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plane geometry

The geometry of planar figures.

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Sci-Tech Encyclopedia: Plane geometry

The branch of mathematics that deals with geometric figures, that is, collections of points that all lie in the same plane (coplanar). Although the words “point” and “plane” are undefined concepts, for elementary applications the intuitive meanings will serve: a point is a location, and a plane is a flat surface. For similar definitions, together with a discussion of the postulates and axioms (assumed truths) used in plane geometry, See also Euclidean geometry.

Dimensions and measures

There are three spatial dimensions; geometric figures are classified as being zero, one, two, or three dimensional. Plane geometry deals only with geometric figures having fewer than three dimensions. Three-dimensional geometric figures, called solids, are dealt with in another branch of euclidean geometry. See also Solid (geometry).

A dimension is any measurement associated with a geometric figure that has units of length. The measure of a geometric figure is a number multiplied by a power of a length, with the result giving information about the size of the figure. The power of the length will be 1, 2, or 3, depending on whether the figure is one, two, or three dimensional. See also Units of measurement.

Lines, line segments, and rays

Exactly one line passes through two given points. The part of a line between (and including) two points is called a line segment, with the two defining points being the end points of the segment. That part of a line that lies on one side of a point (together with that point) is called a ray.

Angles

An angle is the geometric figure formed by joining two rays having a common end point. Each ray is a side of the angle; the common end point is the vertex of the angle.

The concept of the measure of an angle may be understood by imagining that one ray of an angle is held fixed but the other ray is hinged at the vertex and allowed to rotate in the plane. A measure of the angle is a number, together with some unit of angular measure, that tells how much the hinged ray would need to be rotated so that it would overlie the fixed ray. If a ray were to be rotated exactly one revolution, it would return to its original position. The measure of an angle can be what fraction of one revolution would enable one side of the angle to become coincident with the other.

The revolution is a convenient unit of angular measure for many applications. However, a more commonly used unit is the degree, which is defined by 360° = 1 revolution. In the modern-day use of calculators and computers, fractions of degrees are most conveniently expressed by using decimal fractions; however, another subdivision of degrees is firmly entrenched in many applications: 1 degree = 60 minutes (1° = 60′), and 1 minute = 60 seconds (1′ = 60″).

An angle of measure 1/4 revolution, or 90°, is called a right angle. Two lines (or rays or segments) that intersect so as to form a right angle are said to be perpendicular. Two angles are complementary if their measures have a sum of 90°. An angle of measure 1/2 revolution, or 180°, is called a straight angle. Two angles are supplementary if their measures have a sum of 180°.

Polygons

A polygon is the geometric figure formed when line segments are joined end to end so as to enclose a region of the plane. The polygon having the least number of sides (three) is the triangle. An angle whose sides are two of those segments and whose arc lies inside the triangle is an interior angle of the triangle; an exterior angle of the triangle is any angle that is adjacent and supplementary to an interior angle. The interior angles of any triangle have measures that sum to 180°. This fact allows the determination of the measures of all angles formed by the intersections of three line if the measures of only two angles with different vertices are known. See also Polygon.

Congruent and similar geometric figures

Two geometric figures are congruent (≃) if they have exactly the same shape and size. If two geometric figures are congruent, one figure could be made to overlie the other by a combination of these types of motion: translation (sliding), rotation (twisting), and reflection about a line (flipping over). The parts of the geometric figures that would then coincide are called corresponding parts, where a part of a geometric figure is any set of points associated with that figure.

Two geometric figures are similar (∼) if they have the same shape but (perhaps) have different sizes. If two geometric figures are similar, one figure could be made to overlie the other by a combination of translation, rotation, reflection, and either expanding or shrinking. The parts that would then coincide are corresponding.

Circles

A circle is a collection of points in the plane, all of which are the same distance from another point, called the center. The region bounded by a circle sometimes is called a disk. Circumference means either the circle that is the boundary curve of a disk or the distance around that circle. A radius of a circle is any line segment that joins the center and a point of a circle. A chord is any line segment whose end points lie on the circle. A diameter is any chord that contains the center. A secant is a line that intersects a circle in two points. A tangent is a line that intersects a circle in only one point, called the point of tangency.

Associated with a circle are several important dimensions, including the lengths of a radius and a diameter, usually denoted, respectively, by r and d. These symbols appear in almost all formulas involving the length of the circumference C or the area A enclosed by a circle. Relationships between these variables for any circle are given by the equations $d=2r\qquad C=\pi d \qquad A=\pi r^2$

In these formulas, π (the lowercase Greek letter pi) represents the irrational number (value 3.141592 …) that is usually defined as the ratio of the circumference to the diameter of any circle. See also Circle.

Science Dictionary: plane geometry

The study of two-dimensional figures (figures that are confined to a plane).

Plane geometry is one of the oldest branches of mathematics.

The Greek mathematician Euclid was the first to study plane geometry carefully. His book Elements was the standard plane geometry textbook for centuries.

Wikipedia: plane (mathematics)

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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. Informally it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some 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. A lot of mathematics can be and has been performed in the plane, notably in the areas of geometry, trigonometry, graph theory and graphing.

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.

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.

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

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 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.

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

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.

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.

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:

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

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

$\, ax_3 + by_3 + cz_3 + d = 0$ .

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.

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 $\bold p_3$ .

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 $\vec n_1 \times \vec n_2$ .

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 $\bold r_0 = h_1\vec n_1 + h_2\vec n_2$ .

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:

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

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.

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.

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 is one dimension of space and one of time.

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Mathworld: Plane

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		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
		Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Read more
		Science Dictionary. The New Dictionary of Cultural Literacy, Third Edition Edited by E.D. Hirsch, Jr., Joseph F. Kett, and James Trefil. Copyright © 2002 by Houghton Mifflin Company. Published by Houghton Mifflin. All rights reserved. Read more
		Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Plane (mathematics)". Read more

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