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Encyclopedia > Parallel (geometry)

Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate. Table of Geometry, from the 1728 Cyclopaedia. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ... Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ... Euclid (Greek: ), also known as Euclid of Alexandria, was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC). ... a and b are parallel, the transversal t produces congruent angles. ...

Contents

Euclidean Parallelism

As shown by the tick marks, lines a and b are parallel. We can prove this because the transversal t produces congruent angles.
As shown by the tick marks, lines a and b are parallel. We can prove this because the transversal t produces congruent angles.

Given straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean space: Two parallel lines cut by a transversal, with plenty of labels. ... Two parallel lines cut by a transversal, with plenty of labels. ... Transversal t cuts two parallel lines, a and b. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

  1. Every point on line m is located exactly the same minimum distance from line l ('equidistant lines', including the degenerate case where m = l).
  2. Line m is on the same plane as line l but does not intersect l (even assuming that lines extend to infinity in either direction).
  3. Lines m and l are both intersected by a third straight line (a transversal) in the same plane, and the corresponding angles of intersection with the transversal are equal.

boobs are very big. my cock is small. i have sex with dennis harden, sssssseeexxxx is the best,by a. Rossi. and is a prerequisite for the latter two. In other words, parallel lines must be located in the same plane, and parallel planes must be located in the same three-dimensional space. A parallel combination of a line and a plane may be located in the same three-dimensional space. Lines parallel to each other have the same gradient. Compare to perpendicular. This article or section is not written in the formal tone expected of an encyclopedia article. ... Transversal t cuts two parallel lines, a and b. ... Fig. ...


Construction

The three definitions above lead to three different methods of construction of parallel lines.

The problem: Draw a line through a parallel to l.
The problem: Draw a line through a parallel to l.
Definition 1: Line m has everywhere the same distance to line l.
Definition 1: Line m has everywhere the same distance to line l.


Image File history File links Par-prob. ... Image File history File links Par-prob. ... Image File history File links Par-equi. ... Image File history File links Par-equi. ...

Definition 2: Take a random line through a that intersects l in x. Move point x to infinity.
Definition 2: Take a random line through a that intersects l in x. Move point x to infinity.
Definition 3: Both l and m share a transversal line through a that intersect them at 90°.
Definition 3: Both l and m share a transversal line through a that intersect them at 90°.


Another definition of parallel line that's often used is that two lines are parallel if they do not intersect. Image File history File links Par-para. ... Image File history File links Par-para. ... Image File history File links Par-perp. ... Image File history File links Par-perp. ...


Extension to non-Euclidean geometry

In Euclidean geometry it is more common to talk about geodesics than (straight) lines. A geodesic is the path that a particle follows if no force is applied to it. In non-Euclidean geometry (spherical or hyperbolic) the above three definitions are not equivalent: only the second one is useful in other geometries. In general, equidistant lines are not geodesics so the equidistant definition cannot be used. In the Euclidean plane, when two geodesics (straight lines) are intersected with the same angles by a transversal geodesic (see image), every (non-parallel) geodesic intersects them with the same angles. In both the hyperbolic and spherical plane, this is not the case. E.g. geodesics sharing a common perpendicular only do so at one point (hyperbolic space) or at two (antipodal) points (spherical space). Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... Lines through a given point P and hyperparallel to line l. ... personal space, proxemics. ...


In general geometry it is useful to distinguish the three definitions above as three different types of lines, respectively equidistant lines, parallel geodesics and geodesics sharing a common perpendicular.


While in Euclidean geometry two geodesics can either intersect or be parallel, in general and in hyperbolic space in particular there are three possibilities. Two geodesics can be either:

  1. intersecting: they intersect in a common point in the plane
  2. parallel: they do not intersect in the plane, but do in the limit to infinity
  3. ultra parallel: they do not even intersect in the limit to infinity

In the literature ultra parallel geodesics are often called parallel. Geodesics intersecting at infinity are then called limit geodesics.


Spherical

On the spherical plane there is no such thing as a parallel line. Line a is a great circle, the equivalent of a straight line in the spherical plane. Line c is equidistant to line a but is not a great circle. It is a parallel of latitude. Line b is another geodesic which intersects a in two antipodal points. They share two common perpendiculars (one shown in blue).
On the spherical plane there is no such thing as a parallel line. Line a is a great circle, the equivalent of a straight line in the spherical plane. Line c is equidistant to line a but is not a great circle. It is a parallel of latitude. Line b is another geodesic which intersects a in two antipodal points. They share two common perpendiculars (one shown in blue).

In the spherical plane, all geodesics are great circles. Great circles divide the sphere in two equal hemispheres and all great circles intersect each other. By the above definitions, there are no parallel geodesics to a given geodesic, all geodesics intersect. Equidistant lines on the sphere are called parallels of latitude in analog to latitude lines on a globe. These lines are not geodesics. An object traveling along such a line has to accelerate away from the geodesic it is equidistant to avoid intersecting with it. When embedded in Euclidean space a dimension higher, parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center. Image File history File links SphereParallel. ... Image File history File links SphereParallel. ... For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the... Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the sphere into two equal hemispheres. ... A sphere is a perfectly symmetrical geometrical object. ... Latitude, usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ... Acceleration is the time rate of change of velocity, and at any point on a v_t graph, it is given by the gradient of the tangent to that point In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. ... :For other senses of this word, see dimension (disambiguation). ...


Hyperbolic

Intersecting, parallel and ultra parallel lines trough a with respect to l in the hyperbolic plane. The parallel lines appear to intersect l just off the image. This is an artifact of the visualisation. It is not possible to isometrically embed the hyperbolic plane in three dimension. In a real hyperbolic space the line will get closer to each other and 'touch' in infinity.
Intersecting, parallel and ultra parallel lines trough a with respect to l in the hyperbolic plane. The parallel lines appear to intersect l just off the image. This is an artifact of the visualisation. It is not possible to isometrically embed the hyperbolic plane in three dimension. In a real hyperbolic space the line will get closer to each other and 'touch' in infinity.

In the hyperbolic plane, there are two lines through a given point that intersect a given line in the limit to infinity. While in Euclidean geometry a geodesic intersects its parallels in both directions in the limit to infinity, in hyperbolic geometry both directions have their own line of parallelism. When visualized on a plane a geodesic is said to have a left handed parallel and a right handed parallel through a given point. The angle the parallel lines make with the perpendicular from that point to the given line is called the angle of parallelism. The angle of parallelism depends on the distance of the point to the line with respect to the curvature of the space. The angle is also present in the Euclidean case, there it is always 90° so the left and right handed parallels coincide. The parallel lines divide the set of geodesics through the point in two sets: intersecting geodesics that intersect the given line in the hyperbolic plane, and ultra parallel geodesics that do not intersect even in the limit to infinity (in either direction). In the Euclidean limit the latter set is empty. Image File history File links HyperParallel. ... Image File history File links HyperParallel. ... Lines through a given point P and hyperparallel to line l. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ... Coincident is a geometric term that pertains to the relationship between two vectors. ...


External links


  Results from FactBites:
 
Parallel (geometry) - Wikipedia, the free encyclopedia (1037 words)
Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these.
When embedded in Euclidean space a dimension higher, parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center.
The angle of parallelism depends on the distance of the point to the line with respect to the curvature of the space.
Euclidean geometry - Wikipedia, the free encyclopedia (2331 words)
In hyperbolic geometry the sum of the three angles are always less than 180° and can approach zero, while in elliptic geometry the sum is greater than 180°.
Absolute geometry, formed by removing the parallel postulate, is also a consistent theory, as is non-Euclidean geometry, formed by alterations of the parallel postulate.
For example, geometry on the surface of a sphere is a model of an elliptical geometry, carried out within a self-contained subset of a three-dimensional Euclidean space.
  More results at FactBites »

 
 

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