Plane (mathematics): Wikis

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Two intersecting planes in three-dimensional space

In mathematics, a plane is any flat, two-dimensional surface. A plane is the two dimensional analogue of a point (zero-dimensions), a line (one-dimension) and a space (three-dimensions). Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

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.

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Euclidean geometry

Euclid set forth the first known axiomatic treatment of geometry[citation needed]. He selected a small core of undefined terms (called common notions) and postulates (or axioms) 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, it may be thought of as part of the common notions.[1] In his work Euclid never makes use of numbers to measure length, angle, or area[citation needed]. In this way the Euclidean plane is not quite the same as the Cartesian plane.

Three parallel planes.

Planes embedded in ℝ3

This section is specifically concerned with planes embedded in three dimensions: specifically, in 3.

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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, 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.
  • Two planes perpendicular to the same line must be parallel to each other.

Definition 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 r0 be the position vector of some known point P0 in the plane, and let n be a nonzero vector normal to the plane. The idea is that a point P with position vector r is in the plane if and only if the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be expressed as the set of all points r such that

\bold n\cdot (\bold r-\bold r_0)=0.

(The dot here means a dot product, not multiplication.) Expanded this becomes

 n_x (x-x_0)+ n_y(y-y_0)+ n_z(z-z_0)=0,\,

which is the familiar equation for a plane.[2][3]

Define 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

\bold r = \bold {r}_0 + s \bold{v} + t \bold{w},

where s and t range over all real numbers, v and w are given vectors defining the plane, and r0 is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors v and w can be visualized as vectors starting at r0 and pointing in different directions along the plane. Note that v and w can be perpendicular, but cannot be parallel.

Define a plane through three points

Let p1=(x1, y1, z1), p2=(x2, y2, z2), and p3=(x3, y3, z3) be non-colinear points.

Method 1

The plane passing through p1, p2, and p3 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.

Method 2

To describe the plane as an equation in the form ax + by + cz + 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.

This system can be solved using Cramer's Rule and basic matrix manipulations. 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.

Method 3

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

\bold n = ( \bold p_2 - \bold p_1 ) \times ( \bold p_3 - \bold p_1 ),

and the point r0 can be taken to be any of the given points p1,p2 or p3.[3]

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 | .[2]

Line of intersection between two planes

Given intersecting planes described by \Pi_1 : \bold {n}_1 \cdot \bold r = h_1 and \Pi_2 : \bold {n}_2 \cdot \bold r = h_2, the line of intersection is perpendicular to both \bold {n}_1 and \bold {n}_2 and thus parallel to \bold {n}_1 \times \bold {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\bold {n}_1 + c_2\bold {n}_2 + c_3(\bold {n}_1 \times \bold {n}_2), since \{ \bold {n}_1, \bold {n}_2, (\bold {n}_1 \times \bold {n}_2) \} is a basis. In this equation, c3 is the line's parameter, and c1 and c2 are constants. By taking the dot product of this equation against \bold {n}_1 and \bold {n}_2, and by noting that \bold {n}_i \cdot \bold r = h_i, we obtain two scalar equations that may be solved for {c1,c2}.

If we further assume that \bold {n}_1 and \bold {n}_2 are orthonormal then the closest point on the line of intersection to the origin is \bold r_0 = h_1\bold {n}_1 + h_2\bold {n}_2. If that is not the case, then a more complex procedure must be used [1].

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. Each level of abstraction corresponds to a specific category.

At one extreme, all geometrical and metric concepts may be dropped to leave the topological plane, which may be thought of as an idealized 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 neighborhood 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.

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.

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 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 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 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.

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 are two spatial dimensions and one time dimension. (The hyperbolic plane is a timelike hypersurface in three-dimensional Minkowski space.)

Topological and differential geometric notions

The one-point compactification of the plane is homeomorphic to a sphere (see stereographic projection); the open disk is homeomorphic to a sphere with the "north pole" missing; adding that point completes the (compact) sphere. The result of this compactification is a manifold referred as the Riemann sphere or the complex projective line. The projection from the Euclidean plane to a sphere without a point is a diffeomorphism and even a conformal map.

The plane itself is homeomorphic (and diffeomorphic) to an open disk. For the Lobachevsky plane such diffeomorphism is conformal, but for the Euclidean plane it is not.

See also

References

  1. ^ Joyce, D. E. (1996). "Euclid's Elements, Book I, Definition 7". Clark University. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI7.html. Retrieved 8 August 2009. 
  2. ^ a b Weisstein, Eric W. (2009), "Plane", MathWorld--A Wolfram Web Resource, http://mathworld.wolfram.com/Plane.html, retrieved 2009-08-08 
  3. ^ a b Dawkins, Paul, "Equations of Planes", Calculus III, http://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx 

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Simple English

Plane can also refer to airplane.

[[File:|thumb|Two intersecting planes in three-dimensional space]] A plane is a perfectly flat surface extending in all directions. Try to imagine the ceiling of a room, only extended into all directions infinitely. A plane has two dimensions : length and width. All planes are flat surfaces. If a surface is not flat, it is called a curved surface.

In geometry, a plane is made up of an infinite number of lines (or points). It has no thickness.

A plane figure is part of a plane. It is named by the capital letters (e.g. A, B, C, ...X, Y, Z) that are put at its corners.

Mathematicians say two planes can be:

  • Parallel = they never meet.
  • Concurrent = they meet forming a line.
  • Coincident = they are one and the same.

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More mathematical

In mathematics, a plane is a fundamental two-dimensional object. Intuitively, it looks like a flat infinite sheet of paper. There are several definitions of the plane. They are equivalent in the sense of Euclidean geometry, but they can be extended in different ways to define objects in other areas of mathematics. The only 2 dimensional figure in our 3 dimensional world is a shadow.

In some areas of mathematics, such as plane geometry or 2D computer graphics, the whole space in which the work is carried out is a single plane. In such situations, the definite article is used: the plane. Many fundamental tasks in geometry, trigonometry, and graphing are performed in the two dimensional space, or in other words, in the plane.

Euclidean geometry

A plane is a surface such that, given any three distinct points on the surface, the surface also contains all of the straight lines that pass through any two of them. One can introduce a Cartesian coordinate system on a given plane in order to label every point on it with a unique ordered pair, which is composed of two numbers and is the coordinate of the point.

Within any Euclidean space, a plane is uniquely determined by any of the following combinations:

  • three points which are not lying on the same line
  • a line and a point not on the line
  • two different lines which intersect
  • two different lines which are parallel
  • a vector normal to the plane and a distance from the origin.

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