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In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called ''skew lines''. Line segments and Euclidean vectors are parallel if they have the same direction or opposite direction (not necessarily the same length). Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism. Symbol The parallel symbol is \parallel. For example, AB \paral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel (PSF)
Parallel may refer to: Mathematics * Parallel (geometry), two lines in the Euclidean plane which never intersect * Parallel (operator), mathematical operation named after the composition of electrical resistance in parallel circuits Science and engineering * Parallel (latitude), an imaginary east–west line circling a globe * Parallel of declination, used in astronomy * Parallel, a geometric term of location meaning "in the same direction" * Series and parallel circuits#Parallel circuits, Parallel electrical circuits Computing * Parallel (software), a UNIX utility for running programs in parallel Language * Parallelism (grammar), a balance of two or more similar words, phrases, or clauses * Parallelism (rhetoric) Entertainment * Parallel (manga), ''Parallel'' (manga) * Parallel (2018 film), ''Parallel'' (2018 film), a Canadian science fiction thriller film * Parallel (2024 film), ''Parallel'' (2024 film) an American science fiction thriller film * Parallel (video), ''P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Playfair's Axiom
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior ...): ''In a plane (mathematics), plane, given a line and a point not on it, at most one line parallel (geometry), parallel to the given line can be drawn through the point.'' It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry and was named after the Scottish mathematician John Playfair. The "at most" clause is all that is needed since it can be proved from the first four axioms that at least one parallel line exists given a line ''L'' and a point ''P'' not on ''L'', as follows: # ''Construct a perpendicular'': Using the axioms and previously established theorems, you can constr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( ) is a mathematics, mathematical treatise written 300 BC by the Ancient Greek mathematics, Ancient Greek mathematician Euclid. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus and Theaetetus (mathematician), Theaetetus, the ''Elements'' is a collection in 13 books of definitions, postulates, propositions and mathematical proofs that covers plane and solid Euclidean geometry, elementary number theory, and Commensurability (mathematics), incommensurable lines. These include Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the Compass-and-straightedge construction, construction of regular polygons and Regular polyhedra, polyhedra. Often referred to as the most successful textbook ever written, the ''Elements'' has continued to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Fifth Axiom
In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. This postulate does not specifically talk about parallel lines; it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates. ''Euclidean geometry'' is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. The postulate was long considered to be obvious or inevitable, but proofs were elusive. Eventually, it was discovered that inverting the postulate gave valid, albeit different geometries. A geometry where the parallel postulate do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore, two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted. In elementary geometry the word ''congruent'' is often used as follows. The word ''equal'' is often used in place of ''congruent'' for these objects. *Two line segments are congruent if they have the same length. *Two angles are congruent if they have the same measure. *Two circles are congruent if they have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corresponding Angles
In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether two or more other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: vertical angles, consecutive interior angles, consecutive exterior angles, corresponding angles, alternate interior angles, alternate exterior angles, and linear pairs. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive angles and linear pairs are supplementary, while corresponding angles, alternate angles, and vertical angles are equal. Angles of a transversal A transversal produces 8 angles, as shown in the graph at the above left: *4 with each of the two lines, namely α, β, γ and δ and then α1, β1, γ1 and δ1; and *4 of which are interior (between the two lines), namely α, β, γ1 and δ1 and 4 of which are exterior, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transversal (geometry)
In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether two or more other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: vertical angles, consecutive interior angles, consecutive exterior angles, corresponding angles, alternate interior angles, alternate exterior angles, and linear pairs. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive angles and linear pairs are supplementary, while corresponding angles, alternate angles, and vertical angles are equal. Angles of a transversal A transversal produces 8 angles, as shown in the graph at the above left: *4 with each of the two lines, namely α, β, γ and δ and then α1, β1, γ1 and δ1; and *4 of which are interior (between the two lines), namely α, β, γ1 and δ1 and 4 of which are exterior ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including Guillaume de l'Hôpital, l'Hôpital and Johann Bernoulli, Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or Magnitude (mathematics), magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equidistant
A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal. In two-dimensional Euclidean geometry, the locus of points equidistant from two given (different) points is their perpendicular bisector. In three dimensions, the locus of points equidistant from two given points is a plane, and generalising further, in n-dimensional space the locus of points equidistant from two points in ''n''-space is an (''n''−1)-space. For a triangle the circumcentre is a point equidistant from each of the three vertices. Every non-degenerate triangle has such a point. This result can be generalised to cyclic polygons: the circumcentre is equidistant from each of the vertices. Likewise, the incentre of a triangle or any other tangential polygon is equidistant from the points of tangency of the polygon's sides with the circle. Every point on a perpendicular bisector of the side of a triangle or other polygon is equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Transversal
Parallel may refer to: Mathematics * Parallel (geometry), two lines in the Euclidean plane which never intersect * Parallel (operator), mathematical operation named after the composition of electrical resistance in parallel circuits Science and engineering * Parallel (latitude), an imaginary east–west line circling a globe * Parallel of declination, used in astronomy * Parallel, a geometric term of location meaning "in the same direction" * Parallel electrical circuits Computing * Parallel (software), a UNIX utility for running programs in parallel Language * Parallelism (grammar), a balance of two or more similar words, phrases, or clauses * Parallelism (rhetoric) Entertainment * ''Parallel'' (manga) * ''Parallel'' (2018 film), a Canadian science fiction thriller film * ''Parallel'' (2024 film) an American science fiction thriller film * ''Parallel'' (video), a compilation of music videos by R.E.M. * ''Parallel'' (The Black Dog album), 1995 * ''Parallel'' ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unicode
Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Character (computing), characters and 168 script (Unicode), scripts used in various ordinary, literary, academic, and technical contexts. Unicode has largely supplanted the previous environment of a myriad of incompatible character sets used within different locales and on different computer architectures. The entire repertoire of these sets, plus many additional characters, were merged into the single Unicode set. Unicode is used to encode the vast majority of text on the Internet, including most web pages, and relevant Unicode support has become a common consideration in contemporary software development. Unicode is ultimately capable of encoding more than 1.1 million characters. The Unicode character repertoire is synchronized with Univers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
