![Perpendicular-coloured](https://upload.wikimedia.org/wikipedia/commons/8/84/Perpendicular-coloured.svg)
In
mathematics, orthogonality is the generalization of the geometric notion of ''
perpendicularity
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry.
Etymology
The word comes from the
Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic p ...
('), meaning "upright", and ('), meaning "angle".
The Ancient Greek (') and
Classical Latin
Classical Latin is the form of Literary Latin recognized as a literary standard by writers of the late Roman Republic and early Roman Empire. It was used from 75 BC to the 3rd century AD, when it developed into Late Latin. In some later period ...
' originally denoted a
rectangle. Later, they came to mean a
right triangle
A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle.
Mathematics
Physics
* In
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
,
polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal
linear polarization
In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. The term ''linear polarizati ...
or right- and left-handed
circular polarization
In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to ...
.
* In
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
, a time axis determined by a
rapidity of motion is
hyperbolic-orthogonal
In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperb ...
to a space axis of simultaneous events, also determined by the rapidity. The theory features
relativity of simultaneity.
* In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, a sufficient (but not necessary) condition that two
eigenstates
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
of a
Hermitian operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to it ...
,
and
, are orthogonal is that they correspond to different eigenvalues. This means, in
Dirac notation
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety of ...
, that
if
and
correspond to different eigenvalues. This follows from the fact that
Schrödinger's equation is a
Sturm–Liouville equation (in Schrödinger's formulation) or that observables are given by Hermitian operators (in Heisenberg's formulation).
Art
In art, the
perspective (imaginary) lines pointing to the
vanishing point
A vanishing point is a point on the image plane of a perspective drawing where the two-dimensional perspective projections of mutually parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicul ...
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 as
Piet Mondrian and
Burgoyne 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 at the
web site
A website (also written as a web site) is a collection of web pages and related content that is identified by a common domain name and published on at least one web server. Examples of notable websites are Google, Facebook, Amazon, and Wikip ...
of the
Thyssen-Bornemisza Museum
The Thyssen-Bornemisza National Museum (in Spanish, the Museo Nacional Thyssen-Bornemisza (), named after its founder), or simply the Thyssen, is an art museum in Madrid, Spain, located near the Prado Museum on one of the city's main boulevards. I ...
states that "Mondrian ... dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours."
Computer science
Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results. This usage was introduced by
Van Wijngaarden in the design of
Algol 68
ALGOL 68 (short for ''Algorithmic Language 1968'') is an imperative programming language that was conceived as a successor to the ALGOL 60 programming language, designed with the goal of a much wider scope of application and more rigorously d ...
:
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.
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 the
separation of concerns
In computer science, separation of concerns is a design principle for separating a computer program into distinct sections. Each section addresses a separate '' concern'', a set of information that affects the code of a computer program. A concern ...
and
encapsulation, 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.
An
instruction 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) and is designed such that instructions can use any
register
Register or registration may refer to:
Arts entertainment, and media Music
* Register (music), the relative "height" or range of a note, melody, part, instrument, etc.
* ''Register'', a 2017 album by Travis Miller
* Registration (organ), th ...
in any
addressing mode
Addressing modes are an aspect of the instruction set architecture in most central processing unit (CPU) designs. The various addressing modes that are defined in a given instruction set architecture define how the machine language instructions i ...
. 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. An
orthogonal instruction set
In computer engineering, an orthogonal instruction set is an instruction set architecture where all instruction types can use all addressing modes. It is " orthogonal" in the sense that the instruction type and the addressing mode vary independe ...
uniquely encodes all combinations of registers and addressing modes.
Telecommunications
In
telecommunications
Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
,
multiple access
In telecommunications and computer networks, a channel access method or multiple access method allows more than two terminals connected to the same transmission medium to transmit over it and to share its capacity. Examples of shared physical med ...
schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using different
basis functions. One such scheme is
time-division multiple access (TDMA), where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots").
Another scheme is
orthogonal 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, and n) versions of
802.11
IEEE 802.11 is part of the IEEE 802 set of local area network (LAN) technical standards, and specifies the set of media access control (MAC) and physical layer (PHY) protocols for implementing wireless local area network (WLAN) computer com ...
Wi-Fi
Wi-Fi () is a family of wireless network protocols, based on the IEEE 802.11 family of standards, which are commonly used for local area networking of devices and Internet access, allowing nearby digital devices to exchange data by radio wav ...
;
WiMAX
Worldwide Interoperability for Microwave Access (WiMAX) is a family of wireless broadband communication standards based on the IEEE 802.16 set of standards, which provide physical layer (PHY) and media access control (MAC) options.
The WiMAX ...
;
ITU-T
The ITU Telecommunication Standardization Sector (ITU-T) is one of the three sectors (divisions or units) of the International Telecommunication Union (ITU). It is responsible for coordinating standards for telecommunications and Information Comm ...
G.hn
G.hn is a specification for home networking with data rates up to 2 Gbit/s and operation over four types of legacy wires: telephone wiring, coaxial cables, power lines and plastic optical fiber. A single G.hn semiconductor device is able to n ...
,
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 of
ADSL
Asymmetric digital subscriber line (ADSL) is a type of digital subscriber line (DSL) technology, a data communications technology that enables faster data transmission over copper telephone lines than a conventional voiceband modem can provide. ...
.
In OFDM, the
subcarrier frequencies are chosen 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.
Statistics, econometrics, and economics
When performing statistical analysis,
independent variables
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or deman ...
that affect a particular
dependent variable are said to be orthogonal if they are uncorrelated, 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 with
simple regression or simultaneously with
multiple regression
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
. If
correlation 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 the
expected 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).
One
econometric
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
formalism that is alternative to the
maximum likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stat ...
framework, the
Generalized Method of Moments, relies on orthogonality conditions. In particular, the
Ordinary Least Squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
estimator may be easily derived from an orthogonality condition between the explanatory variables and model residuals.
Taxonomy
In
taxonomy
Taxonomy is the practice and science of categorization or classification.
A taxonomy (or taxonomical classification) is a scheme of classification, especially a hierarchical classification, in which things are organized into groups or types. ...
, an orthogonal classification is one in which no item is a member of more than one group, that is, the classifications are mutually exclusive.
Chemistry and biochemistry
In
synthetic organic chemistry
Organic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms.Clayden, J ...
orthogonal
protection
Protection is any measure taken to guard a thing against damage caused by outside forces. Protection can be provided to physical objects, including organisms, to systems, and to intangible things like civil and political rights. Although th ...
is a strategy allowing the deprotection of
functional group
In organic chemistry, a functional group is a substituent or moiety in a molecule that causes the molecule's characteristic chemical reactions. The same functional group will undergo the same or similar chemical reactions regardless of the re ...
s independently of each other. 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.
Bioorthogonal chemistry
The term bioorthogonal chemistry refers to any chemical reaction that can occur inside of living systems without interfering with native biochemical processes. The term was coined by Carolyn R. Bertozzi in 2003. Since its introduction, the concept ...
refers to chemical reactions occurring inside living systems without reacting with naturally present cellular components. In
supramolecular chemistry
Supramolecular chemistry refers to the branch of chemistry concerning chemical systems composed of a discrete number of molecules. The strength of the forces responsible for spatial organization of the system range from weak intermolecular forces ...
the notion of orthogonality refers to the possibility of two or more supramolecular, often
non-covalent
In chemistry, a non-covalent interaction differs from a covalent bond in that it does not involve the sharing of electrons, but rather involves more dispersed variations of electromagnetic interactions between molecules or within a molecule. The ...
, interactions being compatible; reversibly forming without interference from the other.
In
analytical chemistry
Analytical chemistry studies and uses instruments and methods to separate, identify, and quantify matter. In practice, separation, identification or quantification may constitute the entire analysis or be combined with another method. Separati ...
, 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 the
etymologic origin of ''orthogonality''. Orthogonal testing is often required as a part of a
new drug application.
System reliability
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.
Neuroscience
In
neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, developme ...
, a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map.
Gaming
In board games such as
chess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...
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". In the ancient Chinese board game
Go a player can capture the stones of an opponent by occupying all orthogonally-adjacent points.
Other examples
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.
[For an illustration, se]
YouTube
A pure horizontal motion corresponds to a mono signal, equivalent to a stereo signal in which both channels carry identical (in-phase) signals.
See also
*
Orthogonal ligand-protein pair
References
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