query: the shortest distance between two points on the earth 's surface is

pos: Great-circle distance great circles, and all great circle arcs between antipodal points have a length of half the circumference of the circle, or formula_1, where "r" is the radius of the sphere. The Earth is nearly spherical (see Earth radius), so great-circle distance formulas give the distance between points on the surface of the Earth correct to within about 0.5%. (See .) Let formula_2 and formula_3 be the geographical latitude and longitude in radians of two points 1 and 2, and formula_4 be their absolute differences; then formula_5, the central angle between them, is given by the spherical law of cosines if

pos: Great-circle distance apply to a limited area). Using the mean earth radius, formula_21 (for the WGS84 ellipsoid) means that in the limit of small flattening, the mean square relative error in the estimates for distance is minimized. Great-circle distance The great-circle distance or orthodromic distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In

pos: Geographical distance two points on the surface is along the great-circle which contains the two points. The great-circle distance article gives the formula for calculating the distance along a great-circle on a sphere about the size of the Earth. That article includes an example of the calculation. A tunnel between points on Earth is defined by a line through three-dimensional space between the points of interest. The great circle chord length may be calculated as follows for the corresponding unit sphere: The tunnel distance between points on the surface of a spherical Earth is formula_48. For short distances (formula_49), this underestimates the

pos: Great-circle distance Great-circle distance The great-circle distance or orthodromic distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called "great circles". The determination of

pos: Vincenty's formulae 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

pos: Sphere The shortest distance along the surface between two distinct non-antipodal points on the sphere is the length of the smaller of the two arcs on the unique great circle that includes the two points. Equipped with this "great-circle distance", a great circle becomes the Riemannian circle. If a particular point on a sphere is (arbitrarily) designated as its "north pole", then the corresponding antipodal point is called the "south pole", and the equator is the great circle that is equidistant to them. Great circles through the two poles are called lines (or meridians) of longitude, and the line connecting the

neg: Geographical distance great circle distance by formula_50. An ellipsoid approximates the surface of the earth much better than a sphere or a flat surface does. The shortest distance along the surface of an ellipsoid between two points on the surface is along the geodesic. Geodesics follow more complicated paths than great circles and in particular, they usually don't return to their starting positions after one circuit of the earth. This is illustrated in the figure on the right where "f" is taken to be 1/50 to accentuate the effect. Finding the geodesic between two points on the earth, the so-called inverse geodetic

neg: Expansion of the universe long-haul airline flights where the distance between two points is measured based upon a great circle, rather than the straight line one might plot on a two-dimensional map of the Earth's surface. In general, such shortest-distance paths are called "geodesics". In Euclidean geometry, the geodesic is a straight line, while in non-Euclidean geometry such as on the Earth's surface, this is not the case. Indeed, even the shortest-distance great circle path is always longer than the Euclidean straight line path which passes through the interior of the Earth. The difference between the straight line path and the shortest-distance great circle

neg: Geodesics on an ellipsoid on a surface. However, it is frequently more useful to define them as paths with zero geodesic curvature—i.e., the analogue of straight lines on a curved surface. This definition encompasses geodesics traveling so far across the ellipsoid's surface (somewhat more than half the circumference) that other distinct routes require less distance. Locally, these geodesics are still identical to the shortest distance between two points. By the end of the 18th century, an ellipsoid of revolution (the term spheroid is also used) was a well-accepted approximation to the figure of the Earth. The adjustment of triangulation networks entailed reducing all the

neg: Geographical distance Geographical distance Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem. Calculating the distance between geographical coordinates is based on some level of abstraction; it does not provide an "exact" distance, which is unattainable if one attempted to account for every irregularity in the surface of the earth. Common abstractions for the surface between two geographic points are: All abstractions above ignore changes in

neg: Geodesics on an ellipsoid for example , A line traced in the manner we have now been describing, or deduced from trigonometrical measures, by the means we have indicated, is called a "geodetic" or "geodesic line:" it has the property of being the shortest which can be drawn between its two extremities on the surface of the Earth; and it is therefore the proper itinerary measure of the distance between those two points. In its adoption by other fields "geodesic line", frequently shortened to "geodesic", was preferred. This section treats the problem on an ellipsoid of revolution (both oblate and prolate). The problem on

neg: Geographical distance of distance calculations using this approximation become increasingly inaccurate as: The shortest distance between two points in plane is a straight line. The Pythagorean theorem is used to calculate the distance between points in a plane. Even over short distances, the accuracy of geographic distance calculations which assume a flat Earth depend on the method by which the latitude and longitude coordinates have been projected onto the plane. The projection of latitude and longitude coordinates onto a plane is the realm of cartography. The formulae presented in this section provide varying degrees of accuracy. This formula takes into account the

neg: Triangle inequality inequality can be viewed intuitively in either or . The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a angle and two angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on

neg: Geographical distance variation in distance between meridians with latitude: This approximation is very fast and produces fairly accurate result for small distances . Also, when ordering locations by distance, such as in a database query, it is much faster to order by squared distance, eliminating the need for computing the square root. The FCC prescribes the following formulae for distances not exceeding : If we are willing to accept a possible error of 0.5%, we can use formulas of spherical trigonometry on the sphere that best approximates the surface of the earth. The shortest distance along the surface of a sphere between

neg: Geodesics on an ellipsoid Geodesics on an ellipsoid The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an "oblate ellipsoid", a slightly flattened sphere. A "geodesic" is the shortest path between two points on a curved surface, i.e., the analogue of a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry . If the Earth is treated as a sphere, the geodesics are great circles (all of which are closed)

neg: Hilbert's fourth problem 2 and 3 was given by Armenian mathematician Rouben V. Ambartzumian. Hilbert discusses the existence of non-Euclidean geometry and non-Archimedean geometry, as well as the idea that a 'straight line' is defined as the shortest path between two points. He mentions how congruence of triangles is necessary for Euclid's proof that a straight line in the plane is the shortest distance between two points. He summarizes as follows: The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only

neg: Great circle distance between two points on a surface of a sphere in Riemannian geometry where such great circles are called Riemannian circles. These great circles are the geodesics of the sphere. In higher dimensions, the great circles on the "n"-sphere are the intersection of the "n"-sphere with 2-planes that pass through the origin in the Euclidean space R. To prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider the class of all regular paths from a point formula_1 to

neg: Geodesic two concepts are closely related. The difference is that geodesics are only "locally" the shortest distance between points, and are parameterized with "constant speed". Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map "t" → "t" from the unit interval on the real number line to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant. Geodesics are commonly seen in the study of

neg: Extreme points of Earth land. The pairs of cities (with a population over 100,000) with the greatest distance between them are: These are the longest straight lines that can be drawn between any two points on the surface of the Earth and remain exclusively over land or water; the points need not lie on the same latitude or longitude. Extreme points of Earth This is a list of extreme points of Earth, the geographical locations that are farther north or south than, higher or lower in elevation than, or farthest inland or out to sea from, any other locations on the landmasses, continents or

neg: Spherical Earth visible. For example, New York City is about 40° north of the equator. The apparent motion of the Sun blots out slightly different parts of the sky from day to day, but over the course of the entire year it sees a dome of 280° (360° - 80°). So for example, both Orion and the Big Dipper are visible during at least part of the year. Making stellar observations from a representative set of points across the Earth, combined with knowing the shortest on-the-ground distance between any two given points makes an approximate sphere the only possible shape for the

neg: Rhumb line or by solving a nonlinear system of two equations in the two unknowns and . There are infinitely many solutions; the shortest one is that which covers the actual longitude difference, i.e. does not make extra revolutions, and does not go "the wrong way around". The distance between two points Δ, measured along a loxodrome, is simply the absolute value of the secant of the bearing (azimuth) times the north–south distance (except for circles of latitude for which the distance becomes infinite): where is one of the earth average radii. Its use in navigation is directly linked to the style,

neg: Jessica Watson definition states in part "The shortest orthodromic track of the vessel must be at least in length." The analysis suggests that Watson's published distance logs are based on sailed distances, including tacks and strategic weather detours, rather than the shortest orthodromic track between islands and capes as defined. The rule is based on the older rule, followed by current record-holder Jesse Martin, that during a circumnavigation the sailor must pass two points on opposite sides of the earth (antipodes). For example, if starting in southern England, a place near the start will be opposite to the track near New Zealand.

neg: Expansion of the universe it. The metric expansion of space is described using the mathematics of metric tensors. The coordinate system we use is called "comoving coordinates", a type of coordinate system which takes account of time as well as space and the speed of light, and allows us to incorporate the effects of both general and special relativity. For example, consider the measurement of distance between two places on the surface of the Earth. This is a simple, familiar example of spherical geometry. Because the surface of the Earth is two-dimensional, points on the surface of the Earth can be specified by two

neg: Euclidean shortest path from the obstacles or (in an approach called the "continuous Dijkstra" method) propagating a wavefront from one of the points until it meets the other. In three (and higher) dimensions the problem is NP-hard in the general case, but there exist efficient approximation algorithms that run in polynomial time based on the idea of finding a suitable sample of points on the obstacle edges and performing a visibility graph calculation using these sample points. There are many results on computing shortest paths which stays on a polyhedral surface. Given two points s and t, say on the surface of a

neg: Differential calculus is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface. If the surface is a plane, then the shortest curve is a line. But if the surface is, for example, egg-shaped, then the shortest path is not immediately clear. These paths are called geodesics, and one of the simplest problems in the calculus of variations is finding geodesics. Another example is: Find the smallest area surface filling in a closed curve in space. This surface is called a minimal surface and it, too, can be found using the calculus

neg: Rhumb line or "projection" of certain navigational maps. A rhumb line appears as a straight line on a Mercator projection map. The name is derived from Old French or Spanish respectively: ""rumb"" or "rumbo", a line on the chart which intersects all meridians at the same angle. On a plane surface this would be the shortest distance between two points. Over the Earth's surface at low latitudes or over short distances it can be used for plotting the course of a vehicle, aircraft or ship. Over longer distances and/or at higher latitudes the great circle route is significantly shorter than the rhumb

neg: Spherical Earth Earth. Knowing the difference in angle between two points on the Earth's surface and the surface distance between them allows a calculation of the Earth's size. Using observations at Rhodes (in Greece) and Alexandria (in Egypt) and the distance between them, the Ancient Greek philosopher Posidonius actually did use this technique to calculate the circumference of the planet to within perhaps 4% of the correct value (though modern equivalents of his units of measure are not precisely known). Since the 1500s, many people have sailed or flown completely around the world in all directions, and none have discovered an edge

neg: Land's End to John o' Groats route passes over a series of stretches of water in the Irish Sea. Google Earth reports a distance of 602.70 miles between the two iconic marker points. According to a 1964 road atlas, the shortest route using classified roads was but in a 2008 road atlas, the shortest route using classified roads was . An online route planner in 2011 also calculated the quickest route by road as , estimating a time of 15 hours 48 minutes for the journey (this uses the A30, M5, M6, A74(M), M74, M73, M80, M9, A9 & A99) but the overall shortest route by

neg: Introduction to general relativity is the following: In geodesy, the science of measuring Earth's size and shape, a geodesic (from Greek "geo", Earth, and "daiein", to divide) is the shortest route between two points on the Earth's surface. Approximately, such a route is a segment of a great circle, such as a line of longitude or the equator. These paths are certainly not straight, simply because they must follow the curvature of the Earth's surface. But they are as straight as is possible subject to this constraint. The properties of geodesics differ from those of straight lines. For example, on a plane, parallel lines

neg: Geodesic Geodesic In differential geometry, a geodesic () is a generalization of the notion of a "straight line" to "curved spaces". The term "geodesic" comes from "geodesy", the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In the presence of an affine connection, a geodesic is

neg: Geodesic can be found in Geodesics serve as the basis to calculate: Geodesic In differential geometry, a geodesic () is a generalization of the notion of a "straight line" to "curved spaces". The term "geodesic" comes from "geodesy", the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a

neg: Mercator 1569 world map in this widely disseminated volume were on the Mercator projection. In Legend 12 Mercator makes careful distinction between great circles ("plaga") and rhumb lines ("directio") and he points out that the rhumb between two given points is always longer than the great circle distance, the latter being the shortest distance between the points. However, he stresses that over short distances (which he quantifies) the difference may be negligible and a calculation of the rhumb distance may be adequate and more relevant since it is the sailing distance on a constant bearing. He gives the details of such a calculation in

neg: Expansion of the universe latitude and longitude, and x-y graphs are common examples of coordinates. A metric is a formula which describes how a number known as "distance" is to be measured between two points. It may seem obvious that distance is measured by a straight line, but in many cases it is not. For example, long haul aircraft travel along a curve known as a "great circle" and not a straight line, because that is a better metric for air travel. (A straight line would go through the earth). Another example is planning a car journey, where one might want the shortest journey

neg: Vincenty's formulae converges to the correct result 19944127.421 m after about 60 iterations; however, in other cases many thousands of iterations are required. Newton's method has been successfully used to give rapid convergence for all pairs of input points (Karney, 2013). 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

neg: Fréchet distance as a polyhedral terrain or some Euclidean space with obstacles, the distance between two points on the curves is most naturally defined as the length of the shortest path between them. The leash is required to be a geodesic joining its endpoints. The resulting metric between curves is called the geodesic Fréchet distance. Cook and Wenk describe a polynomial-time algorithm to compute the geodesic Fréchet distance between two polygonal curves in a simple polygon. If we further require that the leash must move continuously in the ambient metric space, then we obtain the notion of the homotopic Fréchet distance between

neg: Expansion of the universe path is due to the curvature of the Earth's surface. While there is always an effect due to this curvature, at short distances the effect is small enough to be unnoticeable. On plane maps, great circles of the Earth are mostly not shown as straight lines. Indeed, there is a seldom-used map projection, namely the gnomonic projection, where all great circles are shown as straight lines, but in this projection, the distance scale varies very much in different areas. There is no map projection in which the distance between any two points on Earth, measured along the great circle geodesics,

neg: Hilbert's fourth problem formulates some conditions for a wedge metric to generate a measure in the space of planes. The definition of a wedge metrics is based on certain tetrahedral inequalities of combinatorial nature. The latter inequalities replace the usual triangle inequality. A gnomonic map projection of the sphere displays all great circles as straight lines, resulting in any line segment on a gnomonic map showing the shortest route between the segment's two endpoints. This is achieved by casting surface points of the sphere onto a tangent plane, each landing where a ray from the center of the earth passes through the point

neg: Great circle does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere. For most pairs of points on the surface of a sphere, there is a unique great circle through the two points. The exception is a pair of antipodal points, for which there are infinitely many great circles. The minor arc of a great circle between two points is the shortest surface-path between them. In this sense, the minor arc is analogous to “straight lines” in Euclidean geometry. The length of the minor arc of a great circle is taken as the

neg: Incidence geometry the point is not on the line, the pair is called an "anti-flag". There is no natural concept of distance (a metric) in an incidence structure. However, a combinatorial metric does exist in the corresponding incidence graph (Levi graph), namely the length of the shortest path between two vertices in this bipartite graph. The distance between two objects of an incidence structure – two points, two lines or a point and a line – can be defined to be the distance between the corresponding vertices in the incidence graph of the incidence structure. Another way to define a distance again

neg: Geodesics on an ellipsoid and the problems reduce to ones in spherical trigonometry. However, showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid and, in this case, the equator and the meridians are the only simple closed geodesics. Furthermore, the shortest path between two points on the equator does not necessarily run along the equator. Finally, if the ellipsoid is further perturbed to become a triaxial ellipsoid (with three distinct semi-axes), only three geodesics are closed. There are several ways of defining geodesics . A simple definition is as the shortest path between two points

neg: Geodesic minimization). Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy. The resulting shape of the band is a geodesic. It is possible that several different curves between two points minimize the distance, as is the case for two diametrically opposite points on a sphere. In such a case, any of these curves is a geodesic. A contiguous segment of a geodesic is again a geodesic. In general, geodesics are not the same as "shortest curves" between two points, though the

neg: Lambert azimuthal equal-area projection between "S" and "P" in three-dimensional space ("not" the distance along the sphere surface). Then the projection sends "P" to a point "P′" on the plane that is a distance "d" from "S". To make this more precise, there is a unique circle centered at "S", passing through "P", and perpendicular to the plane. It intersects the plane in two points; let "P"′ be the one that is closer to "P". This is the projected point. See the figure. The antipode of "S" is excluded from the projection because the required circle is not unique. The case of "S" is

neg: Geographical distance as follows: Unless specified otherwise, the radius of the earth for the calculations below is: formula_12 = Distance between the two points, as measured along the surface of the earth and in the same units as the value used for radius unless specified otherwise. Longitude has singularities at the Poles (longitude is undefined) and a discontinuity at the ±180° meridian. Also, planar projections of the circles of constant latitude are highly curved near the Poles. Hence, the above equations for delta latitude/longitude (formula_13, formula_14) and mean latitude (formula_15) may not give the expected answer for positions near the Poles or

neg: Ground track overhead, or cross the zenith, in the frame of reference of a ground observer. In air navigation, ground tracks typically approximate an arc of a great circle, this being the shortest distance between two points on the Earth's surface. In order to follow a specified ground track, a pilot must adjust their heading in order to compensate for the effect of wind. Aircraft routes are planned to avoid restricted airspace and dangerous areas, and to pass near navigation beacons. The ground track of a satellite can take a number of different forms, depending on the values of the orbital elements,

neg: Rhombille tiling the tiling. It is possible to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points ("x","y","z") with |"x" + "y" + "z"| ≤ 1, in such a way that two vertices are adjacent if and only if the corresponding lattice points are at unit distance from each other, and more strongly such that the number of edges in the shortest path between any two vertices of the tiling is the same as the Manhattan distance between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite

neg: Gnomonic projection to find the shortest route between start and destination. The gnomonic projection is used extensively in photography, where it is called "rectilinear projection". The gnomonic projection is used in astronomy where the tangent point is centered on the object of interest. The sphere being projected in this case is the celestial sphere, "R" = 1, and not the surface of the Earth. Gnomonic projection A gnomonic map projection displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achieved by

neg: Conical coordinates for the two conical coordinates are and An alternative set of (non-orthogonal) conical coordinates have been derived formula_13 where formula_14 are spherical polar coordinates. The corresponding inverse relations are formula_15 The infinitesimal Euclidean distance between two points in these coordinates formula_17 and formula_18 are orthogonal coordinates on the surface of the cone given by formula_19. If the path between any two points is constrained to this surface, then the geodesic distance between any two points formula_20 and formula_21 is formula_22 Conical coordinates Conical coordinates are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius ) and

neg: 6th Ring Road a much larger distance than the inner ring roads. Equally large is the distance between two points. For example, the distance between Jingtong Expressway to Jingshen Expressway is approximately 2 kilometres on the 4th Ring Road. It expands to nearly 4 kilometres on the 5th Ring Road. On the 6th Ring Road, 10 kilometres elapse from one expressway to the other—and the Jingtong to Jingshen Expressway (on the 6th Ring Road, the Jingha to Jingshen Expressway) is one of the shortest distances between expressways in Beijing. Anything up to 30 - 35 kilometres can lapse between the Jingcheng Expressway and

neg: Photogrammetry of impact). There exist many software packages for photogrammetry; see comparison of photogrammetry software. Photogrammetry Photogrammetry is the science of making measurements from photographs, especially for recovering the exact positions of surface points. Photogrammetry is as old as modern photography, dating to the mid-19th century and in the simplest example, the distance between two points that lie on a plane parallel to the photographic image plane, can be determined by measuring their distance on the image, if the scale ("s") of the image is known. Photogrammetric analysis may be applied to one photograph, or may use high-speed photography and remote

neg: COROT-7b Although COROT-7b's mass remains uncertain, with a range between 2 and 8 Earth masses, its radius and orbital period are well known from COROT photometry: it orbits very close to its star (1/23rd the distance from the Sun to Mercury) with an orbital period of 20 hours, 29 minutes, and 9.7 seconds and has a radius of 1.58 Earth radii. COROT-7b has the shortest orbit of any planet known at the time of its discovery. COROT-7b has a maximum surface temperature between 1800 and 2600 °C (3300 to 4700 degrees Fahrenheit). Due to the high temperature, it may be covered

neg: Isomap weighted graph. To be specific, the classical scaling of metric MDS performs low-dimensional embedding based on the pairwise distance between data points, which is generally measured using straight-line Euclidean distance. Isomap is distinguished by its use of the geodesic distance induced by a neighborhood graph embedded in the classical scaling. This is done to incorporate manifold structure in the resulting embedding. Isomap defines the geodesic distance to be the sum of edge weights along the shortest path between two nodes (computed using Dijkstra's algorithm, for example). The top "n" eigenvectors of the geodesic distance matrix, represent the coordinates in the

neg: Great-circle distance the cross product over the dot product. A line through three-dimensional space between points of interest on a spherical Earth is the chord of the great circle between the points. The central angle between the two points can be determined from the chord length. The great circle distance is proportional to the central angle. The great circle chord length, formula_14, may be calculated as follows for the corresponding unit sphere, by means of Cartesian subtraction: The central angle is: The shape of the Earth closely resembles a flattened sphere (a spheroid) with equatorial radius formula_17 of 6378.137 km; distance formula_18

neg: Franz-Erich Wolter computations of geodesic Voronoi diagrams and geodesic medial axis on general parametrized curved surfaces. In the surface case the length of a shortest geodesic join defines the distance between two points. In 2007, Wolter extended the computations of geodesic Voronoi diagrams and geodesic medial axis (inverse) transform to Riemannian 3D-manifolds. Wolter's early works on computing Riemannian Laplace Beltrami spectra for surfaces and images lead to a patent application in (2005) for a method using those spectra as Shape DNA for recognizing and retrieving surfaces, solids and images from data repositories. His works used the heat trace of a Riemannian Laplace

neg: Fréchet distance Fréchet distance In mathematics, the Fréchet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. It is named after Maurice Fréchet. Imagine a man traversing a finite curved path while walking his dog on a leash, with the dog traversing a separate one. Assume that the dog varies its speed to keep as much slack in her leash as possible: the Fréchet distance between the two curves is the length of the shortest leash sufficient for both to traverse their separate paths. Note that the definition is

neg: Photogrammetry Photogrammetry Photogrammetry is the science of making measurements from photographs, especially for recovering the exact positions of surface points. Photogrammetry is as old as modern photography, dating to the mid-19th century and in the simplest example, the distance between two points that lie on a plane parallel to the photographic image plane, can be determined by measuring their distance on the image, if the scale ("s") of the image is known. Photogrammetric analysis may be applied to one photograph, or may use high-speed photography and remote sensing to detect, measure and record complex 2-D and 3-D motion fields by feeding

neg: Segment addition postulate Segment addition postulate In geometry, the Segment Addition Postulate states that given 2 points A and C, a third point B lies on the line segment AC if and only if the distances between the points satisfy the equation AB + BC = AC. This is related to the triangle inequality, which states that AB + BC formula_1 AC with equality if and only if A, B, and C are collinear (on the same line). This in turn is equivalent to the proposition that the shortest distance between two points lies on a straight line. The segment addition postulate is

neg: Segment addition postulate often useful in proving results on the congruence of segments. Segment addition postulate In geometry, the Segment Addition Postulate states that given 2 points A and C, a third point B lies on the line segment AC if and only if the distances between the points satisfy the equation AB + BC = AC. This is related to the triangle inequality, which states that AB + BC formula_1 AC with equality if and only if A, B, and C are collinear (on the same line). This in turn is equivalent to the proposition that the shortest distance between two points

neg: Distance (graph theory) Distance (graph theory) In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite. In the case of a directed graph the distance formula_1 between two vertices formula_2 and formula_3 is defined as the

neg: Descriptive geometry in 3D space) of one of the lines: SU in this example, projection 3 is drawn with hinge line H parallel to SU. To get an end view of SU, projection 4 is drawn with hinge line H perpendicular to SU. The perpendicular distance "d" gives the shortest distance between PR and SU. To get points Q and T on these lines giving this shortest distance, projection 5 is drawn with hinge line H parallel to PR, making both PR and SU true views (any projection of an end view is a true view). Projecting the intersection of these lines,

neg: Gravity of Earth the force due to gravitational attraction between two bodies (the Earth and the object being weighed) varies inversely with the square of the distance between them, an object at the Equator experiences a weaker gravitational pull than an object at the poles. In combination, the equatorial bulge and the effects of the surface centrifugal force due to rotation mean that sea-level gravity increases from about 9.780 m/s at the Equator to about 9.832 m/s at the poles, so an object will weigh approximately 0.5% more at the poles than at the Equator. Gravity decreases with altitude as one rises above

neg: Longitude from 1 at the equator to 0 at the poles, which measures how circles of latitude shrink from the equator to a point at the pole, so the length of a degree of longitude decreases likewise. This contrasts with the small (1%) increase in the length of a degree of latitude (north-south distance), equator to pole. The table shows both for the WGS84 ellipsoid with = and = . Note that the distance between two points 1 degree apart on the same circle of latitude, measured along that circle of latitude, is slightly more than the shortest (geodesic) distance between

neg: 3200 Phaethon approaches to the Sun. Phaethon is categorized as a potentially hazardous asteroid (PHA), but that does not mean there is a near-term threat of an impact. It is a potentially hazardous asteroid merely as a result of its size (absolute magnitude ) and Earth minimum orbit intersection distance ). The Earth minimum orbit intersection distance (E-MOID) is , which is defined by the shortest distance between the orbit of Phaethon and the orbit of Earth. With a 30+ year observation arc, the orbit of Phaethon is very well understood with very small uncertainties. Close approaches of Phaethon are well constrained

neg: Spherical geometry Spherical geometry Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean. Two practical applications of the principles of spherical geometry are navigation and astronomy. In plane (Euclidean) geometry, the basic concepts are points and (straight) lines. On a sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" in Euclidean geometry, but in the sense of "the shortest paths between points", which are called geodesics. On a sphere, the geodesics are the great circles;

neg: Geometric spanner it possible to compute the greedy graph on larger point sets. Chew's main result was that for a set of points in the plane there is a triangulation of this pointset such that for any two points there is a path along the edges of the triangulation with length at most formula_14 the Euclidean distance between the two points. The result was applied in motion planning for finding reasonable approximations of shortest paths among obstacles. The best upper bound known for the Euclidean Delaunay triangulation is that it is a formula_15-spanner for its vertices. The lower bound has been increased

neg: Net (polyhedron) consider several nets to see which gives the shortest path. For example, in the case of a cube, if the points are on adjacent faces one candidate for the shortest path is the path crossing the common edge; the shortest path of this kind is found using a net where the two faces are also adjacent. Other candidates for the shortest path are through the surface of a third face adjacent to both (of which there are two), and corresponding nets can be used to find the shortest path in each category. The geometric concept of a net can be

neg: Differential geometry of surfaces of the area-expanding properties of the Gauss map, a map from the surface to a 2-dimensional sphere. However, before obtaining a more intrinsic definition in terms of the area and angles of small triangles, Gauss needed to make an in-depth investigation of the properties of geodesics on the surface, i.e. paths of shortest length between two fixed points on the surface (see below). The Gaussian curvature at a point on an embedded smooth surface given locally by the equation in Euclidean space (), is defined to be the product of the principal curvatures at the point; the mean curvature is

neg: Expansion of the universe in terms of travel time - in that case a straight line is a poor choice of metric because the shortest distance by road is not normally a straight line, and even the path nearest to a straight line will not necessarily be the quickest. A final example is the internet, where even for nearby towns, the quickest route for data can be via major connections that go across the country and back again. In this case the metric used will be the shortest time that data takes to travel between two points on the network. In cosmology, we cannot

neg: Pathfinding Pathfinding Pathfinding or pathing is the plotting, by a computer application, of the shortest route between two points. It is a more practical variant on solving mazes. This field of research is based heavily on Dijkstra's algorithm for finding a shortest path on a weighted graph. Pathfinding is closely related to the shortest path problem, within graph theory, which examines how to identify the path that best meets some criteria (shortest, cheapest, fastest, etc) between two points in a large network. At its core, a pathfinding method searches a graph by starting at one vertex and exploring adjacent nodes until

neg: Taxicab geometry causes the shortest path a car could take between two intersections in the borough to have length equal to the intersections' distance in taxicab geometry. The geometry has been used in regression analysis since the 18th century, and today is often referred to as LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski. The taxicab distance, formula_2, between two vectors formula_3 in an "n"-dimensional real vector space with fixed Cartesian coordinate system, is the sum of the lengths of the projections of the line segment between the points onto the coordinate

neg: Systolic geometry 2-sphere, S ⊂ R, is a Riemannian circle S of length 2π and diameter π. More precisely, the Riemannian distance function of S is the restriction of the ambient Riemannian distance on the sphere. This property is "not" satisfied by the standard imbedding of the unit circle in the Euclidean plane, where a pair of opposite points are at distance 2, not π. We consider all fillings of S by a surface, such that the restricted metric defined by the inclusion of the circle as the boundary of the surface is the Riemannian metric of a circle of length 2π.

neg: Minimum orbit intersection distance Minimum orbit intersection distance Minimum orbit intersection distance (MOID) is a measure used in astronomy to assess potential close approaches and collision risks between astronomical objects. It is defined as the distance between the closest points of the osculating orbits of two bodies. Of greatest interest is the risk of a collision with Earth. Earth MOID is often listed on comet and asteroid databases such as the JPL Small-Body Database. MOID values are also defined with respect to other bodies as well: Jupiter MOID, Venus MOID and so on. An object is classified as a potentially hazardous object (PHO) –

neg: Low Earth orbit orbit is about 7.8 km/s, but reduces with increased orbital altitude. Calculated for circular orbit of 200 km it is 7.79 km/s and for 1500 km it is 7.12 km/s. The delta-v needed to achieve low Earth orbit starts around 9.4 km/s. Atmospheric and gravity drag associated with launch typically adds 1.3–1.8 km/s to the launch vehicle delta-v required to reach normal LEO orbital velocity of around . The pull of gravity in LEO is only slightly less than on the earth's surface. This is because the distance to LEO from the earth's surface is far less than the earth's

neg: Tensile structure intersect other lines of principal curvature at right angles. A geodesic or geodetic line is usually the shortest line between two points on the surface. These lines are typically used when defining the cutting pattern seam-lines. This is due to their relative straightness after the planar cloths have been generated, resulting in lower cloth wastage and closer alignment with the fabric weave. In a pre-stressed but unloaded surface "w" = 0, so formula_2. In a soap film surface tensions are uniform in both directions, so "R" = −"R". It is now possible to use powerful non-linear numerical analysis programs (or

neg: Breakover angle Breakover angle Breakover angle is the maximum possible supplementary angle (usually expressed in degrees) that a vehicle, with at least one forward wheel and one rear wheel, can drive over without the apex of that angle touching any point of the vehicle other than the wheels. This definition is contingent upon the wheels being in continuous contact with the supporting surface(s). Breakover angle may also be referred to as "break-over angle" or "break over angle." Breakover angle is different from ground clearance in that ground clearance is the shortest distance between the ground and lowest point on the vehicle. Breakover

neg: Geographical distance elevation. Calculation of distances which account for changes in elevation relative to the idealized surface are not discussed in this article. Distance, formula_1 is calculated between two points, formula_2 and formula_3. The geographical coordinates of the two points, as (latitude, longitude) pairs, are formula_4 and formula_5 respectively. Which of the two points is designated as formula_2 is not important for the calculation of distance. Latitude and longitude coordinates on maps are usually expressed in degrees. In the given forms of the formulae below, one or more values "must" be expressed in the specified units to obtain the correct result. Where

neg: Appulse Appulse Appulse is an astronomical term that refers to the shortest apparent distance between one celestial object and another, as seen from a third body during a given period. The term refers to apparent motion typical of two planets together in the sky, or of the Moon to a star or planet while the Moon orbits Earth, as seen from Earth. An appulse is an apparent phenomenon caused by perspective only; the two objects involved are not near in physical space. An appulse is related to a conjunction, but the definitions differ in detail. While an appulse occurs when the

neg: James Cook an instrument such as a backstaff or quadrant. Longitude was more difficult to measure accurately because it requires precise knowledge of the time difference between points on the surface of the earth. The Earth turns a full 360 degrees relative to the sun each day. Thus longitude corresponds to time: 15 degrees every hour, or 1 degree every 4 minutes. Cook gathered accurate longitude measurements during his first voyage by his navigational skills, the help of astronomer Charles Green, and by using the newly published Nautical Almanac tables, via the lunar distance method—measuring the angular distance from the moon to

neg: Riemannian manifold the Riemannian manifold "M", then we define its length "L"("γ") in analogy with the example above by With this definition of length, every connected Riemannian manifold "M" becomes a metric space (and even a length metric space) in a natural fashion: the distance between the points "x" and "y" of "M" is defined as Even though Riemannian manifolds are usually "curved", there is still a notion of "straight line" on them: the geodesics. These are curves which locally join their points along shortest paths. Assuming the manifold is complete, any two points "x" and "y" can be connected with a

neg: Geodesic R to the space), and then minimizing this length between the points using the calculus of variations. This has some minor technical problems, because there is an infinite dimensional space of different ways to parameterize the shortest path. It is simpler to restrict the set of curves to those that are parameterized "with constant speed" 1, meaning that the distance from "f"("s") to "f"("t") along the curve equals |"s"−"t"|. Equivalently, a different quantity may be used, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of

neg: Mercator projection RF = , 1 cm corresponds to 3,000 km. On any other parallel the scale factor is sec "φ" so that For the above model 1 cm corresponds to 1,500 km at a latitude of 60°. This is not the shortest distance between the chosen endpoints on the parallel because a parallel is not a great circle. The difference is small for short distances but increases as "λ", the longitudinal separation, increases. For two points, A and B, separated by 10° of longitude on the parallel at 60° the distance along the parallel is approximately 0.5 km greater than the

neg: Minimum orbit intersection distance of 0.00001 AU in the Minor Planet Center database, and is the smallest MOID calculated for an Apollo asteroid. It is even smaller at the more precise JPL Small Body Database (0.0000078 AU). Minimum orbit intersection distance Minimum orbit intersection distance (MOID) is a measure used in astronomy to assess potential close approaches and collision risks between astronomical objects. It is defined as the distance between the closest points of the osculating orbits of two bodies. Of greatest interest is the risk of a collision with Earth. Earth MOID is often listed on comet and asteroid databases such as the

neg: Food desert 0.5 km square grids. The distance from the geographic center of each grid to the nearest grocery store gauges food accessibility for the people living in that grid. Health Canada divides areas into buffer zones with people's homes, schools or workplaces as the center. The Euclidean distance is the shortest distance between the two points of interest, which is measured for gaining food access data. Different factors are excluded or included that affect the scale of distance. The USDA maintains an online interactive mapping tool for the U.S., the "Food Access Research Atlas," which applies four different measurement standards to

neg: Quantum graph an interval formula_4 so that formula_5 is the coordinate on the interval, the vertex formula_6 corresponds to formula_7 and formula_8 to formula_9 or vice versa. The choice of which vertex lies at zero is arbitrary with the alternative corresponding to a change of coordinate on the edge. The graph has a natural metric: for two points formula_10 on the graph, formula_11 is the shortest distance between them where distance is measured along the edges of the graph. Open graphs: in the combinatorial graph model edges always join pairs of vertices however in a quantum graph one may also consider semi-infinite

neg: Solar eclipse four types of solar eclipses: The Sun's distance from Earth is about 400 times the Moon's distance, and the Sun's diameter is about 400 times the Moon's diameter. Because these ratios are approximately the same, the Sun and the Moon as seen from Earth appear to be approximately the same size: about 0.5 degree of arc in angular measure. A separate category of solar eclipses is that of the Sun being occluded by a body other than the Earth's moon, as can be observed at points in space away from the Earth's surface. Two examples are when the crew of

neg: Great ellipse Great ellipse A great ellipse is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid. Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve formed by intersecting the spheroid by a plane through its center. For points that are separated by less than about a quarter of the circumference of the earth, about formula_1, the length of the great ellipse connecting the points is close (within one part in 500,000) to the geodesic distance. The great ellipse therefore is sometimes

neg: Distance (graph theory) often hard to calculate. In most circumstances a pseudo-peripheral vertex can be used. A pseudo-peripheral vertex can easily be found with the following algorithm: Distance (graph theory) In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance

neg: Introduction to the mathematics of general relativity four dimensions of curved coordinates instead of three as used to describe a curved 2D surface. In a Euclidean space, the separation between two points is measured by the distance between the two points. The distance is purely spatial, and is always positive. In spacetime, the separation between two events is measured by the "invariant interval" between the two events, which takes into account not only the spatial separation between the events, but also their temporal separation. The interval, , between two events is defined as: where is the speed of light, and and denote differences of the space and

neg: Levenshtein distance of the shortest string, and, in the worst case, no more than quadratic in the length of the shortest string. The Levenshtein distance between two strings of length can be approximated to within a factor where is a free parameter to be tuned, in time . It has been shown that the Levenshtein distance of two strings of length cannot be computed in time for any ε greater than zero unless the strong exponential time hypothesis is false. By its definition, the Levenshtein distance is one-dimensional (it is used to measure string distance between only one pair of strings each

neg: Euclidean shortest path Euclidean shortest path The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles. In two dimensions, the problem can be solved in polynomial time in a model of computation allowing addition and comparisons of real numbers, despite theoretical difficulties involving the numerical precision needed to perform such calculations. These algorithms are based on two different principles, either performing a shortest path algorithm such as Dijkstra's algorithm on a visibility graph derived

neg: Skip distance Skip distance A skip distance is the distance a radio wave travels, usually including a hop in the ionosphere. A skip distance is a distance on the Earth's surface between the two points where radio waves from a transmitter, refracted downwards by different layers of the ionosphere, fall. It also represents how far a radio wave has travelled per hop on the Earth's surface, for radio waves such as the short wave (SW) radio signals that employ continuous reflections for transmission. Radio waves from a particular transmitting antenna do not all get refracted by a particular layer of the ionosphere;

neg: Great-circle navigation σ is expressed in radians. Using the mean earth radius, "R" = "R" ≈ yields results for the distance "s" which are within 1% of the geodesic distance for the WGS84 ellipsoid. To find the way-points, that is the positions of selected points on the great circle between "P" and "P", we first extrapolate the great circle back to its "node" "A", the point at which the great circle crosses the equator in the northward direction: let the longitude of this point be λ — see Fig 1. The azimuth at this point, α, is given by Let the angular

neg: Space (mathematics) spaces. A hyperbolic space is also a Riemann space. A curve in a Riemann space has a length, and the length of the shortest curve between two points defines a distance, such that the Riemann space is a metric space. The angle between two curves intersecting at a point is the angle between their tangent lines. Waiving positivity of inner products on tangent spaces, one obtains pseudo-Riemann spaces, including the Lorentzian spaces that are very important for general relativity. Waiving distances and angles while retaining volumes (of geometric bodies) one reaches measure theory. Besides the volume, a measure generalizes the

neg: Earth ellipsoid places at nearly the same longitude by measuring a base line and a chain of triangles. (Suitable stations for the end points are rarely at the same longitude). The distance Δ along the meridian from one end point to a point at the same latitude as the second end point is then calculated by trigonometry. The surface distance Δ is reduced to Δ', the corresponding distance at mean sea level. The intermediate distances to points on the meridian at the same latitudes as other stations of the survey may also be calculated. The geographic latitudes of both end points, φ

neg: Girth (functional analysis) shortest path on the unit sphere between two opposite points cannot be shorter than the length-two line segment connecting them through the origin of the space. A Banach space for which it is exactly four is said to be "flat". There exist flat Banach spaces of infinite dimension in which the girth is achieved by a minimum-length curve; an example is the space "C"[0,1] of continuous functions from the unit interval to the real numbers, with the sup norm. The unit sphere of such a space has the counterintuitive property that certain pairs of opposite points have the same distance

neg: HD 8574 b eye of an observer on Earth. HD 8574 is a large planet that orbits its host star every 227.55 days at a distance of 0.77 AU, or 77% the mean distance between the Earth and the Sun. HD 8574 b has, of the six planets announced in a November 2003 discovery paper, the shortest orbit. The planet has an estimated mass of 2.11 times Jupiter's mass. HD 8574 b has a measured orbital eccentricity of 0.288, denoting an elliptical orbit. Jupiter, in comparison, orbits the Sun with an orbital eccentricity of 0.016, which is far more circular. HD 8574 b

neg: Asteroid impact prediction the cryogen ran out. There are therefore currently no active or planned thermal infrared sky surveys which are focused on discovering near-Earth objects. The minimum orbit intersection distance (MOID) between an asteroid and the Earth is the distance between the closest points of their orbits. This first check is a coarse measure that does not allow an impact prediction to be made, but is based solely on the orbit parameters and gives an initial measure of how close to Earth the asteroid could come. If the MOID is large then the two objects never come near each other. In this

neg: Skip distance some are absorbed, some refracted while a portion escapes to the next layer. At this higher layer, there is a possibility of this radio wave being bent downwards to earth again. This bending happens because each layer of the ionosphere has a refractive index that varies from that of the others. Because of the differing heights of refraction, or apparent reflection, the radio waves hit the earth surface at different points hence generating the skip distance. Skip distance is greatest during the night when the ionosphere is the highest. Maximum usable frequency Skip distance A skip distance is the distance

neg: Earth mover's distance in 1781, and anchors the field of transportation theory. The use of the EMD as a distance measure for monochromatic images was described in 1989 by S. Peleg, M. Werman and H. Rom. The name "earth movers' distance" was proposed by J. Stolfi in 1994, and was used in print in 1998 by Y. Rubner, C. Tomasi and L. G. Guibas. Earth mover's distance In statistics, the earth mover's distance (EMD) is a measure of the distance between two probability distributions over a region "D". In mathematics, this is known as the Wasserstein metric. Informally, if the distributions are interpreted

neg: Orbital perturbation analysis moon was that the shape of the Earth is actually an oblate spheroid due to its spin, and he used the perturbations of the lunar orbit to estimate the oblateness of the Earth. In Newton's Philosophiæ Naturalis Principia Mathematica, he demonstrated that the gravitational force between two mass points is inversely proportional to the square of the distance between the points, and he fully solved the corresponding "two-body problem" demonstrating that the radius vector between the two points would describe an ellipse. But no exact closed analytical form could be found for the three-body problem. Instead, mathematical models called "orbital

neg: A* search algorithm track of its predecessor. After this algorithm is run, the ending node will point to its predecessor, and so on, until some node's predecessor is the start node. As an example, when searching for the shortest route on a map, might represent the straight-line distance to the goal, since that is physically the smallest possible distance between any two points. If the heuristic satisfies the additional condition for every edge of the graph (where denotes the length of that edge), then is called monotone, or consistent. In such a case, A* can be implemented more efficiently—roughly speaking, no node needs