Plane (geometry)

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Two intersecting planes in three­dimensional space

Three parallel planes.

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In mathematics, a plane is a flat, two­dimensional surface. A plane is the two­

dimensional analogue of a point (zero­dimensions), a line (one­dimension) and a solid

(three­dimensions). Planes can arise as subspaces of some higher­dimensional 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 exclusively in two­dimensional Euclidean space, the definite article is

used, so, the plane refers to the whole space. Many fundamental tasks in mathematics,

geometry, trigonometry, graph theory and graphing are performed in a two­dimensional

space, or in other words, in the plane.

Contents [hide]

1 Euclidean geometry

2 Planes embedded in 3­dimensional Euclidean space

2.1 Properties

2.2 Point­normal form and general form of the equation of a plane

2.3 Describing a plane with a point and two vectors lying on it

2.4 Describing a plane through three points

2.4.1 Method 1

2.4.2 Method 2

2.4.3 Method 3

2.5 Distance from a point to a plane

2.6 Line of intersection between two planes

2.7 Dihedral angle

3 Planes in various areas of mathematics

4 Topological and differential geometric notions

5 See also

6 Notes

7 References

8 External links

Euclidean geometry [edit]

Main article: Euclidean geometry

Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry.[1] 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.[2] In his work Euclid never makes use of numbers to measure length, angle, or area.

In this way the Euclidean plane is not quite the same as the Cartesian plane.

Planes embedded in 3­dimensional Euclidean space [edit]

This section is solely concerned with planes embedded in three dimensions: specifically, in R3.

Properties [edit]

The following statements hold in three­dimensional Euclidean space but not in higher dimensions,

though they have higher­dimensional analogues:

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.

Point­normal form and general form of the equation of a plane [edit]

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Vector description of a plane

In a manner analogous to the way lines in a two­dimensional space are described using a point­slope form for their equations,

planes in a three dimensional space have a natural description using a point in the plane and a vector (the normal vector) to

indicate its "inclination".

Specifically, let be the position vector of some point , and let be a nonzero vector. The

plane determined by this point and 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 product, not scalar multiplication.) Expanded this becomes

which is the point­normal form of the equation of a plane.[3] This is just a linear equation:

Conversely, it is easily shown that if a, b, c and d are constants and a, b, and c are not all zero, then the graph of the equation

is a plane having the vector as a normal.[4] This familiar equation for a plane is called the general form of the

equation of the plane.[5]

Describing a plane with a point and two vectors lying on it [edit]

Alternatively, a plane may be described parametrically as the set of all points of the form

where s and t range over all real numbers, v and w are given linearly independent

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 wcan be perpendicular, but cannot be parallel.

Describing a plane through three points [edit]

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

Method 1 [edit]

The plane passing through p1, p2, and p3 can be described as the set of all points (x,y,z) that satisfy the following determinantequations:

Method 2 [edit]

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 and basic matrix manipulations. Let

.

If D is non­zero (so for planes not through the origin) the values for a, b and c can be calculated as follows:

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These equations are parametric in d. Setting d equal to any non­zero number and substituting it into these equations will yield

one solution set.

Method 3 [edit]

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

and the point r0 can be taken to be any of the given points p1,p2 or p3.[6]

Distance from a point to a plane [edit]

For a plane and a point not necessarily lying on the plane, the shortest

distance from to the plane is

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

If meaning that a, b, and c are normalized[7] then the equation becomes

Another vector form for the equation of a plane, known as the Hesse normal form relies on the parameter D. This form is:[5]

where is a unit normal vector to the plane, a position vector of a point of the plane and D0 the distance of the plane from

the origin.

The general formula for higher dimensions can be quickly arrived at using vector notation. Let the hyperplane have equation

, where the is a normal vector and is a position vector to a point in the

hyperplane. We desire the perpendicular distance to the point . The hyperplane may also be

represented by the scalar equation , for constants . Likewise, a corresponding may be represented

as . We desire the scalar projection of the vector in the direction of . Noting that

(as satisfies the equation of the hyperplane) we have

.

Line of intersection between two planes [edit]

The line of intersection between two planes and where are normalized is given by

where

This is found by noticing that the line must be perpendicular to both plane normals, and so parallel to their cross product

(this cross product is zero if and only if the planes are parallel, and are therefore non­intersecting or entirely

coincident).

The remainder of the expression is arrived at by finding an arbitrary point on the line. To do so, consider that any point in

space may be written as , since is a basis. We wish to find a

point which is on both planes (i.e. on their intersection), so insert this equation into each of the equations of the planes to get

two simultaneous equations which can be solved for and .

If we further assume that and are orthonormal then the closest point on the line of intersection to the origin is

. If that is not the case, then a more complex procedure must be used.[8]

Dihedral angle [edit]

Given two intersecting planes described by and

, the dihedral angle between them is defined to be the angle between their normal

directions:

Planes in various areas of mathematics [edit]

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 2­manifolds) classified in low­dimensional

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 non­singular

linear maps. From this viewpoint there are no distances, but collinearity and ratios of distances on any line are preserved.

Differential geometry views a plane as a 2­dimensional 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, one­dimensional (over the complex

numbers) complex manifold, sometimes called the complex line. However, this viewpoint contrasts sharply with the case of the

plane as a 2­dimensional 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 three­dimensional Minkowski space.)

Topological and differential geometric notions [edit]

The one­point 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 to 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 hyperbolic plane such diffeomorphism is

conformal, but for the Euclidean plane it is not.

See also [edit]

Half­plane

Hyperplane

Line­plane intersection

Plane of rotation

Point on plane closest to origin

Projective plane

Notes [edit]

1. ^ Eves 1963, pg. 19

2. ^ Joyce, D. E. (1996), Euclid's Elements, Book I, Definition 7 , Clark University, retrieved 8 August 2009

3. ^ Anton 1994, p. 155

4. ^ Anton 1994, p. 156

5. a b Weisstein, Eric W. (2009), "Plane" , MathWorld­­A Wolfram Web Resource, retrieved 2009­08­08

6. ^ Dawkins, Paul, "Equations of Planes" , Calculus III

7. ^ To normalize arbitrary coefficients, divide each of a, b, c and d by (which can not be 0). The "new"

coefficients are now normalized and the following formula is valid for the "new" coefficients.

8. ^ Plane­Plane Intersection ­ from Wolfram MathWorld . Mathworld.wolfram.com. Retrieved on 2013­08­20.

References [edit]

Anton, Howard (1994), Elementary Linear Algebra (7th ed.), John Wiley & Sons, ISBN 0­471­58742­7

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Eves, Howard (1963), A Survey of Geometry I, Boston: Allyn and Bacon, Inc.

External links [edit]

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

Categories: Euclidean geometry Surfaces Mathematical concepts