = (x₀,y₀,z₀), and letn = (a,b,c) be a nonzero vector. The plane determined by the pointP₀ and the vectorn consists of those pointsP, with position vectorr, such that the vector drawn fromP₀ toP is perpendicular ton. 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 pointsr such that$${\displaystyle \mathit{n}\cdot (\mathit{r}-{\mathit{r}}_0)=0.}$$The dot here means adot (scalar) product.
Expanded this becomes$${\displaystyle a(x-x_0)+b(y-y_0)+c(z-z_0)=0,}$$which is thepoint–normal form of the equation of a plane.^[4] This is just alinear equation$${\displaystyle ax+by+cz+d=0,}$$where$${\displaystyle d=-(a{x}_0+b{y}_0+c{z}_0),}$$which is the expanded form of${\displaystyle -\mathit{n}\cdot {\mathit{r}}_0.}$

In mathematics it is a common convention to express the normal as aunit vector, but the above argument holds for a normal vector of any non-zero length.

Conversely, it is easily shown that ifa,b,c, andd are constants anda,b, andc are not all zero, then the graph of the equation$${\displaystyle ax+by+cz+d=0,}$$is a plane having the vectorn = (a,b,c) as a normal.^[5] This familiar equation for a plane is called thegeneral form of the equation of the plane or just theplane equation.^[6]

Thus for example aregression equation of the formy =d +ax +cz (withb = −1) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.

Alternatively, a plane may be described parametrically as the set of all points of the form$${\displaystyle \mathit{r}={\mathit{r}}_0+s\mathit{v}+t\mathit{w},}$$

wheres andt range over all real numbers,v andw are givenlinearly independentvectors defining the plane, andr₀ is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectorsv andw can be visualized as vectors starting atr₀ and pointing in differentdirections along the plane. The vectorsv andw can beperpendicular, but cannot be parallel.

Letp₁ = (x₁,y₁,z₁),p₂ = (x₂,y₂,z₂), andp₃ = (x₃,y₃,z₃) be non-collinear points.

The plane passing throughp₁,p₂, andp₃ can be described as the set of all points (x,y,z) that satisfy the followingdeterminant equations:

To describe the plane by an equation of the form${\displaystyle ax+by+cz+d=0}$, solve the following system of equations:$${\displaystyle a{x}_1+b{y}_1+c{z}_1+d=0}$$$${\displaystyle a{x}_2+b{y}_2+c{z}_2+d=0}$$$${\displaystyle a{x}_3+b{y}_3+c{z}_3+d=0.}$$

This system can be solved usingCramer's rule and basic matrix manipulations. Let

IfD is non-zero (so for planes not through the origin) the values fora,b andc can be calculated as follows:

These equations are parametric ind. Settingd equal to any non-zero number and substituting it into these equations will yield one solution set.

This plane can also be described by the§ Point–normal form and general form of the equation of a plane prescription above. A suitable normal vector is given by thecross product$${\displaystyle \mathit{n}=({\mathit{p}}_2-{\mathit{p}}_1)\times ({\mathit{p}}_3-{\mathit{p}}_1),}$$and the pointr₀ can be taken to be any of the given pointsp₁,p₂ orp₃^[7] (or any other point in the plane).

This section is an excerpt fromDistance from a point to a plane.[edit]

InEuclidean space, thedistance from a point to a plane is the distance between a given point and itsorthogonal projection on the plane, theperpendicular distance to the nearest point on the plane.

It can be found starting with achange of variables that moves the origin to coincide with the given point then finding the point on the shiftedplane${\displaystyle ax+by+cz=d}$ that is closest to theorigin. The resulting point hasCartesian coordinates${\displaystyle (x,y,z)}$:

${\displaystyle {\displaystyle x=\frac{ad}{a^2+b^2+c^2},{\displaystyle y=\frac{bd}{a^2+b^2+c^2},{\displaystyle z=\frac{cd}{a^2+b^2+c^2}}}}}$.

The distance between the origin and the point${\displaystyle (x,y,z)}$ is${\displaystyle \sqrt{x^2+y^2+z^2}}$.

This section is an excerpt fromLine–plane intersection.[edit]

In analyticgeometry, the intersection of aline and aplane inthree-dimensional space can be theempty set, apoint, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.

Distinguishing these cases, and determining equations for the point and line in the latter cases, have use incomputer graphics,motion planning, andcollision detection.

This section is an excerpt fromPlane–plane intersection.[edit]

In analyticgeometry, the intersection of twoplanes inthree-dimensional space is aline.

A plane serves as a mathematical model for many physical phenomena, such asspecular reflection in aplane mirror orwavefronts in atraveling plane wave.Thefree surface of undisturbed liquids tends to be nearly flat (seeflatness).The flattest surface ever manufactured is a quantum-stabilized atom mirror.^[11]In astronomy, variousreference 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 ofimaging, such as thefocal plane,picture plane, andimage plane.

The attitude of alattice plane is the orientation of the line normal to the plane,^[12] and is described by the plane'sMiller indices. In three-space a family of planes (a series of parallel planes) can be denoted by itsMiller indices (hkl),^[13][14] so the family of planes has an attitude common to all its constituent planes.

Many features observed in geology are planes or lines, and their orientation is commonly referred to as theirattitude. These attitudes are specified with two angles.

For a line, these angles are called thetrend and theplunge. The trend is the compass direction of the line, and the plunge is the downward angle it makes with a horizontal plane.^[15]

For a plane, the two angles are called itsstrike (angle) and itsdip (angle). Astrike line is the intersection of a horizontal plane with the observed planar feature (and therefore a horizontal line), and the strike angle is thebearing of this line (that is, relative togeographic north or frommagnetic 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.

• Anton, Howard (1994),Elementary Linear Algebra (7th ed.), John Wiley & Sons,ISBN 0-471-58742-7
• Eves, Howard (1963),A Survey of Geometry, vol. I, Boston: Allyn and Bacon, Inc.
• Hobbs, Charles Austen (1921).Solid Geometry. Cambridge, G.H. Kent. pp. 396-400.LCCN 21016427.

Wikimedia Commons has media related toEuclidean planes in three-dimensional space.

• "Plane",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Weisstein, Eric W."Plane".MathWorld.
• "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.

• Planes (geometry)

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