Plane (mathematics) - Citizendia

This article is about the mathematical construct. For other uses, see Plane.

Two intersecting planes in three-dimensional space

In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry Informally, it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some space. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another Formally, it is an affine space of dimension two. In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space.

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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. A lot of mathematics can be and has been performed in the plane, notably in the areas of geometry, trigonometry, graph theory and graphing. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) All two-dimensional figures are assumed to be on a plane, even on the plane, unless otherwise specified.

1 Euclidean geometry
- 1.1 Orientation
2 Planes embedded in R³
3 Planes in various areas of mathematics
4 Planes in Fiction
5 See also
6 External links

Euclidean geometry

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. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume

The fundamental structure of two such planes will always be the same. In mathematics this is described as topological equivalence. Topological equivalence redirects here see also Topological equivalence (dynamical systems. Informally though, it means that any two planes look the same.

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

three non-collinear points (i. e. , 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

Orientation

Three parallel planes.

Like lines, planes can be parallel or intersecting. Differing from lines, however, planes cannot be skew. Skew or skew lines lie on different planes They are neither parallel nor intersecting Lines drawn on two parallel planes will either be parallel or skew, but will not intersect. Intersecting planes may be perpendicular, or may form any number of other angles. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent

Planes embedded in R³

This section is specifically concerned with planes embedded in three dimensions: specifically, in ℝ³. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

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 perpendicular to the same plane must be parallel to each other. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent
Two planes perpendicular to the same line must be parallel to each other.

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.

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 $\vec n\cdot (r-\bold p)=0.$

If we write $\vec n = \begin{bmatrix}a\\ b\\ c\end{bmatrix}$ , $\bold r = (x, y, z)$ and d as the dot product $\vec n\cdot \bold p=-d$ , 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 an operation which takes two vectors over the Real numbers R In Mathematics, the real numbers may be described informally in several different ways

Alternatively, a plane may be described parametrically as the set of all points of the form $\vec{u} + s\vec{v} + t\vec{w},$ 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 (but they cannot be collinear).

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 defined as the set of all points (x,y,z) that satisfy the following determinant equations:

$\begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1& z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} =\begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x - x_2 & y - y_2 & z - z_2 \\ x - x_3 & y - y_3 & z - z_3 \end{vmatrix} = 0.$

To describe the plane as an equation in the form $a x + b y + c z + d = 0$ , solve the following system of equations:

$\, ax_1 + by_1 + cz_1 + d = 0$

$\, ax_2 + by_2 + cz_2 + d = 0$

$\, ax_3 + by_3 + cz_3 + d = 0$ . In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

This system can be solved using Cramer's Rule and basic matrix manipulations. Cramer's rule is a Theorem in Linear algebra, which gives the solution of a System of linear equations in terms of Determinants It is named after Let $D = \begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{vmatrix}$ . Then,

$a = \frac{-d}{D} \begin{vmatrix} 1 & y_1 & z_1 \\ 1 & y_2 & z_2 \\ 1 & y_3 & z_3 \end{vmatrix}$

$b = \frac{-d}{D} \begin{vmatrix} x_1 & 1 & z_1 \\ x_2 & 1 & z_2 \\ x_3 & 1 & z_3 \end{vmatrix}$

$c = \frac{-d}{D} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$ .

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.

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 Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

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 $\Pi_1 : \vec n_1\cdot \bold r = h_1$ and $\Pi_2 : \vec n_2\cdot \bold r = h_2$ , the line of intersection is perpendicular to both $\vec n_1$ and $\vec n_2$ and thus parallel to $\vec n_1 \times \vec n_2$ . This cross product is zero only if the planes are parallel, and are therefore non-intersecting or coincident.

Any point in space may be written as $\bold r = c_1\vec n_1 + c_2\vec n_2 + c_3(\vec n_1 \times \vec n_2)$ , since $\{ \vec n_1, \vec n_2, (\vec n_1 \times \vec n_2) \}$ is a basis. In this equation, $c 3$ is the line's parameter, and $c 1$ and $c 2$ are constants. By taking the dot product of this equation against $\vec n_1$ and $\vec n_2$ , and by noting that $\vec n_i \cdot \bold r = h_i$ , we obtain two scalar equations that may be solved for ${c 1, c 2}$ .

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 $\bold r_0 = h_1\vec n_1 + h_2\vec n_2$ . In Linear algebra, two vectors in an Inner product space are orthonormal if they are orthogonal and both of unit length

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:

$\cos\alpha = \hat n_1\cdot \hat n_2 = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}.$

Planes in various 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. In Aerospace engineering, the Dihedral is the Angle between the two wings see Dihedral. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective For the Mechanical engineering and Architecture usage see Isometric projection. 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 Each level of abstraction corresponds to a specific category. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships

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. 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, "study" is the branch of Mathematics that studies the properties of In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical 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. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Mathematics, low-dimensional topology is the branch of Topology that studies Manifolds of four or fewer dimensions Isomorphisms of the topological plane are all continuous bijections. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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 and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects Kuratowski's and Wagner's theorems The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of Forbidden 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 plane may also be viewed as an affine space, whose isomorphisms are combinations of translations and non-singular linear maps. In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. From this viewpoint there are no distances, but colinearity and ratios of distances on any line are preserved.

Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, an n -dimensional differential structure (or differentiable structure on a set M makes it into an n -dimensional Differential 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). In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability The isomorphisms in this case are bijections with the chosen degree of differentiability.

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. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex The complex field has only two isomorphisms that leave the real line fixed, the identity and conjugation. 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. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, 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 Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane

In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry The plane may be given a spherical geometry by using the stereographic projection. Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane 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.

Alternatively, the plane can also be given a metric which gives it constant negative curvature giving the hyperbolic plane. In The latter possibility finds an application in the theory of special relativity in the simplified case where there are two spatial dimensions and one time dimension. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial (The hyperbolic plane is a timelike hypersurface in three-dimensional Minkowski space. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Mathematics, a hypersurface is a kind of Submanifold. For Differential geometry usage see Glossary of differential geometry and In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity )

Planes in Fiction

The 1884 novel Flatland by Edwin A. For other uses see Flatland (disambiguation Flatland A Romance of Many Dimensions is an 1884 Science fiction Abbott features the concept of a geometric, two dimensional infinite plane inhabited by living geometric figures (triangles, squares, circles, etc. ). It has been described by Isaac Asimov, in his foreword to the Signet Classics 1984 edition, as "the best introduction one can find into the manner of perceiving dimensions. "

External links

In Geometry, a half-space is either of the two parts into which a plane divides the three-dimensional space A hyperplane is a concept in Geometry. It is a higher-dimensional generalization of the concepts of a line in Euclidean plane geometry and a In analytic Geometry, the intersection of a line and a plane can be the Empty set, a point, or a line Here we will find the point on an arbitrary plane that is closest to the origin using Lagrange multipliers.

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