Orthogonality
Orthogonality is a term with various meanings depending on the context.
In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity. Although many authors use the two terms perpendicular and orthogonal interchangeably, the term perpendicular is more specifically used for lines and planes that intersect to form a right angle, whereas orthogonal is used in generalizations, such as orthogonal vectors or orthogonal curves.
The term is also used in other fields like physics, art, computer science, statistics, and economics.
Etymology
The word comes from the Ancient Greek, meaning "upright", and, meaning "angle".The Ancient Greek and Classical Latin originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle.
Physics
Optics
In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right-handed and left-handed circular polarization.Special relativity
In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of simultaneous events, also determined by the rapidity. The theory features relativity of simultaneity.Quantum mechanics
In quantum mechanics, a sufficient condition that two eigenstates of a Hermitian operator, and, are orthogonal is that they correspond to different eigenvalues. This means, in Dirac notation, that if and correspond to different eigenvalues. This follows from the fact that Schrödinger's equation is a Sturm–Liouville equation or that observables are given by Hermitian operators.Art
In art, the perspective lines pointing to the vanishing 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 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 of the Thyssen-Bornemisza Museum 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:
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 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.
Orthogonal instruction set
An instruction set is said to be orthogonal if it lacks redundancy and is designed such that instructions can use any register in any addressing 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. An orthogonal instruction set uniquely encodes all combinations of registers and addressing modes.Telecommunications
In telecommunications, multiple access 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, where the orthogonal basis functions are nonoverlapping rectangular pulses.Orthogonal frequency-division multiplexing
Another scheme is orthogonal frequency-division multiplexing, 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 versions of 802.11 Wi-Fi; WiMAX; ITU-T G.hn, DVB-T, the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT, the standard form of ADSL.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 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. 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, uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data and as random variables.One econometric formalism that is alternative to the maximum likelihood framework, the Generalized Method of Moments, relies on orthogonality conditions. In particular, the Ordinary Least Squares estimator may be easily derived from an orthogonality condition between the explanatory variables and model residuals.