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Distance

Distance

:This article is about distance in the mathematical / physical sense. For other uses, see distance (disambiguation). The distance between two points is the length of a straight line segment between them. In the case of two locations on Earth, usually the distance along the surface is meant: either "as the crow flies" (along a great circle) or by road, railroad, etc. Distance is sometimes expressed in terms of the time to cover it, for example walking or by car. Sometimes a distance thus indicated is ambiguous because the means of transport is neither mentioned nor obvious. Distance as mentioned above is sometimes not symmetric, hence not a metric (see below): this applies to distance by car in the case of one-way streets, and also in the case the distance is expressed in terms of the time to cover it (a road may be more crowded in one direction than in the other, for a ship upstream and downstream makes a difference). As opposed to a position coordinate, a distance can not be negative. Distance is a scalar quantity, containing only a magnitude, whereas displacement is an equivalent vector quantity containing both magnitude and direction. In the study of complicated geometries, we call the most common type of distance Euclidean distance, as we define it from the Pythagorean theorem.

Distance covered

Pythagorean theorem The distance covered by a vehicle (often recorded by a odometer), person, animal, object, etc. should be distinguished from the distance from starting point to end point, even if latter is taken to mean e.g. the shortest distance along the road, because a detour could be made, and the end point can even coincide with the starting point.

Formal definition

A distance between two points P and Q in a metric space is d(P,Q), where d is the distance function that defines the given metric space. We can also define the distance between two sets A and B in a metric space as being the minimum (or infimum) of distances between any two points P in A and Q in B. Alternatively, the distance between sets may indicate "how different they are", by taking the supremum over one set of the distance from a point in that set to the other set, and conversely, and taking the larger of the two values (Hausdorff distance).

The distance formula

The (Euclidean) distance, d, between two points expressed in Cartesian coordinates equals the square root of the sum of the squares of the changes of each coordinate. Thus, in a two-dimensional space :d = \sqrt, and in a three-dimensional space: :d = \sqrt. Here, "Δ" (delta) refers to the change in a variable. Thus, Δx is the change in x, pronounced as such, or as "delta-x". In mathematical terms, \Delta x = \left|x_1 - x_0\right|, and so (\Delta x)^2 = (x_1 - x_2)^2. This distance formula can be seen as a specialized form of the Pythagorean theorem; it can also be expanded into the arc-length formula.

Generalized distance in arbitrary dimensions: Norms

In the Euclidean space Rn, the distance between two points is usually given by the Euclidean distance (2-norm distance). Other distances, based on other norms, are sometimes used instead. For a point (x1, x2, ...,xn) and a point (y1, y2, ...,yn), the Minkowski distance of order p (p-norm distance) is defined as: p need not be an integer, but it cannot be less than 1, because then the triangle inequality does not hold. There is no such thing as a negative distance. The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean theorem to more than two coordinates. It is what would be obtained if the distance between two points were measured with a ruler: the "intuitive" idea of distance. The 1-norm distance is more colourfully called the taxicab norm or Manhattan distance, because it is the distance a car would drive in a city laid out in square blocks (if there are no one-way streets). The infinity norm distance is also called Chebyshev distance. In 2D it represents the distance kings, queens, and bishops must travel between two squares on a chessboard. Also, if you measure the strength of each of the n links in a chain (where larger numbers mean weaker links), then because a chain is only as strong as its weakest link, the strength of the chain will be the infinity-norm distance from the list of measurements to the origin. The p-norm is rarely used for values of p other than 1, 2, and infinity, but see super ellipse. In physical space the Euclidean distance is in a way the most natural one, because in this case the length of a rigid body does not change with rotation.

Distances in other spaces


- Mahalanobis distance is used in statistics.
- Hamming distance is used in coding theory.
- Levenshtein distance

See also


- astronomical units of length
- cosmic distance ladder
- comoving distance
- distance geometry
- distance (graph theory)
- distance-based road exit numbers
- Distance Measuring Equipment (DME)
- great-circle distance
- length
- milestone
- Metric (mathematics)
- Metric space
- neighborhood
- orders of magnitude (length) Category:Length Category:Elementary mathematics ja:距離 simple:Distance

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also


- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle

Bibliography


- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links


- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์



Distance (disambiguation)

Distance:
- In the mathematical / physical sense, see distance.
- Between people, see personal space, proxemics, and social distance.
- An obsolete unit of measure, see list of strange units of measurement.
- Japanese pop star Hikaru Utada's sophomore album, Distance.

Length

:This article is about the concept and measurement of distance. For usage in cricket, see line and length. In general English usage, length (symbols: l, L) is but one particular instance of distance – an object's length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with "distance". Height is vertical distance; width (or breadth) is a lateral distance; an object's width is less than its length. No one speaks of "the length from here to Alpha Centauri", but rather of "the distance from here to Alpha Centauri," but when one speaks of distance more abstractly, one says "A kilometre or a mile, is a unit of length" or "...of distance", and the two statements are synonymous. Likewise, a mountain might be a mile in height. Length is the metric of one dimension of space. The metric of space itself is volume, or (length)3. Length is commonly considered to be one of the fundamental units, meaning that it cannot be defined in terms of other dimensions. However, a set of units can be constructed where units of length can be derived from fundamental physical constants - see Planck units and Planck length. Colloquially length sometimes refers to duration, especially when used in context of music.

Units of length(SI)

The SI unit of Length is the metre (U.S. spelling: meter), from which can be derived:from the regular basis of the foundation of the whole world
- centimetre
- kilometre

Other units of length

The Imperial and US customary units of length


- inch
- foot
- yard
- mile

Units are used in astronomy


- Astronomical unit
- Light year
- Parsec

See also


- Curve
- Metric space
- Orders of magnitude
- Distance
- Planck length
- International standard ISO 31-1: Quantities and units – Space and time

External links


- [http://www.unitconversion.org/unit_converter/length.html Length Converter: convert between units of length, such as meter, yard, mile, and so on]
- [http://www.unitconversion.org/unit_converter/length-v.html Length Conversion table: convert selected unit to all other units of length]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- Category:Norm ko:길이 ja:長さ

Surface

:For other senses of this word, see surface (disambiguation). surface (disambiguation) In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, surface chemistry, surface energy, roughness.

Definition

In what follows, all surfaces are considered to be second-countable 2-dimensional manifolds. More precisely: a topological surface (with boundary) is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open subset of E2 (Euclidean 2-space) or an open subset of the closed half of E2. The set of points which have an open neighbourhood homeomorphic to En is called the interior of the manifold; it is always non-empty. The complement of the interior, is called the boundary; it is a (1)-manifold, or union of closed curves. A surface with empty boundary is said to be closed if it is compact, and open if it is not compact.

Classification of closed surfaces

There is a complete classification of closed (i.e compact without boundary) connected, surfaces up to homeomorphism. Any such surface falls into one of two infinite collections:
- Spheres with g handles attached (called g-fold tori). These are orientable surfaces with Euler characteristic 2-2g, also called surfaces of genus g.
- Spheres with k projective planes attached. These are non-orientable surfaces with Euler characteristic 2-k. Therefore Euler characteristic and orientability describe a compact surfaces up to homeomorphism (and if surfaces are smooth then up to diffeomorphism).

Compact surfaces

Compact surfaces with boundary are just these with one or more removed open disks whose closures are disjoint.

Embeddings in R3

A compact surface can be embedded in R3 if it is orientable or if it has nonempty boundary. It is a consequence of the Whitney embedding theorem that any surface can be embedded in R4.

Differential geometry

A simple review of the embedding of a surface in n dimensions, and a computation of the area of such a surface, is provided in the article volume form. Metric properties of Riemann surfaces are briefly reviewed in the article Poincaré metric.

Some models

To make some models of various surfaces, attach the sides of these squares (A with A, B with B) so that the directions of the arrows match: Image:SphereAsSquare.png|sphere Image:ProjectivePlaneAsSquare.png|real projective plane Image:KleinBottleAsSquare.png|Klein bottle Image:TorusAsSquare.png|torus

Fundamental polygon

Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of its edges. This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or -1. The exponent -1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon. The above models can be described as follows:
- sphere: A A^
- projective plane: A A
- Klein bottle: A B A^ B
- torus: A B A^ B^ (See the main article fundamental polygon for details.)

Connected sum of surfaces

Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly formed boundary components. We use the following notation.
- sphere: S
- torus: T
- Klein bottle: K
- Projective plane: P Facts:
- S # S = S
- S # M = M
- P # P = K
- P # K = P # T We use a shorthand natation: nM = M # M # ... # M (n-times) with 0M = S. Closed surfaces are classified as follows:
- gT (g-fold torus): orientable surface of genus g, for g \ge 0.
- gP (g-fold projective plane): non-orientable surface of genus g, for g \ge 1.

Algebraic surface

This notion of a surface is distinct from the notion of an algebraic surface. A non-singular complex projective algebraic curve is a smooth surface. Algebraic surfaces over the complex number field have dimension 4 when considered as a real manifold.

See also


- minimal surface
- Riemann surface
- algebraic surface
- Klein bottle
- torus
- sphere
- cylinder
- Möbius strip
- projective plane

External links


- [http://xahlee.org/surface/gallery.html Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing] Category:Surfaces Category:Geometric topology ja:表面

Road

:This page is related to transport; you may be looking for the 2002 Bollywood movie Road. Road Road A road is a strip of land, smoothed, paved, or otherwise prepared to allow easy travel, connecting two or more destinations. Some roads are streets, chiefly in urban areas. In the context of railways (railroads in American English), a road is a single track, which may be part of a multi-track system or may be an isolated line. In the context of sea transport, a road is an anchorage.

Usage and etymology

anchorage In original usage, a "road" was simply any pathway fit for riding ("road" is cognate with "ride", e.g.: ships ride at anchor in roads). The word “street,” whose origin is the Latin strata, was kept for paved pathways that had been prepared to ease travel in some way. Thus, many "Roman Roads" have the word "street" as part of their street name. However, modern usage does not usually make this distinction and it is only important since place names often hold the earlier usage in them; these days, roads are also prepared in some way. This includes, at the least, the removal of trees and smoothing of the ground. In some dialects, lower grade roads are called trails and tracks and it is uncertain where "road" begins and trail ends. Roads are a prerequisite for road transport of goods on wheeled vehicles. The word “road” emphasizes its function of transportation along its length, while a “street” may be considered to have activity and commerce taking place on it (see street life). street life.]]

History

The first pathways were the trails made by migrating animals. By about 10,000 BC, these rough pathways were used by human hunter nomads following these herds. Street paving has been found from the first human settlements around 4,000 BC. The oldest engineered road discovered is the Sweet Track causeway in England, dating from the 3800 BC. The Ancient Egyptians constructed a stone paved road to help move materials for the building of the Great Pyramid in about 3000 BC. The ancient Chinese constructed an extensive system of roads, some paved, from about 1100 BC onwards. By 20 AD, the Chinese road network extended over 40,000km. The Incas built fine highways for couriers through the Andes, and the Mayans built an extensive network of paved roads in Mexico before the European discovery of the New World. In ancient times, transport by river was far easier and faster than travel by road, especially considering the cost of road construction and the difference in carrying capacity between carts and river barges. A hybrid of road transport and ship transport is the horse-drawn boat in which the horse follows a cleared path along the river bank. From about 300 BC, the Roman Empire built straight strong stone Roman roads throughout Europe and North Africa, in support of its military campaigns. Road construction and maintenance in Britain was traditionally done on a local parish basis. The poor and variable state of the roads that resulted lead to the first of the 'Turnpike Trusts' around 1706. These were formed to build good roads and collect tolls from passing vehicles. Eventually there were approximately 1,100 Trusts in Britain and some 38,000 km of engineered roads. Engineered roads in the age of horse drawn transport aimed for a maximum gradient of 1 in 30 on a macadamized surface since this was the steepest a horse could exert to pull a load up hill which it could manage easily on the flat. Notable road engineers from this period are Pierre Marie Jérôme Trésaguet (1716-1796) in France and John Loudon McAdam (1756-1836) in England. During the industrial revolution,the railway developed as a solution to the problem of rutting of the road surface by heavy carts. Instead of trying to build a strong surface across the whole road the cart was constrained to run either on rails or grooves which could be made of much stronger , wear resistant material. Today, roads are almost exclusively built to enable travel by car and other wheeled vehicles. In most countries, road transport is the most utilized way to move goods. Also, in most developed countries, roads are formally divided into lanes to ensure the safe and smooth movement of traffic.

Funding

Road building and maintenance is an area of economic activity (compare military spending) that remains dominated by the public sector (though often through private contractors). Roads (except those on private property not accessible to the general public) are typically paid for by taxes (often raised through levies on fuel), though some public roads, especially highways are funded by tolls.

Driving on the right or the left

Traffic drives on the right or on the left side of the road depending on the country. See Rules of the road. In countries where traffic drives on the right, traffic signs are mostly on the right side of the road, roundabouts (traffic circles) go counter-clockwise, and pedestrians crossing a two-way road should watch out for traffic from the left first. In countries where traffic drives on the left, the reverse is true. Traffic flow and road design in both cases are each other's mirror image.

Design

Road design consists of two important technical aspects:
- geometrical road design
- structural road design Besides these two technical sides of the design, environmental issues, planning issues and juridical issues are important.

Construction

structural road design Road construction requires the creation of a continuous right-of-way, overcoming geographic obstacles and having grades low enough to permit vehicle or foot travel. Removal of earth and rock by digging or blasting, construction of bridges and tunnels, and removal of vegetation (this may involve deforestation) are often needed. A variety of road building equipment is employed in road building. The soil is tested to see if it will support weight and if not, a layer of soil is removed and replaced. The soil is compacted to form what is known as a "base course". On top of the base course is placed a wearing course which consists of asphalt concrete or concrete. While the main purpose of the wearing course is to prevent moisture from entering the road, for safety reasons this wearing course must also be constructed to ensure adequate grip (and skid resistance) with vehicles. Modern roads, and indeed many ancient ones, such as those built by the Romans, feature a convex lateral surface known as camber. This is designed to allow water to drain away from the road to its edges. Water is then carried away by gutters to drains placed at intervals. Some roads don't have gutters and water simply drains away to a naturally porous verge, or into ditches. Modern roads that carry motor traffic also employ camber in curves to aid traffic stability by allowing them to "bank into" the bend to some extent. On the side of the road there may be retroreflectors on pegs, rocks or crash barriers, white toward the direction of the traffic on that side of the road, and red toward the other direction. In the road surface there may be cat's eyes: retroreflectors that protrude slightly, but which can be driven over without damage. Road signs are often also made retroreflective or even illuminated in rare circumstances. For greater visibility of road signs at daytime, sometimes fluorescence is applied to get very bright colors.

Maintenance

retroreflective]] Like all outdoor structures, roads deteriorate over time. They may develop cracks or potholes, or be washed away altogether by floods. Cracks can be filled with various sealants and potholes can be filled with fresh asphalt, but eventually a whole new surface is needed. Lack of maintenance speeds up the deterioration, especially in frost-prone areas, as water enters the cracks, and freezes under the road. The resulting ice has a bigger volume than the water, which causing a localized rising and falling (when the ice melts again) of the wearing course which can severely damage the road. Most European countries have strict standards for road construction that ensure that most roads should be able to go 30 years or longer between major resurfacings. The United States and many other countries have less stringent standards under which most roads last only 20 to 25 years. However, even those countries with stricter standards suffer from increasing levels of truck traffic, which is mainly responsible for road damage (see below). On any road, the load per vehicle axle passing over it is mainly responsible for the amount of wear. According to a series of experiments carried out in the late 1950s, called the AASHO Road Test, it was empirically determined that the effective wear done to the road is roughly proportional to the 4th power of vehicle weight. As a result, truck traffic almost always is the exclusive 'real' cause of road damage. In an example, a hypothetical car weighs half a ton per axle. A 6-axle, 38-ton truck also travelling on the same road weighs in at over 6 tons per axle. The truck causes 20,736 times the wear of the car (12 times the car's axle load, with a power of 4, yielding 12^4 = 20,736). Actual trucks can have even higher axle loads, though there is a wide variation in the configuration of trucks, with some having larger, wider tyres, or multiple tyres per axle, which will cause the exact figures to vary. While such figures sound dramatic, it should be realised that a single car causes almost no wear at all, so 20,000 times this figure still may not be very high. The wear is only measurable over an extended period.

Terminology

AASHO Road Test]
- arterial road
- asphalt
- autobahn
- autoroute
- autostrasse
- bitumen
- byway
- bypass
- bottleneck
- boulevard
- cat's eye
- chicane
- concrete
- corduroy road
- corniche
- cul-de-sac
- curb extension
- dirt road
- divided highway
- expressway
- farm to market
- freeway
- gravel road
- guard rail
- green lane green lane
- hard shoulder
- highway
- Interchange
- Intersection
- Interstate
- lane
- median
- mountain pass
- milestone
- motorway
- off-ramp
- on-ramp
- Parkway
- pavement
- pavement markings
- pedestrian crossing
- performance
- plank road
- private highway
- private road
- public road
- public space
- ranch road
- range road
- ridge road
- road number
- road safety
- road junction
- roadworks roadworks, England. A British Airways Boeing 777-200 is being towed across a public road on its way to the maintenance hangars.]]
- roundabout intersection
- rural route
- state highway
- street
- super-highway
- toll road
- traffic calming
- traffic circle
- traffic light
- traffic sign
- US highway
- winter road

See also


- Inca road system
- List of roads and highways
- Public road
- Reclaim the Streets
- Road movie
- Trade route

References


- Lay MG, Ways of the world. Rutgers University Press, New Brunswick, New Jersey, (1992). ISBN 0813517583 .

External links


- [http://www.2pass.co.uk/goodluck.htm List of countries where traffic drives on the left, as well as historical background.]
- [http://www.travel-library.com/general/driving/drive_which_side.html Which side of the road do they drive on?]
- [http://www.ce.ksu.edu/facultystaff/stefan/ce777/docs/L02.pdf Kansas State University Department of Civil Engineering - History of Concrete Road Building] (PDF file) Category:Road infrastructure Category:Road transport ja:道路 simple:Road th:ถนน

Transport

:For other article subjects named transport, see Transport (disambiguation). Transportation redirects here, for other uses, see Transportation (disambiguation). Transport or transportation is the movement of people, goods, signals and information from one place to another. The term is derived from the Latin trans ("across") and portare ("to carry").

Aspects of transport

The field of transport has several aspects: loosely they can be divided into a triad of infrastructure, vehicles, and operations. Infrastructure includes the transport networks (roads, railways, airways, canals, pipelines, etc.) that are used, as well as the nodes or terminals (such as airports, railway stations, bus stations and seaports). The vehicles generally ride on the networks, such as automobiles, bicycles, buses, trains, airplanes. The operations deal with the control of the system, such as traffic signals and ramp meters, railroad switches, air traffic control, etc, as well as policies, such as how to finance the system (for example, the use of tolls or gasoline taxes). Broadly speaking, the design of networks are the domain of civil engineering and urban planning, the design of vehicles of mechanical engineering and specialized subfields such as nautical engineering and aerospace engineering, and the operations are usually specialized, though might appropriately belong to operations research or systems engineering.

Modes of transport

Modes are combinations of networks, vehicles, and operations, and include walking, the road transport system, rail transport, ship transport and modern aviation.

Categories of transport


- (Non-human) Animal-powered transport
- Aviation
- Cable transport
- Conveyor transport
- Human-powered transport
- Hybrid transport
- Ship transport
- Space transport
- Transport on other planets
- Proposed future transport

Transport and communications

Transport and communication are both substitutes and complements. Though it might be possible that sufficiently advanced communication could substitute for transport, one could telegraph, telephone, fax, or email a customer rather than visiting them in person, it has been found that those modes of communication in fact generate more total interactions, including interpersonal interactions. The growth in transport would be impossible without communication, which is vital for advanced transportation systems, from railroads which want to run trains in two directions on a single track, to air traffic control which requires knowing the location of aircraft in the sky. Thus, it has been found that the increase of one generally leads to more of the other.

Transport and land use

There is a well-known relationship between the density of development, and types of transportation. Intensity of development is often measured by area of Floor Area Ratio (FAR), the ratio of useable floorspace to area of land. As a rule of thumb, FARs of 1.5 or less are well suited to automobiles, those of six and above are well suited to trains. The range of densities from about two up to about four is not well served by conventional public or private transport. Many cities have grown into these densities, and are suffering traffic problems. Personal rapid transit could provide a solution to this problem. Land uses support activities. Those activities are spatially separated. People need transport to go from one to the other (from home to work to shop back to home for instance). Transport is a "derived demand," in that transport is unnecessary but for the activities pursued at the ends of trips. Good land use keeps common activities close (e.g. housing and food shopping), and places higher-density development closer to transportation lines and hubs. Poor land use concentrates activities (such as jobs) far from other destinations (such as housing and shopping). There are economies of agglomeration. Beyond transportation some land uses are more efficient when clustered. Transportation facilities consume land, and in cities, pavement (devoted to streets and parking) can easily exceed 20 percent of the total land use. An efficient transport system can reduce land waste.

Transport, energy, and the environment

Transport is a major use of energy, and transport burns most of the world's petroleum. Hydrocarbon fuels produce carbon dioxide, a greenhouse gas widely thought to be the chief cause of global climate change, and petroleum-powered engines, especially inefficient ones, create air pollution, including nitrous oxides and particulates (soot). Although vehicles in the United States have been getting cleaner because of environmental regulations, this has been offset by an increase in the number of vehicles and more use of each vehicle. Other environmental impacts of transport systems include traffic congestion, toxic runoff from roads and parking lots that can pollute water supplies and aquatic ecosystems, and automobile-oriented urban sprawl, which can consume natural habitat and agricultural lands. Low-pollution fuels can reduce pollution. Low pollution fuels may have a reduced carbon content, and thereby contribute less in the way of carbon dioxide emissions, and generally have reduced sulfur, since sulfur exhaust is a cause of acid rain. The most popular low-pollution fuel at this time is liquified natural gas. Hydrogen is an even lower-pollution fuel that produces no carbon dioxide, but producing and storing it economically is currently not feasible. Other alternative renewable energy sources such as biodiesel are being researched heavily. Another strategy is to make vehicles more efficient, which reduces pollution and waste by reducing the energy use. Electric vehicles use efficient electric motors, but their range is limited by either the extent of the electric transmission system or by the storage capacity of batteries. Electrified public transport generally uses overhead wires or third rails to transmit electricity to vehicles, and is used for both rail and bus transport. Battery electric vehicles store their electric fuel onboard in a battery pack. Another method is to generate energy using fuel cells, which may eventually be two to five times as efficient as the internal combustion engines currently used in most vehicles. Another effective method is to streamline ground vehicles, which spend up to 75% of their energy on air-resistance, and to reduce their weight. Regenerative braking is possible in all electric vehicles and recaptures the energy normally lost to braking, and is becoming common in rail vehicles. In internal combustion automobiles and buses, regenerative braking is not possible, unless electric vehicle components are also a part of the powertrain, these are called hybrid electric vehicles. Shifting travel from automobiles to well-utilized public transport can reduce energy consumption and traffic congestion. Use of non-motorized modes walking and bicycling also reduces the consumption of fossil fuels. However, as most areas get wealthier, the use of these modes declines. There are a few wealthy cities where bicycling comprises a significant share of trips, including Copenhagen, Denmark and Groningen, Netherlands. A number of other cities, including London, Paris, New York, Bogotá, Chicago, and San Francisco, are creating networks of bicycle lanes and bicycle paths to encourage bicycling by increasing safety from traffic.

Transport Research

Transport research facilities are mainly attached to universities or are steered by the state. In most countries (not in France and Spain) one can see now how laboratories are brought into PPP-operation, where industry takes over part of the share. Some major players in Europe:
- Transport Research Laboratory [http://www.trl.co.uk/ TRL UK]
- [http://www.vtt.fi/transport/ VTT FI]
- [http://www.lcpc.fr LCPC FR]
- [http://www.inrets.fr INRETS FR]
- [http://www.certu.fr CERTU FR]
- [http://www.dlr.de/dlr/Verkehr DLR DE]
- [http://www.crf.it CRF IT]
- [http://www.vv.tno.nl TNO NL]
- [http://www.cedex.es/ CEDEX ES]
- [http://www.cemt.org/jtrc/ Joint OECD-ECMT Transport Research Centre]
- [http://www.cemt.org/index.htm European Conference of Ministers of Transport] USA:
- http://www.its.berkeley.edu Institute of Transportation Studies, University of California, Berkeley
- National Transportation Research Center
- [http://www.trb.org/ Transportation Research Board] The European Commission supports the co-operation and collaboration amongst the transport laboratories by funding projects like EXTR@Web and [http://www.intransnet.org Intransnet]. Especially the transition from planned economy to achieving a stable position on the market will be a challenge for laboratories in the new member states. Another EU-project [http://www.etra.cc etra.cc]is coping with those problems.

See also


- List of transport topics
- Transportation reference tables
- Historic transport Category:Commercial item transport and distribution
- Category:Technology ko:교통 ja:交通 simple:Transport th:การขนส่ง



Scalar

Scalar is a concept that has meaning in mathematics, physics, and computing. A simple definition is that a scalar is a quantity which only specifies magnitude (i.e. a numerical value and unit), unlike a vector which has both magnitude and direction. For example, speed (180 km/h) is a scalar, while velocity (180 km/h north) is a vector. While this is a useful definition, it is not quite complete. A scalar is more completely defined as a magnitude which does not change under a change of coordinate system. In the above example, suppose the velocity vector has two components (e.g, 180 km/h north and 0 km/h east). Each component has a magnitude, yet they are not scalars, because they change when the coordinate system used to calculate them changes. (e.g to 180/√2 km/h northwest and 180/√2 km/h northeast.) Similar considerations hold for the mathematical definition. It is only for the computer definition that "scalar" simply means a single number. The word scalar derives from the English word "scale" for a range of numbers, which in turn is derived from scala (Latin for "ladder"). According to a citation in the Oxford English Dictionary the first usage of the term (by W. R. Hamilton in 1846) described it as: :"The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression of numbers from negative to positive infinity; we shall call it therefore the scalar part." Hamilton's usage actually describes his quaternion-based notation, which (in modern terms) represented rotations by a scalar, the real part of the quaternion, and vectors by the other three parts. Quaternions are widely used in spacecraft attitude determination and control, because they are not subject to the singularities of Euler angles and have only four components, while a rotation matrix has nine components to represent only three angles.

In physics

In physics, a scalar is a physical quantity which assumes a single value which is independent of the coordinate system being used to describe the physical system. In this sense it is a "real" quantity and not an artifact of the coordinate system. For example, the distance between two points in space is a scalar. It does not depend on one's choice of coordinate system. A physical quantity is expressed as the product of a numerical value and a physical unit, not just a number. It does not depend on the unit distance (1 km is the same as 1000 m), although the number depends on the unit. Thus distance does not depend on the length of the base vectors of the coordinate system. Also, other changes of the coordinate system may affect the formula for computing the scalar (for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormal), but not the scalar itself. In this sense, physical distance deviates from the definition of metric in not being just a real number; however it satisfies all other properties. The same applies for other physical quantities which are not dimensionless. A scalar field is a scalar-valued function of position, again independent of the coordinate system. A vector is a physical entity which has a magnitude which is a scalar, but in addition, in contrast with a scalar, has a direction. The components of a vector as such are not scalars, since they change with a change of coordinate system; a scalar field may however for one choice of the coordinate system be equal to a particular component. Examples of scalar quantities:
- electric charge and charge density (the latter nonrelativistically; in relativity it must be combined with current density to comprise a 4-vector)
- relativistic distance
- mass and mass density (the latter nonrelativistically; in relativity it must be made part of the energy tensor in combination with momentum density and pressure)
- speed, but not velocity or momentum
- temperature
- energy and energy density (the latter nonrelativistically) A related concept is a pseudoscalar, which is invariant under proper rotations but (like a pseudovector) flips sign under improper rotations. One example is the scalar triple product (see vector), and thus the signed volume. Another example is magnetic charge (as it is mathematically defined, regardless of whether it exists physically).

In mathematics

In mathematics, the meaning of scalar depends on the context; it can refer to real numbers or complex numbers or rational numbers, or to members of some other specified field. Generally, when a vector space over the field F is studied, then F is called the field of scalars and members of F are called scalars. More generally, a scalar for a module over a ring, is simply an element of the ring. This happens in manifold theory, where the tangent bundle forms a module over the algebra of real functions on the manifold. Since spacetime is supposed to be a manifold, the physical and mathematical concepts agree. A scalar is a tensor of rank zero.

In computing

In computing scalar refers to variables that can hold only one value at a time, as distinct from arrays, list or other containers which are variables that can hold many values at the same time.

See also


- Scalar field Category:Abstract algebra Category:Linear algebra Category:Introductory physics Category:Fundamental physics concepts ko:스칼라 ms:Skalar ja:スカラー

Displacement

The term displacement can have one of several meanings, depending on context:
- Displacement (distance), a physical quantity in kinematics
- another term for a congruent transformation in geometry
- Electric displacement field, a physical quantity in electrodynamics
- Engine displacement, a property of an internal combustion engine
- Displacement (fencing)
- Displacement (fluid), a different physical quantity, used in fluid mechanics and navigation; used as a measure of a ship's size
- Displacement hull, where the moving hull's weight is supported by buoyancy alone and it must displace water from its path rather than planing on the water's surface
- A ship's displacement, also known as tonnage
- Displacement aka offset used in relative addressing of computer memory
- Particle displacement, acoustics of sound in air
- Displacement of people by persecution or violence
- Displacement (psychology)
- Single or double displacement reaction, a chemical reaction concerning the exchange of ions
- Displacement in Orthopedic surgery refers to change in alignment of the fracture fragments.

Direction

The term direction can be applied to various topics.
- For the management, supervision, or guidance of a film, see film director
- For the guidance and cueing of a group of musicians during performance, see conducting
- A complex number x on the unit circle. x\bar x=1
- A direction can be specified by a unit vector; the components are direction cosines; together with magnitude, direction can be specified by a vector in general
- For determination of position, see navigation, compass or left and right
- For finding the direction to a radio source, see radio direction finder
- Direction (Alexander Technique), a principal tool used in practicing the Alexander Technique
- Direction - Social Democracy, a major political party in Slovakia simple:Direction

Euclidean distance

In mathematics the Euclidean distance or Euclidean metric is the "ordinary" distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. By using this formula as distance, Euclidean space becomes a metric space (even a Hilbert space). Older literature refers to this metric as Pythagorean metric.

Definition

The Euclidean distance between two points P=(p_1,p_2,...,p_n)\, and Q=(q_1,q_2,...,q_n)\,, in Euclidean n-space, is defined as: :\sqrt = \sqrt

One-dimensional distance

For two 1D points, P=(p_x)\, and Q=(q_x)\,, the distance is computed as: :\sqrt = |p_x-q_x| The absolute value signs are used, since distance is normally considered to be an unsigned scalar value.

Two-dimensional distance

For two 2D points, P=(p_x,p_y)\, and Q=(q_x,q_y)\,, the distance is computed as: :\sqrt

2D approximations for computer applications

A fast approximation of 2D distance based on an octagonal boundary can be computed as follows. Let dx = | p_x - q_x | (absolute value) and dy = | p_y - q_y|. If dy > dx, approximated distance is 0.41 dx + 0.941246 dy . (If dy < dx, swap these values.) The difference from the exact distance is between -6% and +3%; more than 85% of all possible differences are between -3% to +3%. The following Maple code implements this approximation and produces the plot on the right, with a true circle in black and the octagonal approximate boundary in red: 
fasthypot :=
  unapply(piecewise(abs(dx)>abs(dy), 
                    abs(dx)
 - 0.941246+abs(dy)
 - 0.41,
                    abs(dy)
 - 0.941246+abs(dx)
 - 0.41),
          dx, dy):
hypot := unapply(sqrt(x^2+y^2), x, y):
plots[display](
  plots[implicitplot](fasthypot(x,y) > 1, 
                      x=-1.1..1.1, 
                      y=-1.1..1.1,
                      numpoints=4000),
  plottools[circle]([0,0], 1),
  scaling=constrained,thickness=2
);
Other approximations exist as well. They generally try to avoid the square root, which is an expensive operation in terms of processing time, and provide various error:speed ratio. Using the above notation, dx + dy - (1/2)×min(dx,dy) yields error in interval 0% to 12% (attributed to Alan Paeth). A better approximation in term of RMS error is: dx + dy - (5/8)×min(dx,dy) and yields error in interval -3% to 7%. Also note that when comparing distances (for which is greatest, not for the actual difference), it isn't necessary to take the square root at all. If distance A is greater than distance B, then A^2 will also be greater than B^2. Or, when checking to see if distance A is greater than 2B, that is the same as comparing A^2 with (2B)^2 or 4B^2, etc. An example of the first case might be when trying to determine which nearest grid point an arbitrary point should "snap to" in a 2D CAD/CAM system. This isn't really an approximation, however, as the results are exact.

Three-dimensional distance

For two 3D points, P=(p_x,p_y,p_z)\, and Q=(q_x,q_y,q_z)\,, the distance is computed as: :\sqrt

3D approximations for computer applications

As noted in the 2D approximation section, when comparing distances (for which is greatest, not for the actual difference), it isn't necessary to take the square root at all. If distance A is greater than distance B, then A^2 will also be greater than B^2. An example is when searching for the minimum distance between two surfaces in 3D space, using a 3D CAD/CAM system. One way to start would be to build a point grid on each surface, and compare the distance of every grid point on the first surface with every grid point on the second surface. It isn't necessary to know the actual distances, but only which distance is the least. Once the closest two points are located, a much smaller point grid could be created around those closest points on each surface, and the process repeated. After several iterations, the closest two points could then be fully evaluated, including the square root, to give an excellent approximation of the minimum distance between the two surfaces. Thus, the square root only needs to be taken once, instead of thousands (or even millions) of times.

See also


- Mahalanobis distance category:metric geometry

Odometer

An odometer is a device used for indicating distance traveled by an automobile or other vehicle. It may be electronic or mechanical. The word derives from the Greek words hodōs, meaning "path" or "way", and mētron, "measure". Mechanical odometers usually appear as a row of wheels with the edge of the wheels towards the person viewing it. There are digits written on the edge of these wheels. A mask obscures these wheels from view, except for one row of digits which can be seen through a window in the mask. On older cars, odometers could only indicate up to a value of 99,999. At 100,000, the odometer would restart from zero. This is known as odometer rollover. Newer cars usually have odometers that can indicate up to a value of 999,999. A common form of fraud is to tamper with the reading on an odometer. This is done to make a car appear to have been used less than it actually has been, to get a higher price for the car. Many new cars sold today use digital odometers that store the mileage in the vehicle's engine control module which makes it even easier to manipulate the mileage by simply reprogramming it.

History

An odometer for measuring distance is described by Vitruvius around 27 and 23 BC. The actual invention may have been by Archimedes during the First Punic War. Hero of Alexandria describes a similar device in chapter 34 of his Dioptra. The device was also invented in ancient China by Zhang Heng (78 – 139). The odometer of Vitruvius was based on chariot wheels of 4 feet (1.2 m) diameter turning 400 times in one Roman mile (about 1400 m). For each revolution a pin on the axle engaged a 400 tooth cogwheel thus turning it one complete revolution per mile. This engaged another gear with holes along the circumference, where pebbles (calculus) were located, that were to drop one by one into a box. The distance travelled would thus be given simply by counting the number of pebbles. Whether this instrument was ever built at the time is disputed. Leonardo da Vinci tried to build it according to the description but failed. In modern times, however, Andre Sleeswyk was able to make a working model using gears similar to the Antikythera mechanism as opposed to the traditional cogwheel. The odometer as used in most modern systems, where separate gears control each digit, was invented by William Clayton with help from Orson Pratt. Clayton, a Mormon Pioneer, developed the odometer (dubbed the "roadometer") to keep track of wheel revolutions on the pioneer carts. The odo