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Encyclopedia > Plane (mathematics)
Two intersecting planes in three-dimensional space

In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. Informally it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some space. Look up plane in Wiktionary, the free dictionary. ... Image File history File links PlaneIntersection. ... Image File history File links PlaneIntersection. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. ... Space has been an interest for philosophers and scientists for much of human history. ...

When working in two-dimensional Euclidean space, the definite article is used, the plane, to refer to the whole space. Many fundamental tasks in geometry, trigonometry, and graphing are performed in two-dimensional space, or in other words, in the plane. A lot of mathematics can be and has been performed in the plane, notably in the areas of geometry, trigonometry, graph theory and graphing. Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigÅnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigÅnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ...

In Euclidean space a plane is a surface such that, given any two distinct points on the surface, the surface also contains the unique straight line that passes through those points. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... An open surface with X-, Y-, and Z-contours shown. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...

The fundamental structure of two such planes will always be the same. In mathematics this is described as topological equivalence. Informally though, it means that any two planes look the same. This word should not be confused with homomorphism. ...

A plane can be uniquely determined by any of the following (sets of) objects:

• three non-collinear points (ie. not lying on the same line)
• a line and a point not on the line
• two lines with one point of intersection
• two parallel lines

A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... â€œLineâ€ redirects here. ...

## Planes embedded in ℝ3

This section is specifically concerned with planes embedded in three dimensions: specifically, in 3. In mathematics, the Cartesian product is a direct product of sets. ...

### Properties

In three-dimensional Euclidean space, we may exploit the following facts that do not hold in higher dimensions:

• Two planes are either parallel or they intersect in a line.
• A line is either parallel to a plane or intersects it at a single point or is contained in the plane.
• Two lines normal (perpendicular) to the same plane must be parallel to each other.
• Two planes normal to the same line must be parallel to each other.

Fig. ... Fig. ...

### Define a plane with a point and a normal vector

In a three-dimensional space, another important way of defining a plane is by specifying a point and a normal vector to the plane. Fig. ...

Let $bold p$ be the point we wish to lie in the plane, and let $vec n$ be a nonzero normal vector to the plane. The desired plane is the set of all points $bold r$ such that $vec ncdot(vec r-vec p)=0.$

If we write $vec n = begin{bmatrix}a b cend{bmatrix}$, $bold r = (x, y, z)$ and d as the dot product $vec ncdot bold p=-d$, then the plane Π is determined by the condition $ax + by + cz + d = 0,$, where a, b, c and d are real numbers and a,b, and c are not all zero. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Alternatively, a plane may be described parametrically as the set of all points of the form $vec{u} + svec{v} + tvec{w},$ where s and t range over all real numbers, and $vec{u}$, $vec{v}$ and $vec{w}$ are given vectors defining the plane. $vec{u}$ points from the origin to an arbitrary point on the plane, and $vec{v}$ and $vec{w}$ can be visualized as starting at $vec{u}$ and pointing in different directions along the plane. $vec{v}$ and $vec{w}$ can, but do not have to be perpendicular. A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...

### Define a plane through three points

• The plane passing through three points $bold p_1 = (x_1,y_1,z_1)$, $bold p_2 = (x_2,y_2,z_2)$ and $bold p_3 = (x_3,y_3,z_3)$ can be determined by the following determinant equations:
$begin{vmatrix} x - x_1 & y - y_1 & z - z_1 x_2 - x_1 & y_2 - y_1& z_2 - z_1 x_3 - x_1 & y_3 - y_1 & z_3 - z_1 end{vmatrix} =begin{vmatrix} x - x_1 & y - y_1 & z - z_1 x - x_2 & y - y_2 & z - z_2 x - x_3 & y - y_3 & z - z_3 end{vmatrix} = 0.$
• This plane can also be described by the "point and a normal vector" prescription above.

A suitable normal vector is given by the cross product $vec n = ( bold p_2 - bold p_1 ) times ( bold p_3 - bold p_1 ),$ and the point $bold p$ can be taken to be any of given points $bold p_1, bold p_2$ or $bold p_3$. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... For the cross product in algebraic topology, see KÃ¼nneth theorem. ...

### Distance from a point to a plane

For a plane $Pi : ax + by + cz + d = 0,$ and a point $bold p_1 = (x_1,y_1,z_1)$ not necessarily lying on the plane, the shortest distance from $bold p_1$ to the plane is

$D = frac{left | a x_1 + b y_1 + c z_1+d right |}{sqrt{a^2+b^2+c^2}}.$

It follows that $bold p_1$ lies in the plane if and only if D=0.

If $sqrt{a^2+b^2+c^2}=1$ meaning that a, b and c are normalized then the equation becomes

$D = | a x_1 + b y_1 + c z_1+d | .$

### Line of intersection between two planes

Given intersecting planes described by $Pi_1 : vec n_1cdot bold r = h_1$ and $Pi_2 : vec n_2cdot bold r = h_2$, the line of intersection is perpendicular to both $vec n_1$ and $vec n_2$ and thus parallel to $vec n_1 times vec n_2$ .

If we further assume that $vec n_1$ and $vec n_2$ are orthonormal then the closest point on the line of intersection to the origin is $bold r_0 = h_1vec n_1 + h_2vec n_2$ . In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ...

### Dihedral angle

Given two intersecting planes described by $Pi_1 : a_1 x + b_1 y + c_1 z + d_1 = 0,$ and $Pi_2 : a_2 x + b_2 y + c_2 z + d_2 = 0,$, the dihedral angle between them is defined to be the angle α between their normal directions: In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ...

$cosalpha = hat n_1cdot hat n_2 = frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{sqrt{a_1^2+b_1^2+c_1^2}sqrt{a_2^2+b_2^2+c_2^2}}.$

## The plane areas of mathematics

In addition to its familiar geometric structure, with isomorphisms that are isometries with respect to the usual inner product, the plane may be viewed at various other levels of abstraction. Each level of abstraction corresponds to a specific category. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ...

At one extreme, all geometrical and metric concepts may be dropped to leave the topological plane, which may be thought of as an idealised homotopically trivial infinite rubber sheet, which retains a notion of proximity, but has no distances. The topological plane has a concept of a linear path, but no concept of a straight line. The topological plane, or its equivalent the open disc, is the basic topological neighbourhood used to construct surfaces (or 2-manifolds) classified in low-dimensional topology. Isomorphisms of the topological plane are all continuous bijections. The topological plane is the natural context for the branch of graph theory that deals with planar graphs, and results such as the four color theorem. In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... Topology (Greek topos = place and logos = word) is a branch of mathematics concerned with the study of topological spaces. ... The two bold paths shown above are homotopic relative to their endpoints. ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... A bijective function. ... A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ... In graph theory, a planar graph is a graph that can be embedded in a plane so that no edges intersect. ... Example of a four-colored map The four color theorem (also known as the four color map theorem) states that given any plane separated into regions, such as a political map of the states of a country, the regions may be colored using no more than four colors in such...

The plane may also be viewed as an affine space, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, but colinearity and ratios of distances on any line are preserved. In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...

Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a differentiable or smooth path (depending on the type of differential structure applied). The isomorphisms in this case are bijections with the chosen degree of differentiability. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes it into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ...

In the opposite direction of abstraction, we may apply a compatible field structure to the geometric plane, giving rise to the complex plane and the major area of complex analysis. The complex field has only two isomorphisms that leave the real line fixed, the identity and conjugation. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

In the same way as in the real case, the plane may also be viewed as the simplest, one-dimensional (over the complex numbers) complex manifold, sometimes called the complex line. However, this viewpoint contrasts sharply with the case of the plane as a 2-dimensional real manifold. The isomorphisms are all conformal bijections of the complex plane, but the only possibilities are maps that correspond to the composition of a multiplication by a complex number and a translation. In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, a conformal map is a function which preserves angles. ...

In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. The plane may be given a spherical geometry by using the stereographic projection. This can be thought of as placing a sphere on the plane (just like a ball on the floor), removing the top point, and projecting the sphere onto the plane from this point). This is one of the projections that may be used in making a flat map of part of the Earth's surface. The resulting geometry has constant positive curvature. In mathematics, curvature refers to a number of loosely related concepts in different areas of geometry. ... Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... Stereographic projection of a circle of radius R onto the x axis. ...

Alternatively, the plane can also be given a metric which gives it constant negative curvature giving the hyperbolic plane. The latter possibility finds an application in the theory of special relativity in the simplified case where there is one dimension of space and one of time. Lines through a given point P and asymptotic to line l. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ...

In geometry, a half-space is any of the two parts into which a hyperplane divides an affine space. ... A hyperplane is a concept in geometry. ... The three possible plane-line intersections: 1. ... Here we will find the point on an arbitrary plane that is closest to the origin using Lagrange multipliers. ...

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