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A discipline that combines the means to know the location and heading of an aircraft with respect to some map or coordinate system, with guidance which means steering the aircraft along some desired path toward a destination. The construction of the…

The branch of mathematics dealing with the solution of equations. These equations involve unknowns, or variables, along with fixed numbers from a specified system (Fig. 1). The origins of algebra were based on the need to develop equations that…

The study of zero sets of polynomial equations. Examples of zero sets of polynomial equations studied in algebraic geometry are the parabola y − x2 = 0, thought of as sitting in the (x, y)-plane, and the locus of all points (t, t2, t3, t4), which is…

The branch of mathematics dealing with two fundamental operations, differentiation and integration, which are carried out on functions. The subject of calculus (Fig. 1), as traditionally developed in college textbooks, is partly an elementary…

A heart-shaped curve generated by a point of a circle that rolls (without slipping) on a fixed circle of the same diameter. In point-wise construction of the curve, let O be a fixed point of a circle C of diameter a, and Q a variable point of C. Lay…

A theory of mathematical structure in which smooth continuous inputs lead to discontinuous responses. Water suddenly boils, ice melts, a building crashes to the ground, or the earth unexpectedly buckles and quakes. The French mathematician René Thom…

The branch of mathematics which studies arrangements of elements (usually a finite number) into sets under certain prescribed constraints. Problems combinatorialists attempt to solve include the enumeration problem (how many such arrangements are…

The theory of complex systems. Complexity theory arises from the need to understand the richness in structure and behavior often seen in large systems. The property that distinguishes complex systems from systems that are large but simple is the…

The solid of revolution obtained by revolving a right triangle about one of its shorter sides is called a cone, or more precisely a right circular cone (see illustration). More generally, the term cone is used in solid geometry to describe a solid…

A special operation in mathematics in which a set of points in one coordinate system is mapped or transformed into a corresponding set in another coordinate system, preserving the angle of intersection between pairs of curves.A mapping or…

Schemes for locating points in a given space by means of numerical quantities specified with respect to some frame of reference. These quantities are the coordinates of a point. To each set of coordinates there corresponds just one point in any…

A form of navigation that determines position of a craft by advancing a previous position to a new one on the basis of assumed distance and direction moved. The name probably stems from the early practice of determining speed by throwing overboard a…

An applied branch of decision theory. Decision analysis offers individuals and organizations a methodology for making decisions; it also offers techniques for modeling decision problems mathematically and finding optimal decisions numerically.…

A broad range of concepts which have been developed to both describe and prescribe the process of decision making, where a choice is made from a finite set of possible alternatives. Normative decision theory describes how decisions should be made in…

A mathematical operation performed on a function to determine the effect of a change in the value of the independent variable. Functions of one variable are considered in this article. For differentiation of functions of several variablesPartial…

Equipment used for determining the direction from which a received signal is being radiated. Direction finders serve a variety of applications. Direction finders can be used for navigation purposes to determine the heading of a vehicle relative to a…

For a polynomial f(x) = an xn + an−1 xn−1 + + a1 x + a0, the discriminant is given by the expression where x1, x2, …, xn are the roots of the equation f(x) = 0. There are n(n − 1)/2 terms (xi − xj)2 in the product corresponding to all possible…

An internationally standardized navigation system which allows an aircraft to measure its distance from a selected ground-based beacon. Such beacons are used throughout the world by all airliners, most of the military aircraft of the West, and a…

A mathematical abstraction corresponding to an expression of the type βγ + δ∊ + · · · , in which the elements (dyad symbols) consist of two vector symbols in juxtaposition without the intervention of either the dot (·) or cross (×). Essentially a…

One of the solutions of an eigenvalue equation. A parameter-dependent equation that possesses nonvanishing solutions only for particular values (eigenvalues) of the parameter is an eigenvalue equation, the associated solutions being the…

One of the values of the parameter in an eigenvalue equation for which the equation has a solution. A linear eigenvalue equation is a parameter-dependent equation of the form (1) 1that possesses nonvanishing solutions, |v(n), only for particular…

Systems that locate suitably equipped users by means of measurements made by electronic sensors. Typical measurements include distance or bearing from a vehicle to a known point, or direct determination of the present position of the vehicle in a…

The chief subject matter of the monumental 13-volume work called The Elements, written about 300 BCE by the Greek mathematician Euclid, who taught and founded a school of geometry at Alexandria. One of the milestones in the history of scientific…

A numerical analysis technique for obtaining approximate solutions to many types of engineering problems. The need for numerical methods arises from the fact that for most engineering problems analytical solutions do not exist. While the governing…

The theory of games of strategy can briefly be characterized as the application of mathematical analysis to abstract models of conflict situations. The first such models analyzed by the theory were parlor games such as chess, poker, and bridge.…

Procedures and techniques for visually representing on paper or a screen information pertaining to data analysis and decision making or relationships between variables by means of diagrams or charts.Structural information is usually depicted by a…

A system carried on many aircraft to warn the pilot that the aircraft may be in danger of inadvertent contact with the ground. It is intended to reduce the occurrence of controlled-flight-into-terrain (CFIT) accidents, in which aircraft with no…

Any nonplanar curve all of whose tangents make the same angle with a fixed line. Other characteristic properties are that all principal normals are parallel to a plane and that the ratio of torsion to curvature is constant. If a helix has constant…

A process of navigation by which a destination is approached by keeping some navigation parameter constant. In its early uses, the most commonly chosen parameter was the relative bearing from the vehicle to destination as determined from a signal…

The application to biological systems of the theory of encoding effects on signal transmission and communications efficiency. In everyday language, information is associated with knowledge, meaning (semantics), and the influence of information on…

A term applied to a curve C′ that cuts at right angles all tangents of a curve C (see illustration). Each curve C has infinitely many involutes and the distance between corresponding points of any two involutes is constant. If xi = xi (s), with i =…

A method, developed originally by A. M. Legendre, of obtaining the best values (the ones with least error) of unknown quantities supposed to satisfy a system of linear equations of the form shown as notation (1),1where n > m. Since there are more…

Solutions to the differential equation (1 − x2)y′ − 2xy′ + v(v + 1)y = 0. Legendre polynomials The most elementary of the Legendre functions, the Legendre polynomial Pn(x) can be defined by the generating function in Eq. (1). More explicit…

A curve shaped like the figure eight (see illustration), referred to by Jacques Bernoulli in 1694. Let F1, F2 be points of a plane π, with F1 F2 = 2a, a > 0. The locus of a point P of π which moves so that PF1 · PF2 = b2, where b is a positive…

A topological group with only countably many connected components whose identity component is open and is an analytic group. An analytic group or connected Lie group is a topological group with the additional structure of a smooth manifold such that…

An area of mathematics concerned with the minimization (or maximization) of a linear function of several variables subject to linear equations and inequalities. Linear programming developed from three main areas: transportation problems, game…

An exponent of a suitably chosen positive number (base) larger than unity. Logarithms are of value in mathematical computation and in the equations and formulas used in expressing natural phenomena.The invention of natural logarithms (base e = 2.718…

The subject that investigates, formulates, and establishes principles of valid reasoning.The first attempt to investigate acceptable modes of reasoning systematically was made by Aristotle, in whose Organon reasoning was recognized as the subject of…

A navigation system from which hyperbolic lines of position are determined by measuring the difference in times of arrival of pulses from widely spaced, synchronized transmitting stations. Since radio waves travel at the speed of light, a line of…

A compass depending for its directive force upon the attraction of the Earth's magnetism on a magnet free to turn in any horizontal direction. A compass is an instrument used for determining horizontal direction. Principle of operation The magnetic…

The process of directing a watercraft to a destination in a safe and expeditious manner. From a known present position, a course is determined that avoids dangers, and on this course estimates are made of time schedules. The task is to make periodic…

The study of matrices and their properties, and of linear transformations on vector spaces, which can be represented by matrices. A matrix is a rectangular array of numbers, with the numbers that appear in the matrix being called entries. For…

A branch of mathematical analysis connected with the theory of integration. In order to discuss this subject, a formal definition of the term measure must be given. Sigma algebras Let X be an arbitrary set. Let m be a fixed collection of subsets of…

An all-weather aircraft landing-guidance system that operates at microwave frequencies and provides deviations from the landing runway centerline using a time-referenced scanning beam (TRSB) technique. The MLS was standardized in 1988 and approved…

A technique for estimating the solution, x, of a numerical mathematical problem by means of an artificial sampling experiment. The estimate is usually given as the average value, in a sample, of some statistic whose mathematical expectation is equal…

A graphical relationship between a set of variables that are related by a mathematical equation or law. The fundamental principle involved in the construction of a nomographic or alignment chart consists of representing an equation containing three…

The area of applied mathematics and operations research concerned with finding the largest or smallest value of a function subject to constraints or restrictions on the variables of the function. Nonlinear programming is sometimes referred to as…

The study of the properties and relationships between integers and other special types of numbers. There are many sets of positive integers of particular interest, such as the primes and the perfect numbers. Number theory, of ancient and continuing…

The form of navigation in which position is determined relative to external reference points, usually fixed points on the Earth. It is the oldest form of navigation. The following remarks are formulated in the context of marine navigation. However,…

The complex of navigational techniques modified from those used in other areas to suit the distinctive regional character of polar areas. Although polar navigation has become routine to a rising number of navigators operating in and through such…