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Inmathematics,orthogonality is the generalization of the geometric notion ofperpendicularity. Although many authors use the two termsperpendicular andorthogonal interchangeably, the termperpendicular is more specifically used for lines and planes that intersect to form aright angle, whereasorthogonal is used in generalizations, such asorthogonal vectors ororthogonal curves.[1][2]
Orthogonality is also used with various meanings that are often weakly related or not related at all with the mathematical meanings.
The word comes from theAncient Greekὀρθός (orthós), meaning "upright",[3] andγωνία (gōnía), meaning "angle".[4]
The Ancient Greekὀρθογώνιον (orthogṓnion) andClassical Latinorthogonium originally denoted arectangle.[5] Later, they came to mean aright triangle. In the 12th century, the post-classical Latin wordorthogonalis came to mean a right angle or something related to a right angle.[6]
Inmathematics,orthogonality is the generalization of the geometric notion ofperpendicularity tolinear algebra ofbilinear forms.
Two elementsu andv of avector space with bilinear form are orthogonal when. Depending on the bilinear form, the vector space may containnull vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced withhyperbolic orthogonality.
In the case offunction spaces, families of functions are used to form an orthogonalbasis, such as in the contexts oforthogonal polynomials,orthogonal functions, andcombinatorics.
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Inoptics,polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontallinear polarization or right-handed and left-handedcircular polarization.
Inspecial relativity, a time axis determined by arapidity of motion ishyperbolic-orthogonal to a space axis of simultaneous events, also determined by the rapidity. The theory featuresrelativity of simultaneity.
Inquantum mechanics, a sufficient (but not necessary) condition that twoeigenstates of aHermitian operator, and, are orthogonal is that they correspond to different eigenvalues. This means, inDirac notation, that if and correspond to different eigenvalues. This follows from the fact thatSchrödinger's equation is aSturm–Liouville equation (in Schrödinger's formulation) or that observables are given by Hermitian operators (in Heisenberg's formulation).[citation needed]
In art, theperspective (imaginary) lines pointing to thevanishing point are referred to as "orthogonal lines". The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such asPiet Mondrian andBurgoyne Diller are noted for their exclusive use of "orthogonal lines" — not, however, with reference to perspective, but rather referring to lines that are straight and exclusively horizontal or vertical, forming right angles where they intersect. For example, an essay of theThyssen-Bornemisza Museum states that "Mondrian [...] dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours."[8]
Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results.[9] This usage was introduced byVan Wijngaarden in the design ofAlgol 68:
The number of independent primitive concepts has been minimized in order that the language be easy to describe, to learn, and to implement. On the other hand, these concepts have been applied “orthogonally” in order to maximize the expressive power of the language while trying to avoid deleterious superfluities.[10]
Orthogonality is a system design property which guarantees that modifying the technical effect produced by a component of a system neither creates nor propagates side effects to other components of the system. Typically this is achieved through theseparation of concerns andencapsulation, and it is essential for feasible and compact designs of complex systems. The emergent behavior of a system consisting of components should be controlled strictly by formal definitions of its logic and not by side effects resulting from poor integration, i.e., non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time because it is easier to verify designs that neither cause side effects nor depend on them.
Aninstruction set is said to be orthogonal if it lacks redundancy (i.e., there is only a single instruction that can be used to accomplish a given task)[11] and is designed such that instructions can use anyregister in anyaddressing mode. This terminology results from considering an instruction as a vector whose components are the instruction fields. One field identifies the registers to be operated upon and another specifies the addressing mode. Anorthogonal instruction set uniquely encodes all combinations of registers and addressing modes.[12]
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Intelecommunications,multiple access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using differentbasis functions. One such scheme istime-division multiple access (TDMA), where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots").
Another scheme isorthogonal frequency-division multiplexing (OFDM), which refers to the use, by a single transmitter, of a set of frequency multiplexed signals with the exact minimum frequency spacing needed to make them orthogonal so that they do not interfere with each other. Well known examples include (a,g, andn) versions of802.11Wi-Fi;WiMAX;ITU-TG.hn,DVB-T, the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT (Discrete Multi Tone), the standard form ofADSL.
In OFDM, thesubcarrier frequencies are chosen[how?] so that the subcarriers are orthogonal to each other, meaning that crosstalk between the subchannels is eliminated and intercarrier guard bands are not required. This greatly simplifies the design of both the transmitter and the receiver. In conventional FDM, a separate filter for each subchannel is required.
When performing statistical analysis,independent variables that affect a particulardependent variable are said to be orthogonal if they are uncorrelated,[13] since the covariance forms an inner product. In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually withsimple regression or simultaneously withmultiple regression. Ifcorrelation is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered by subtracting theexpected value (the mean), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data (i.e., vectors) and as random variables (i.e., density functions).Oneeconometric formalism that is alternative to themaximum likelihood framework, theGeneralized Method of Moments, relies on orthogonality conditions. In particular, theOrdinary Least Squares estimator may be easily derived from an orthogonality condition between the explanatory variables and model residuals.
Intaxonomy, an orthogonal classification is one in which no item is a member of more than one group, that is, the classifications are mutually exclusive.
In chemistry and biochemistry, an orthogonal interaction occurs when there are two pairs of substances and each substance can interact with their respective partner, but does not interact with either substance of the other pair. For example,DNA has two orthogonal pairs: cytosine and guanine form a base-pair, and adenine and thymine form another base-pair, but other base-pair combinations are strongly disfavored. As a chemical example, tetrazine reacts with transcyclooctene and azide reacts with cyclooctyne without any cross-reaction, so these are mutually orthogonal reactions, and so, can be performed simultaneously and selectively.[14]
Inorganic synthesis,orthogonal protection is a strategy allowing the deprotection offunctional groups independently of each other.
Insupramolecular chemistry the notion of orthogonality refers to the possibility of two or more supramolecular, oftennon-covalent, interactions being compatible; reversibly forming without interference from the other.
Inanalytical chemistry, analyses are "orthogonal" if they make a measurement or identification in completely different ways, thus increasing the reliability of the measurement. Orthogonal testing thus can be viewed as "cross-checking" of results, and the "cross" notion corresponds to theetymologic origin oforthogonality. Orthogonal testing is often required as a part of anew drug application.
In the field of system reliability orthogonal redundancy is that form of redundancy where the form of backup device or method is completely different from the prone to error device or method. The failure mode of an orthogonally redundant back-up device or method does not intersect with and is completely different from the failure mode of the device or method in need of redundancy to safeguard the total system against catastrophic failure.
Inneuroscience, a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map.
Inphilosophy, two topics, authors, or pieces of writing are said to be "orthogonal" to each other when they do not substantively cover what could be considered potentially overlapping or competing claims. Thus, texts in philosophy can either support and complement one another, they can offer competing explanations or systems, or they can be orthogonal to each other in cases where the scope, content, and purpose of the pieces of writing are entirely unrelated.[example needed]
In board games such aschess which feature a grid of squares, 'orthogonal' is used to mean "in the same row/'rank' or column/'file'". This is the counterpart to squares which are "diagonally adjacent".[28] In the ancient Chinese board gameGo a player can capture the stones of an opponent by occupying all orthogonally adjacent points.
In law, orthogonality can refer to interests in a proceeding that are not aligned, but also bear no correlation or effect on each other, so as not to create a conflict of interest.
Stereo vinyl records encode both the left and right stereo channels in a single groove. The V-shaped groove in the vinyl has walls that are 90 degrees to each other, with variations in each wall separately encoding one of the two analogue channels that make up the stereo signal. The cartridge senses the motion of the stylus following the groove in two orthogonal directions: 45 degrees from vertical to either side.[29] A pure horizontal motion corresponds to a mono signal, equivalent to a stereo signal in which both channels carry identical (in-phase) signals.
