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Plane (mathematics)
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In mathematics, a plane is a flat surface. 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.
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In mathematics, a plane is a flat surface. 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. All two-dimensional figures are assumed to be on a plane, even on the plane, unless otherwise specified.
Euclidean geometry
Euclid set forth the first known axiomatic treatment of geometry. This means that Euclid selected a small core of undefined terms (called common notions) and postulates (or axioms) and he then uses these to prove the geometrical statements. Euclid's Axioms had minor flaws, which were later corrected by David Hilbert, George Birkhoff, and Alfred Tarski. The plane is not directly given a definition, may be thought of as part of the common notions. More formally it may be regarded as anything that satisfies the axioms for Euclidean geometry. In his work Euclid never makes use of a numbers to measure length, angle, or area. In this way the Euclidean plane is not quite the same as the Cartesian plane.
In higher dimensional Euclidean space, a plane inside this space 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 non-colinear (that is, not exactly identical) lines
In 3 dimensional Euclidean space, like lines, planes can be parallel or intersecting. In this setting planes differ from lines Differing from lines, however, planes cannot be skew. 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 higher dimensional Euclidean space it is possible to have two planes that intersect in a single point.
Planes embedded in R3
This section is specifically concerned with planes embedded in three dimensions: specifically, in R3.
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.
- 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 p be the vector representing the position of any known point in the plane, and let n be a nonzero normal vector which, if placed on the plane, would point away from the origin. The desired plane is the set of all points such that
Given that
then the plane can be expressed by the equation
(which is in the form of ax + by + cz + d = 0)
In English:
- If a line drawn from any point on the plane to a point we want to test is at right angles to the direction the plane is facing, the point being tested is on the plane.
In jargon, without equations:
- Take the dot product of the normal vector and the vector from any point on the plane to the point under consideration. If the result is zero, meaning the angle between the two vectors is a right angle, the point is on the plane.
All points (considered as a vector from the origin) with the same cosine of their angle as measured from the normal vector, and scaled by the distance to the point, are on the same plane. is , the (inverted) dot product of the normal vector and the known point on the plane; is the dot product of the normal and the point being tested. If they cancel each other out, the cosines are the same.
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
where s and t range over all real numbers, and , and are given vectors defining the plane. is the vector representing the position of an arbitrary (but fixed) point on the plane, and and can be visualized as vectors starting at and pointing in different directions along the plane. and can be perpendicular, but cannot be parallel.
Define a plane through three points
- The plane passing through three points , and can be defined as the set of all points (x,y,z) that satisfy the following determinant equations:
- To describe the plane as an equation in the form , solve the following system of equations:
.
This system can be solved using Cramer's Rule and basic matrix manipulations. Let . Then,
.
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
and the point can be taken to be any of given points or .
Distance from a point to a plane
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 then the equation becomes
Line of intersection between two planes
Given intersecting planes described by and , the line of intersection is perpendicular to both and and thus parallel to . 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 , since is a basis. In this equation, is the line's parameter, and and are constants. By taking the dot product of this equation against and , and by noting that , we obtain two scalar equations that may be solved for .
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 .
Dihedral angle
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
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.)
Planes in fiction
The 1884 novel Flatland by Edwin A. 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."
See also
External links
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