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Great-circle Distance
The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path between the two points on the surface of the sphere. (By comparison, the shortest path passing through the sphere's interior is the chord between the points.) On a curved surface, the concept of straight lines is replaced by a more general concept of geodesics, curves which are locally straight with respect to the surface. Geodesics on the sphere are great circles, circles whose center coincides with the center of the sphere. Any two distinct points on a sphere that are not antipodal (diametrically opposite) both lie on a unique great circle, which the points separate into two arcs; the length of the shorter arc is the great-circle distance between the points. This arc length is proportional to the central angle between the points, which if measured in radians can be scaled u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Illustration Of Great-circle Distance
An illustration is a decoration, interpretation, or visual explanation of a text, concept, or process, designed for integration in print and digitally published media, such as posters, flyers, magazines, books, teaching materials, animations, video games and films. An illustration is typically created by an illustrator. Digital illustrations are often used to make websites and apps more user-friendly, such as the use of emojis to accompany digital type. Illustration also means providing an example; either in writing or in picture form. The origin of the word "illustration" is late Middle English (in the sense ‘illumination; spiritual or intellectual enlightenment’): via Old French from Latin">-4; we might wonder whether there's a point at which it's appropriate to talk of the beginnings of French, that is, when it wa ... from Latin ''illustratio''(n-), from the verb ''illustrare''. Illustration styles Contemporary illustration uses a wide range of styles and techniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Angle
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called '' spherical distance''. The size of a central angle is or (radians). When defining or drawing a central angle, in addition to specifying the points and , one must specify whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°). Equivalently, one must specify whether the movement from point to point is clockwise or counterclockwise. Formulas If the intersection points and of the legs of the angle with the circle form a diameter, then is a straight angle. (In radians, .) Let be the min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directorate Of Overseas Surveys
The Ordnance Survey International or Ordnance Survey Overseas Directorate its predecessors built an archive of air photography, map and survey records for the United Kingdom from 1946 to 1999. The Ordnance Survey International Collection (formerly the Ordnance Survey International Library) held mapping records that were acquired outside the UK. Although the international division opened in 1946, the OS had been involved in overseas work for almost a century (notably the 1864-65 Ordnance Survey of Jerusalem). The agency was closed in 2001. History The agency In 1946 the ''Directorate of Colonial Surveys'' (DCS) was established by the Colonial Office to provide a central survey and mapping organisation for British colonies and protectorates. In 1957, with the imminent decolonisation of many British territories, it was renamed the ''Directorate of Overseas Surveys'' (DOS). Government reviews during the 1970s led to it being merging into the Ordnance Survey (OS) in 1984 whence it w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vincenty's Formulae
Vincenty's formulae are two related iterative methods used in geodesy to calculate the distance between two points on the surface of a spheroid, developed by Thaddeus Vincenty (1975a). They are based on the assumption that the figure of the Earth is an oblate spheroid, and hence are more accurate than methods that assume a spherical Earth, such as great-circle distance. The first (direct) method computes the location of a point that is a given distance and azimuth (direction) from another point. The second (inverse) method computes the geographical distance and azimuth between two given points. They have been widely used in geodesy because they are accurate to within 0.5 mm (0.020in) on the Earth ellipsoid. Background Vincenty's goal was to express existing algorithms for geodesics on an ellipsoid in a form that minimized the program length (Vincenty 1975a). His unpublished report (1975b) mentions the use of a Wang 720 desk calculator, which had only a few kilobytes of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haversine
The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'', Section I) trigonometric tables. The versine of an angle is 1 minus its . There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation. Overview The versine[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condition Number
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given f(x) = y, one is solving for ''x,'' and thus the condition number of the (local) inverse must be used. The condition number is derived from the theory of propagation of uncertainty, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haversine Formula
The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, that relates the sides and angles of spherical triangles. The first table of haversines in English was published by James Andrew in 1805, but Florian Cajori credits an earlier use by José de Mendoza y Ríos in 1801. (NB. ISBN and link for reprint of second edition by Cosimo, Inc., New York, 2013.) The term ''haversine'' was coined in 1835 by James Inman. (Fourth edition) These names follow from the fact that they are customarily written in terms of the haversine function, given by . The formulas could equally be written in terms of any multiple of the haversine, such as the older versine function (twice the haversine). Prior to the advent of computers, the elimination of division and multiplication by factors of two proved convenient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IEEE 754
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard #Design rationale, addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and Software portability, portably. Many hardware floating-point units use the IEEE 754 standard. The standard defines: * ''arithmetic formats:'' sets of Binary code, binary and decimal floating-point data, which consist of finite numbers (including signed zeros and subnormal numbers), infinity, infinities, and special "not a number" values (NaNs) * ''interchange formats:'' encodings (bit strings) that may be used to exchange floating-point data in an efficient and compact form * ''rounding rules:'' properties to be satisfied when rounding numbers during arithmetic and conversions * ''operations:'' arithmetic and other operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rounding Error
In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error. When using approximation equations or algorithms, especially when using finitely many digits to represent real numbers (which in theory have infinitely many digits), one of the goals of numerical analysis is to estimate computation errors. Computation errors, also called numerical errors, include both truncation errors and roundoff errors. When a sequence of calculations with an input involving any roundoff error are made, errors may accumulate, sometimes dominating the calculation. In ill-conditioned problems, significant error may accumulate. In short, there are two major facets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating Point
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base. Numbers of this form are called floating-point numbers. For example, the number 2469/200 is a floating-point number in base ten with five digits: 2469/200 = 12.345 = \! \underbrace_\text \! \times \! \underbrace_\text\!\!\!\!\!\!\!\overbrace^ However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits—it needs six digits. The nearest floating-point number with only five digits is 12.346. And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits. In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chord (geometry)
A chord (from the Latin ''chorda'', meaning " bowstring") of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a ''secant line''. The perpendicular line passing through the chord's midpoint is called '' sagitta'' (Latin for "arrow"). More generally, a chord is a line segment joining two points on any curve, for instance, on an ellipse. A chord that passes through a circle's center point is the circle's ''diameter''. In circles Among properties of chords of a circle are the following: # Chords are equidistant from the center if and only if their lengths are equal. # Equal chords are subtended by equal angles from the center of the circle. # A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. # If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arc Length
Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the most basic formulation of arc length for a vector valued curve (thought of as the trajectory of a particle), the arc length is obtained by integrating speed, the magnitude of the velocity vector over the curve with respect to time. Thus the length of a continuously differentiable curve (x(t),y(t)), for a\le t\le b, in the Euclidean plane is given as the integral L = \int_a^b \sqrt\,dt, (because \sqrt is the magnitude of the velocity vector (x'(t),y'(t)), i.e., the particle's speed). The defining integral of arc length does not always have a closed-form expression, and numerical integration may be used instead to obtain numerical values of arc length. Determining the length of an irregular arc segment by approximating the arc segment as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
