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On 12/07/02 Software Scientific's Concept Engine ^TM read 144 documents and considered 4,460 links. From documents of any date using deep mining.

Research based on What is the distance between two points on a great circle? on the area of trigonometry, astronomy, calculus, mathematics

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Answers

Answer	X-Ref
Calculating Distances Between Two Points	Distance Between Two Points
How do we calculate the great circle distance between two airports?	Time and motion
Calculate distance between two points .	Links to Coordinates, Datums and Transformation Information
It is the shortest distance between two points on a sphere.	flights
The shortest distance between two points on the Earth is a great circle route.	MAEL-GPS/Amateur Radio - Glossary
Table 3 - Calculating the Great Circle Distance Between Two Cities	table03
Hence show that the great circle distance g between the two points is given by	NRICH Mathematics Enrichment Club (problems5.html)
Two distinct great circles meet in exactly two antipodal points.	Geometry of the Sphere 2.
The great-circle distance is always less than this.	Positional Astronomy: <br>The terrestrial Sphere
Point-Point Distance--2-Dimensional -- from MathWorld	Point-Point Distance--2-Dimensional -- from MathWorld

Relevant Documents

Document	Summary

NRICH Mathematics Enrichment Club (problems5.html): 91% (1)
http: //www. nrich. maths. org. uk/ MOTIVATE/ conf6/ problems5. html: unknown date

NRICH Mathematics Enrichment Club (problems5.html)

ⁱ

- Hint: If 2 points a & b are on a circle of radius r which has centre o & if the angle aob is q , then the distance between a & b along the circle is p r q /180.
- If a ship sails on a parallel course between 2 points at the same latitude which are separated by a constant difference in longitude (eg^?, 2 points on the same latitude on opposite points of the earth which are separated by 180 o of longitude), show that the distance sailed will decrease as the latitude increases.
- In fact, if the 2 points have latitude f & their distance apart in longitude is d in degrees, then d, the distance sailed, is This distance d is called the departure between the 2 points.

The shortest distance between 2 points on the earth's surface is not a rhumb line but is the distance along a {...}, which is a circle with its centre at the centre of the earth which passes through the 2 points.

- Calculate the departure between the 2 points in terms of f .
- Show that the great circle joining the 2 points passes through the north pole.
- Hence show that the great circle distance g between the 2 points is given by

ⁱ

Now draw a triangle on the earth by connecting 2 points a & b of equal latitude but with a difference in longitude of 90 o to the north pole at point p.

Great Circle -- from MathWorld: 90% (2)
http: //mathworld. wolfram. com/ GreatCircle. html: unknown date

Great Circle -- from MathWorld

A great circle is a section of a sphere which contains a diameter of the sphere (Kern & Bland 1948, p.ⁱ The shortest path between 2 points on a sphere, also known as an orthodrome, is a segment of a great circle. To find the great circle (geodesic) distance between 2 points located at latitude & longitude of & on a sphere of radius a, convert spherical coordinates to Cartesian coordinates using ⁱ The great circle distance is then A great circle becomes a straight line in a gnomonic projection (Steinhaus 1999, pp. The equation of the great circle can be explicitly computed using the geodesic formalism. which is simply a plane passing through the center of the sphere & the 2 points on the surface of the sphere. Geodesic, Great Sphere, Loxodrome, Mikusinski's Problem, Oblate Spheroid Geodesic, Orthodrome, Point-Point Distance--3-Dimensional, Pseudocircle, Small Circle, Sphere Tietze, H. Famous Problems of Mathematics: Solved & Unsolved Mathematics Problems from Antiquity to Modern Times.

Spherical Geometry and Trigonometry: 89% (3)
http: //www. sjsu. edu/ faculty/ watkins/ sphere. htm: unknown date

Spherical Geometry and Trigonometry

Great Circle Distances

ⁱ Let 1 & 2 be the latitudes of the 2 points & 1 & 2 Their longitudes. The basis for the determination of the angular separation of the 2 points on the great circle which connects them is the Law of Cosines for plane triangles. When the points are on a great circle this formula reduces to: Therefore if we can find the straight line (...) we can find their great circle angular separation A. The Euclidean distance between the 2 points can be found from their Euclidean coordinates, (...) & (...), by the formula Thus the formula for the cosine of the great circle angular separation reduces to: Hence the standard formula for great circle angular separation: 1^st consider the area of a lune, the area between 2 great circles as shown below.ⁱ ⁱ Thus a Great Circle divides the sphere exactly in half & hence the sum of all the areas on one side of any Great Circle is exactly equal to half of the area of a sphere.ⁱ

Distance Between Two Points: 89% (4)
http: //www. auslig. gov. au/ geodesy/ datums/ distance. htm: unknown date

Distance Between Two Points

There are many ways to calculate the distance between 2 points on the earth's surface, defined by their latitude & longitude. L1 = latitude at the 1^st point (...) L2 = latitude at the 2^nd point (...) G1 = longitude at the 1^st point (...) G2 = longitude at the 2^nd point (...) DG = longitude of the 2^nd point - longitude of the 1^st point (...) DL = latitude of the 2^nd point - latitude of the 1^st point (...) D = computed. ()

Great circle distance is the shortest distance between 2 points on a sphere. This coincides with the circumference of a circle which passes through both points & the centre of the sphere.
Geodesic distance is the shortest distance between 2 points on a spheroid.
Normal section distance is formed by a plane on a spheroid containing a point at one end of the line & the normal of the point at the other end.ⁱ

Great Circle Distance (...)

This method calculates the great circle distance, is based on spherical trigonometry, & assumes that: ⁱ However, remember that the different methods compute different types of distance (...), & you must decide which type of system (...) you wish to use. ⁱ

Positional Astronomy: <br>The terrestrial Sphere: 88% (5)
http: //star- www. st- and. ac. uk/ ~fv/ webnotes/ chapter1. htm: unknown date

Positional Astronomy: <br>The terrestrial Sphere

ⁱ Every great circle has 2 poles. We can define these: (a) as the points which are 90° away from the circle, on the surface of the sphere. (b) as the points where the perpendicular to the plane of the great circle cuts the surface of the sphere.ⁱ Note that the length of a great-circle arc on the surface of a sphere is the angle between its end-points, as seen at centre of sphere, & is expressed in degrees (...). A great circle is a geodesic (...) on the surface of a sphere, analogous to a straight line on a plane surface. ⁱ Draw a great circle from pole to pole, passing through location X: this is a meridian of longitude . The length of arc of a small circle between 2 meridians of longitude is (...) x cos(...). The great-circle distance is always less than this. Note that a position on the surface of the Earth is fixed using one fundamental circle (...) & one fixed point on it (...).

Time and motion: 87% (6)
http: //plus. maths. org/ issue7/ features/ greatcircles/: 23/5/2002

Time and motion

ⁱ

The shortest route between 2 points on a globe is along part of a great circle, which is a large circle going all the way round the globe with the centre of the Earth at the centre of the circle. In general, on a surface that isn't flat, a line between 2 points on the surface which is as short as possible is called a geodesic. On the Earth, all geodesics are parts of great circles. How do we calculate the great circle distance between 2 airports? So for 2 points on the Earth's surface, their position vectors relative to the centre of the Earth areⁱ So now we know the distance along the great circle route!ⁱ Look up some locations in an atlas & work out the distances between them - eg^?, you could work out the distance from your home town to London.ⁱ The real routes that aeroplanes take aren't always the great circle ones, for various reasons. Wind speed & direction make a difference as well: it can be quicker to deviate from the great circle route in order to pick up a beneficial tailwind! ⁱ

The dot product: 87% (7)
http: //members. tripod. com/ ~Paul_ Kirby/ vector/ Vdotproduct. html: unknown date

The dot product

Finding the angle between 2 vectors

We can now, given the coordinates of any 2 nonzero vectors u & v find the angle q between them: u = a i + b j + c k v = x i + y j + z k u . The distance is d, which,..., d = a cos q => d = n a cos q => d = a . You have 2 sides of a triangle, a & b, & the angle in between, C, - the problem is to find the remaining side c.

Finding the distance between 2 places, along the surface of the Earth

From the latitudes & longitudes of 2 places on the Earth together with the radius of the Earth we can determine the position vectors of the 2 places with the origin at the centre of the Earth. If you have 2 points on the circumference of a circle then the radius of the circle times the angle (...) subtended by the 2 points at the centre of the circle gives the arc distance between the 2 points. Using the dot product we can find the angle subtended by our 2 position vectors, multiply by the radius of the Earth, & hey presto we have the great circle distance. Find out the distance between us using this applet (...).

52A55: Spherical Geometry: 87% (8)
http: //www. math. niu. edu/ ~rusin/ known- math/ index/ 52A55. html: unknown date

52A55: Spherical Geometry

Here is a longish essay on spheres which discusses geometry, trigonometry, & topology.
Many answers -- take your pick -- to the incessantly-asked question, "What's the (...) distance between 2 points on a sphere (...) given their latitude & longitude (...): Clairaut's formula.
Citations & pointers for latitude/longitude calculations
Numerically stable formula for distances on a sphere
Where is the point on a sphere determined as being a certain geodesic distance along a fixed arc away from a given point?
Given 2 points in spherical coordinates, what's the angle between the rays joining them to the center of the sphere?
Given 3 angles as above & 2 points, where's the 3^rd point on the sphere making the appropriate angles?
What is the relation between the angles as shown above & the angles at the vertices of the resulting spherical triangle?
Clairaut's formula: how far north does a great circle pass?

Links to Coordinates, Datums and Transformation Information: 87% (9)
http: //users. netonecom. net/ ~rburtch/ geodesy/ datums. html: 25/1/2001

Links to Coordinates, Datums and Transformation Information

Aviation Formulary by Ed Williams with formulas for computing great circel distances.ⁱ Also available is the Great Circle Calculator for use. Calculate distance between 2 points . This is a part of the GIS faq on how to calculate the great circle distance between 2 points ignoring elevation differences. A Guide to Coordinate Systems in UK^? . The text describes the transformation between NAD83 & the ITRF - International Earth Roatation Service (IERS) Terrestrial Reference Frame. Also includes information on the transformation between the EUREF89 & Finnish systems. The transformation between Finnish National System & WGS84 is also given. ⁱ Geodesy Foundation Classes by Sam Blackburn that has a C++ code that will calculate the distance between 2 points (...).

navigation4: 85% (10)
http: //www. ams. org/ new- in- math/ cover/ navigation4. html: 12/3/2001

navigation4

Navigational Mathematics The calculation of the great circle track between 2 points A & B with given latitude & longitude is an exercise in spherical trigonometry. The points A & B form a spherical triangle with the North Pole C. Each side of this triangle is an arc of a circle centered at the center of the earth, i. a great circle. The length of a great-circle arc can be read off immediately from the corresponding central angle: the measurement of the central angle in minutes of arc gives the length of the arc in nautical miles. The angle C is the difference between the longitudes of A & B. This is enough information to solve for all the elements of the triangle, in particular side c (...) & angle A (...).

Compute the distance & initial course by great circle sailing from a point in Lat.ⁱ

3. The great circle track

navigation3: 83% (11)
http: //www. ams. org/ new- in- math/ cover/ navigation3. html: 12/3/2001

navigation3

Navigational Mathematics Its disadvantage is that the straight line on a Mercator map may not be the shortest path between its endpoints, when measured back on the earth's surface. The earth's surface is (...) a sphere, & the path of shortest length between 2 points on a sphere is the great-circle arc between them. The great circle in question is the intersection of the spherical surface with the plane passing through the 2 points & the center of the earth. The great-circle path is different from the rhumb line unless the 2 points are both on the equator or both on the same meridian. If the points are nearby, say within 50 miles of each other, the difference between the 2 paths is inconsequential.ⁱ

Compute the distance & initial course by great circle sailing from a point in Lat.ⁱ

4. Great circle sailing

The Geometry of the Sphere 1.: 82% (12)
http: //math. rice. edu/ ~pcmi/ sphere/ sphere. html: 15/4/1999

The Geometry of the Sphere 1.

ⁱ The only other thing that can happen is that the line hits the sphere in precisely 2 points. In this case the 2 points of intersection with the sphere are said to be antipodal points.

Planes, spheres, circles, & great circles.

ⁱ Such a circle is called a great circle .ⁱ Great circles become more important when we realize that the shortest distance between 2 points on the sphere is along the segment of the great circle joining them. On any surface the curves that minimize the distance between points are called geodesics .ⁱ A pretty good approximation to a great circle can be drawn through 2 points on a beach ball by holding a piece of string tight to the ball at the 2 points in question.ⁱ However, since the great circles are geodesics on the sphere, just as lines are in the plane, we should consider the great circles as replacements for lines.ⁱ

Great Circle Sailing: 82% (13)
http: //www. info. gov. hk/ mardep/ javascpt/ gcsail. htm: 20/7/2001

Great Circle Sailing

ⁱ

The great circle track is the shortest distance between 2 places on the earth's surface.
The vertices of a great circle are the 2 points nearest to the poles which have a course on the great circle track due EAST / WEST.

ⁱ To follow a great circle track, the navigator needs to adjust the ship's course continuously as the great circle track is a curve when plotted on a Mercator Chart.ⁱ In order to take advantage of the shorter steaming distance of the great circle track, mariners usually divide a great circle track between the initial position & the destination into smaller segments (...) of about one to 2 day's steaming time & make course adjustment at noon.ⁱ Although the great circle track is the shortest route between 2 locations, it also usually enroutes closer to the pole (...) than the 2 places. The ocean passage will thus consist of a 1^st great circle track with vertex at the latitude limit then sailing along that latitude until meeting the vertex of a 2^nd great circle track leading to the destination.ⁱ The figure on the right shows the relationship between the 3 types of route, the rhumb line, the great circle & the composite great circle routes for the same set of locations. The most annoying part of the great circle sailing calculation is to find out the way points along the great circle route.ⁱ

Development Articles - Directions Magazine: 82% (14)
http: //www. directionsmag. com/ devel/ devel_ articles. php? ID= 1: unknown date

Development Articles - Directions Magazine

ⁱ

A user submits their ZIP code, & a distance (...).

The distance between the user & each dealer is found by finding the great circle distance between the 2 lat/long points using the Distance method of my ActiveX component.

We compare the user's submitted distance, with the distance of each dealer.

A list of dealers is returned that are less than or equal to the submitted distance. 2 tables exist in our Access database, one is US_ZIP which includes fields for:

To calculate the great circle distance between 2 points (our ZIP codes) I have created a small ActiveX component using formulas compiled from past discussion on the comp. The component has a single public method called Distance. When calling the Distance function, you pass the lat/long of the 1^st point, the lat/long of the 2^nd point, & an optional measurement argument to return results in miles or kilometers (...).ⁱ

flights: 82% (15)
http: //www. navworld. com/ navcerebrations/ flights. htm: 7/3/2002

flights

ⁱ its ^?the longer distance of the 2 paths discussed between 2 points on the Earth. ⁱ The bomber flies a rhumb line for a distance of 7,641.8 nmi & the tanker flies a great circle for a distance of 21,614.5 nmi (...). The tanker flies a rhumb line for a distance of 7,641.8 nmi & the bomber flies a great circle for a distance of 21,614.5 nmi (...). ⁱ Note the comparable great circle path approximation between the equators intersection with the Greenwich meridian (...) & the North Pole (...) is simply the latitude difference multiplied by 60nmi/ or 90 x 60 nmi/ = 5,400 nmi. This illustrates the advantage of the great circle over the rhumb line in distance savings in the most extreme case ~2,242 nmi (...). ⁱ its ^?interesting to note that in the higher latitudes, the change in magnetic variation & the convergence of the meridians are a close match which enables a magnetic rhumb line to be closer to a great circle in distance (...) than it is to a true rhumb line path. ⁱ Lindberghs flight across the Atlantic between New York & Paris in 1927 illustrates the use of short segmented rhumb lines of 100 miles that were carefully plotted to approx the great circle path (...) resulting in a savings of 140 nmi. Lindbergh used a planning chart with a gnomonic projection where a great circle course is obtained by a straight line between departure & termination. ⁱ

Point-Point Distance--2-Dimensional -- from MathWorld: 81% (16)
http: //mathworld. wolfram. com/ Point- PointDistance2- Dimensional. html: unknown date

Point-Point Distance--2-Dimensional -- from MathWorld

Points

Point-Point Distance--2-Dimensional

Given 2 points in the plane, find the curve which minimizes the distance between them. so the arc length between the points & is Now verify that the arc length is indeed the straight-line distance between the points. The shortest distance between 2 points on a sphere is the so-called great circle distance. Point-Point Distance--1-Dimensional, Point-Point Distance--3-Dimensional

GeoSystems: Map Projections: 81% (17)
http: //www- personal. umich. edu/ ~sarhaus/ tigdd27proj. html: unknown date

GeoSystems: Map Projections

ⁱ Every map projection distorts at least 3, & sometimes all 4, of the following properties: Shape, Area, Distance, & Direction. Equidistant: No map can show distance correctly between all points on the map, but only from one, or at most 2, points to any other point. Azimuthal: This projection correctly shows directions (...) from a single point to all other points on the map.

Visualize the properties of a map projection by comparing the arrangement of its meridians & parallels with the characteristics of the graticule on the globe: Parallels are spaced equal distances apart.

This can be at a point, or along one or 2 lines (...). Distortion increases with increasing distance from the standard point or lines.

ⁱ If parallels are then drawn as equally spaced concentric circles, this projection would be equidistant (...). Gnomonic Projection: Great circle routes (...) appear as straight lines on this Azimuthal

http: //www3. deasy. psu. edu/ projection/ chapter11. html: 81% (18)
http: //www3. deasy. psu. edu/ projection/ chapter11. html: 15/8/1997

"http: //www3. deasy. psu. edu/ projection/ chapter11. html"

For some maps, we may be less concerned about the great-circle path between 2 points than we are about the direction & distance from one point to another.ⁱ In this case all distances can be measured from Washington to any other point by connecting the 2 points with a straightedge & applying the map scale to the measured distance.ⁱ In this case, however, the straight line connecting the center & some other point, although having the correct length, doesn't follow the great-circle route, & its direction cannot be determined readily.ⁱ Then the great-circle route from Washington to any 2^nd point is the straight vertical line intersecting the 2^nd point, while the distance is that directly measured from the top edge.ⁱ Retroazimuthal Projections its ^?also possible to have a projection on which a straight line connecting the center & any other point shows the correct distance, & on which the direction from the 2^nd point to the center relative to north is the same as the angle between the connecting line & a line extending straight up from the 2^nd point.ⁱ Measuring from 2 Points There is one projection on which all directions (...) are correct from not just one but 2 points.ⁱ Since all great circles are straight lines on the original Gnomonic, they are all straight on the 2-Point Azimuthal projection. On a related but different projection, presented by Maurer in 1919, the 2-Point Equidistant (Figure 11-6), all distances (...) are correct from 2 points.ⁱ

Great Circle Distance Calcuations: 81% (19)
http: //www. cryptnet. net/ fsp/ zipdy/ gc/: 12/12/2000

Great Circle Distance Calcuations

A great circle distance calculation is the calculation of the distance between 2 points on the surface of a sphere over the surface of that sphere.

http: //www. indo. com/ distance/ dist. pl: 81% (20)
http: //www. indo. com/ distance/ dist. pl: 16/5/1998

"http: //www. indo. com/ distance/ dist. pl"

usr/bin/perl -w # dist -- find great-circle distance between 2 points on earth's surface # -*- perl -*- # # This code was written in 1998 by Darrell Kindred . # Calculate the great-circle distance & initial heading from one point on # the Earth to another, given latitude & longitude. # For a good discussion of the formula used here for calculating distances, # as well as several more & less accurate techniques, see # http://www.census.gov/cgi-bin/geo/gisfaq?htm require 5.001; use strict; if (scalar(@ARGV) == 2) { my(...) = &parse_location(shift); my(...) = &parse_location(shift); my $meters_per_mile = 1609.344; my $nautical_miles_per_mile = (...); my $dist = &great_circle_distance(...); my $heading =to_degrees(heading($lat1,$long1, $lat2,$.));. ".-.. ", &loc__.($.,$.), " ", &loc__.($.,$.), "\"; #.' #.0f degrees (%s)\n", $heading,string(...); } else { print STDERR "$0: two arguments required, ", scalar(@ARGV), " found\n"; print STDERR "usage: $0 \n"; print STDERR " allowed loc formats: \n"; print STDERR\EndFormats; 40:26:46N,79:56:55W 40:26:46.302N 79:56:55.903W 4026'21"N 79d58'36"W 40d 26' 21" N 79d 58' 36" W 40.446195N 79.948862W 40.446195N -79.948862E EndFormats exit(1); } #

given coordinates of 2 places in radians, compute distance in meters sub great_circle_distance { my (...) = @_; #

This is a simpler formula, but it's subject to rounding errors # for small distances. return an angle in radians, between 0 & pi, whose cosine is x sub acos { my($x) = @_; die "bad acos argument ($x)\n" if (abs($x) > 1.0); return atan2(sqrt(...), $x); } #

Great circle distance: 81% (21)
http: //members. tripod. com/ ~Paul_ Kirby/ appletgreatcircle/ greatc. html: unknown date

Great circle distance

Great circle distances

Given the latitudes & longitudes of 2 points on the surface of a sphere which happens to have a radius of 6369 km, how do you determine the shortest distance between them if you are constrained to travel along the surface of the sphere? The above diagram tries to show the 2 locations as small blue spheres, the red lines being their position vectors from the centre of the sphere. (...). From this you can see that the shortest distance between the 2 points is given by the length of an arc of a circle concentric with the sphere & with the same radius as the sphere. Knowing that arc length is just the product of the radius & the angle subtended at the centre of the circle, you can use your knowledge of spherical polar coordinates & the dot product to find this distance. Challenge: Find the distance between London & New York, N.B The equator is the line of 0 latitude, the north pole has latitude 90 N, & the south pole 90 S. The Greenwich Meridian has longitude 0 & longitude extends up to 180 E (...) on the other side of the Earth.

Geometry of the Sphere 2.: 81% (22)
http: //math. rice. edu/ ~pcmi/ sphere/ gos2. html: 15/4/1999

Geometry of the Sphere 2.

ⁱ This plane passes through C , the center of the sphere, & consequently the intersection of the plane with the sphere is a great circle containing A & B . Thus A & B determine a unique great circle.ⁱ

If A & B are 2 points which aren't antipodal, then there is a unique great circle that contains both of them.ⁱ

ⁱ

2 distinct great circles meet in exactly 2 antipodal points.

If A & B are 2 points on the sphere, then the distance between them is the distance along the great circle connecting them. Since this circle lies totally in a plane, we can figure this distance using the plane figure to our left.ⁱ An isometry of the sphere is a mapping of the sphere to itself which preserves the distance between points.ⁱ Any 2 great circles meet in 2 antipodal points, & divide the sphere into 4 regions each of which has 2 sides which are segments of great circles.ⁱ ⁱ In the case of a lune, the angle between the great circles at either of the vertices is simply the angle between the planes that define the great circles, & so it doesn't matter at which vertex the measurement is made.

Latitude -- from MathWorld: 81% (23)
http: //mathworld. wolfram. com/ Latitude. html: unknown date

Latitude -- from MathWorld

The latitude of a point on a sphere is the elevation of the point from the plane of the equator. More generally, the latitude of a point on an ellipsoid is the angle between a line perpendicular to the surface of the ellipsoid at the given point & the plane of the equator (...). The shortest distance between any 2 points on a sphere is the so-called great circle distance, which can be directly computed from the latitudes & longitudes of the 2 points. Auxiliary Latitude, Colatitude, Conformal Latitude, Great Circle, Isometric Latitude, Latitude, Longitude, Spherical Coordinates

Longitude -- from MathWorld: 81% (24)
http: //mathworld. wolfram. com/ Longitude. html: unknown date

Longitude -- from MathWorld

The shortest distance between any 2 points on a sphere is the so-called great circle distance, which can be directly computed from the latitude & longitudes of 2 points. Great Circle, Latitude, Meridian, Oblate Spheroidal Coordinates, Prolate Spheroidal Coordinates

Explanation: 81% (25)
http: //www. bcca. org/ misc/ qiblih/ explain. html: 2/12/1995

Explanation

Explanation of Great Circle Directions

Since He draws a distinction between the 2 types of prayer, it seems that we really ought to give a bit more thought to the specific orientation for Obligatory Prayers.ⁱ Merely by placing the string on the globe so that it touches the origin & destination of the route, & then tightening the string so that there is no slack, while keeping it touching the origin & destination points, will illustrate quickly the shortest route between the 2 points. The 1^st point to realize is that compass directions have only local relevance.ⁱ its ^?clear that one may take an infinite number of paths between any 2 points on the globe & reach the intended destination, but if one is going to maintain the same sense of direction to the Qiblih which is experienced by someone who is within sight of the structure, one can do so only by following what is called a great-circle path. On the surface of a globe, the shortest (direct) distance between 2 points is along the great-circle route. The initial (local) compass direction of this route is the same as the straight line direction through the earth between the 2 points. In fact, the great-circle route is just the geometric intersection of the plane,..., with the sphere of the Earth.ⁱ

Where Is The Russian-U.S. Boundary?, Alaska Science Forum: 81% (26)
http: //www. gi. alaska. edu/ ScienceForum/ ASF8/ 802. html: unknown date

Where Is The Russian-U.S. Boundary?, Alaska Science Forum

It involves a border dispute between the U.S. & the U.S.S.R. that has existed ever since the purchase of Alaska. ⁱ The surface path would be the shortest route to any other point along the cut, but to follow a great circle route at sea, you'd have to change your magnetic heading constantly, unless you were sailing directly toward one of the poles. The Russians claim that the northeast-southwest trending boundary between the 2 countries in the Bering Sea is a rhumb line. The Americans claim its ^?a great circle.ⁱ The Treaty of Cession reveals that the most important part of the boundary (...) runs for almost a 1000 miles between 2 points. The southernmost point lies halfway between Attu Island on the American side & Copper Island of the Commander group on the Russian side.ⁱ The northernmost point lies halfway between St. Lawrence Island & Cape Chukotskiy on the Russian mainland, at approx 64 degrees N. & 172 degrees W. According to the Treaty of Cession, the boundary has to lie between these 2 points without a break. My calculations show a great circle distance between the 2 points of 935+ statute miles & a rhumb line distance of 938+ statute miles.ⁱ

Definition: great circle: 81% (27)
http: //www. its. bldrdoc. gov/ fs- 1037/ dir- 017/ _2490. htm: 16/9/1996

Definition: great circle

great circle

great circle: A circle defined by the intersection of the surface of the Earth & any plane that passes through the center of the Earth. Note: On the idealized surface of the Earth, the shortest distance between 2 points lies along a great circle.

Universia Portugal: 81% (28)
http: //www. universia. pt/ conteudos/ bibliotecas/ bibliotecas_ mapas_ 4. jsp: unknown date

Universia Portugal

Distance Calculator: Para calcular distâncias entre cidades e pases em todo o mundo.
Direct Line Distance Server (Indo.
Distance Calculator Between Airports (Landings.
Great Circle Distance Between 2 Cities / Surface Distance Between 2 Points: Calcula a distância em superfície entre dois pontos da terra.
Texas Distance Calculator (...) : Calcula a distância entre cidades norte-americanas.

Dome Glossary: 81% (29)
http: //www. bfi. org/ glossary. htm: 30/12/2001

Dome Glossary

ⁱ Replaces "obsolete" vocabulary eg^? "solid," "point," "thing," etc. Fuller points out that frequency never relates to the quantity "one," for it necessarily involves a plurality of experiences. geodesic: "Great circle," or the shortest distance between 2 points or energy events on the surface of a sphere. great circle: A circle on the surface of a sphere, which lies in a plane intersecting the center of that sphere. intertransformability: Phenomenon of significant relationships between systems, allowing transformations from one to another. The "omnisymmetrical" matrix consists of an indefinite expanse of alternating tetrahedra & octahedra, with 60-degree angles between adjacent vectors. spherical triangle: A curved area bounded by 3 connected great-circle arcs. The result of interconnecting 3 points on the surface of a spherical system. ⁱ

Special Relativity: 81% (30)
http: //www. mtnmath. com/ whatth/ node50. html: 25/1/2002

Special Relativity

EG^? it seems obvious that the shortest distance between 2 points is a straight line.ⁱ The earth circles the sun which in turn circles the galaxy which in turn moves away form neighboring galaxies. When the earth is moving in the same direction as the light beam the speed relative to the earth will be slower as light has to travel not only the distance between 2 points on earth but also the distance the earth moved in the time between the measurements at the 2 locations. Take the difference of the 2 speeds & divide by 2 & you have the absolute speed of the earth.ⁱ We can set up a topology or mathematical set of points & then impose on this any distance function we choose. EG^? we can use pairs of real numbers to specify the ordering of points in a 2 dimensional space.ⁱ The standard Euclidean distance function between & is as follows.ⁱ The relativistic distance function is strange as the measurements 2 astronauts make of each others space ships depend on the relative speed of the 2 ships.ⁱ

MAEL-GPS/Amateur Radio - Glossary: 81% (31)
http: //www. grc. nasa. gov/ WWW/ MAEL/ ag/ gloss. htm: unknown date

MAEL-GPS/Amateur Radio - Glossary

Great Circle The intersection of a plane through the center of the Earth & the surface of the Earth. The shortest distance between 2 points on the Earth is a great circle route.ⁱ The only latitude that is a great circle is the Equator. Latitude An angular measurement of a point on the earth, north or south of the equator.Longitude The angular measurement of a point on the earth's surface, east or west of the prime meridian.Nautical Mile A distance of 6076.11549 feet which is one minute of arc of a great circle of the Earth. Packet Radio The radio transmission of data in packets between stations. Statute Mile A distance of 5,280 feet.

table03: 81% (32)
http: //www. ncgia. ucsb. edu/ education/ curricula/ giscc/ units/ u014/ tables/ table03. html: 1/7/1998

table03

Table 3 - Calculating the Great Circle Distance Between 2 Cities

The great circle distance (D) between any 2 points P & A on the sphere is calculated with the following formula: cos D = (sin p sin a) + (cos p cos a cos \| dl \|) \| dl \| is the absolute value of the difference in longitude between P & A
Calculate the great circle distance between Paris (P) & Austin (A): Paris, France (...) Austin, Texas (...)

da Vinci Technologies "Navigare": 81% (33)
http: //www. dvtg. com/ products/ datasheets/ navigation. htm: 23/2/2001

da Vinci Technologies "Navigare"

ⁱ For great circle sailing the earth is assumed to be completely spherical. It allows you to plan great circle passages by selecting various methods & conditions. There are also some independent routines for distance, course, vertex, & intermediate points. This will always involve sailing a greater distance between 2 points than following a great circle, but it’s easier. This section also includes mercator sailing,..., that is, passages over longer distances (...). Navigare can in be connected to a GPS receiver & position speed, course, distance & UTC can be calculated.ⁱ Navigare will then calculate speed, true course, correct course, distance, & total distance. This includes volumes, center of mass, vertical distance from heel to center of gravity (KG) & horizontal distance from center of gravity to midships (KGL) for rooms & tanks, moments of inertia (...) & information about the empty ship.

http: //www3. deasy. psu. edu/ projection/ chapter9. html: 76% (34)
http: //www3. deasy. psu. edu/ projection/ chapter9. html: 14/8/1997

"http: //www3. deasy. psu. edu/ projection/ chapter9. html"

The most useful thing about a great-circle arc is that on the earth's surface, or on a map, it shows the shortest distance between points along that line. The Gnomonic Projection The Gnomonic projection (Figure 9-2) is another member of the azimuthal projection family (...), & it has the distinction of being the only map projection on which any straight line represents a great-circle arc. Whereas on the Azimuthal Equidistant projection only straight lines that originate at the map's central point show great-circle arcs, the Gnomonic projection has no such limitation.ⁱ In the example shown here, the great-circle path can be determined between any pair of points in North America, Europe, & much of Asia & North Africa. These 2 projections, the Azimuthal Equidistant & the Gnomonic, probably are the most versatile & commonly used projections for representing great-circle routes. Great Circles on Cylindrical Projections Some projections show great-circle arcs as straight lines in more limited ways, eg^? in a single direction or along just one or 2 lines on the map.ⁱ its ^?possible to "force" 2 points to lie along one of the straight lines that is a great-circle arc on the projection.ⁱ This shows the normal aspect of another cylindrical projection, the Mercator, with the great-circle route between Miami & Tokyo plotted on it.ⁱ

About the Mileage Calculator: 74% (35)
http: //www. symsys. com/ ~ingram/ mileage/ about. html: 3/7/2002

About the Mileage Calculator

1^st, we approx the distance between 2 cities in nautical miles. A nautical mile is an angular measurement equal to one arc-minute along any great circle of the earth. The program computes distance as though the earth were a sphere,..., but errors should be negligable. After computing the (x,y,z) coordinates for the 2 points, the distance between them is calculated as: The triangle formed by the center of the sphere & the 2 points contains the angle in question. The sides of the triangle are the distance computed above & 2 radii (1).

Sphere -- from MathWorld: 72% (36)
http: //mathworld. wolfram. com/ Sphere. html: unknown date

Sphere -- from MathWorld

A sphere is defined as the set of all points in which are a distance r (the "radius") from a given point (the "center").ⁱ Any cross section through a sphere is a circle (or, in the degenerate case where the slicing plane is tangent to the sphere, a point). The size of the circle is maximized when the plane defining the cross section passes through a diameter. Given 2 points on a sphere, the shortest path on the surface of the sphere which connects them (the sphere geodesic) is an arc of a circle known as a great circle. The equation of the sphere with points & lying on a diameter is given by 4 points are sufficient to uniquely define a sphere. Given the points with i = 1, 2, 3, & 4, the sphere containing them is given by the beautiful determinant equation

Angle -- from MathWorld: 67% (37)
http: //mathworld. wolfram. com/ Angle. html: unknown date

Angle -- from MathWorld

Trigonometry ">

Given 2 intersecting lines or line segments, the amount of rotation about the point of intersection (the vertex) required to bring one into correspondence with the other is called the angle between them.ⁱ It has the property that the arc length around a circle is simply given by the radian angle measure times the circle radius. The radian is also the most useful angle measure in calculus as the derivative of trigonometric functions eg^? doesn't require the insertion of multiplicative constants like. Gradians are sometimes used in surveying (they have the nice property that a right angle is exactly 100 gradians), but are encountered infrequently, if at all, in mathematics. The concept of an angle can be generalized from the circle to the sphere.3.3 in Handbook of Mathematics & Computational Science.

Geodesic -- from MathWorld: 67% (38)
http: //mathworld. wolfram. com/ Geodesic. html: unknown date

Geodesic -- from MathWorld

Calculus & Analysis ">

On the sphere, the geodesics are great circles (...). The geodesics in a space depend on the Riemannian metric, which affects the notions of distance & acceleration. The normal vector to any point of a geodesic arc lies along the normal to a surface at that point (Weinstock 1974, p. Blaschke Conjecture, Ellipsoid Geodesic, Geodesic Curvature, Geodesic Dome, Geodesic Equation, Geodesic Mapping, Geodesic Triangle, Graph Geodesic, Great Circle, Harmonic Map, Oblate Spheroid Geodesic, Paraboloid Geodesic, Wiedersehen Surface, Zoll Surface Tietze, H. Famous Problems of Mathematics: Solved & Unsolved Mathematics Problems from Antiquity to Modern Times. Weinstock, R. Calculus of Variations, with Applications to Physics & Engineering.

Point-Point Distance--1-Dimensional -- from MathWorld: 66% (39)
http: //mathworld. wolfram. com/ Point- PointDistance1- Dimensional. html: unknown date

Point-Point Distance--1-Dimensional -- from MathWorld

Points

Point-Point Distance--1-Dimensional

ⁱ Call the 1^st point & the 2^nd point. Find the distribution of distances d between points. The probability of the points being a (positive) distance d apart (i. The probability distribution of the distance between 2 points randomly picked on a line segment is germane to the problem of determining the access time of computer hard drives. Point-Point Distance--2-Dimensional, Point-Point Distance--3-Dimensional, Point-Quadratic Distance, Sphere Point Picking

Small Circle -- from MathWorld: 61% (40)
http: //mathworld. wolfram. com/ SmallCircle. html: unknown date

Small Circle -- from MathWorld

Circles

Small Circle

Great Circle, Sphere Tietze, H. Famous Problems of Mathematics: Solved & Unsolved Mathematics Problems from Antiquity to Modern Times.

GCD Calculator: 55% (41)
http: //www. mercury. demon. co. uk/ dist/ dodist. html: 26/6/2002

GCD Calculator

Great Circle Distance Calculator

Mathematical Subject Classification: 51% (42)
http: //www. ams. org/ msc/: 18/6/2002

Mathematical Subject Classification

ⁱ The Mathematics Subject Classification (MSC) is used to categorize items covered by the 2 reviewing databases, Mathematical Reviews (MR) & Zentralblatt MATH (Zbl). The MSC is broken down into over 5,000 two-, three-, & five-digit classfications, each corresponding to a discipline of mathematics (e. The current classification system, 2000 Mathematics Subject Classification (MSC2000), is a revision of the 1991 Mathematics Subject Classification, which is the classification that has been used by MR & Zbl since the beginning of 1991.

The Boolean operator, Or , is the default between words within a keyword box; adjacency is not assumed.
To & search terms, enter text into the 2 keyword boxes.

Integral transforms, operational calculus {For fractional derivatives & integrals, see 26A33.

Earth -- from Eric Weisstein's World of Astronomy: 50% (43)
http: //scienceworld. wolfram. com/ astronomy/ Earth. html: unknown date

Earth -- from Eric Weisstein's World of Astronomy

The Earth's average orbital distance from the Sun is m, a distance known as the astronomical unit.ⁱ The crustal thickness varies between 5-15 km under the oceans & 30-50 km under continental shields. The boundary between the crust & mantle, as determined from seismic velocities, is called the Mohorovicic discontinuity. The most common mineral structure in the mantle in perovskite, which is stable between 700-2900 km. There is no correlation between gravity & topography, although geoid highs are found at subduction zones.

Spherical Coordinates -- from MathWorld: 50% (44)
http: //mathworld. wolfram. com/ SphericalCoordinates. html: unknown date

Spherical Coordinates -- from MathWorld

Define to be the azimuthal angle in the xy-plane from the x-axis with (denoted when referred to as the longitude), to be the polar angle from the z-axis with (colatitude, equal to where is the latitude), & r to be distance (radius) from a point to the origin. Colatitude, Great Circle, Helmholtz Differential Equation--Spherical Coordinates, Latitude, Longitude, Oblate Spheroidal Coordinates, Prolate Spheroidal Coordinates

ITRF - GPS: 46% (45)
http: //schubert. ign. fr/ CIAG/ WITRF/ ITRF- GPS. html: unknown date

ITRF - GPS

It represents one over 3 products of the IERS CB, the 2 others are: the determination of Earth rotation parameters & the realization of the International Celestial Reference System.ⁱ T1s, T2s, T3s, Ds, R1s, R2s, & R3s are respectively the 3 translations, the scale factor & the 3 rotations between the ITRF & the individual solution s. - local ties between colocated stations are used with proper variances; * or by differentiating combined coordinates at 2 different epochs;ⁱ TRANSFORMATION PARAMETERS BETWEEN ITRF SOLUTIONS In order to quantify the 4 characteristics of the ITRF datum definition described before, we provide here the 7 transformation parameters & their rates (...) between the successive ITRF solutions.ⁱ

Corpscon 5.x for Windows: 45% (46)
http: //crunch. tec. army. mil/ software/ corpscon/ corpscon. html: 28/3/2002

Corpscon 5.x for Windows

08 (...), is a MS-Windows-based program which allows the user to convert coordinates between Geographic, State Plane & Universal Transverse Mercator (UTM) systems on the North American Datum of 1927 (NAD 27), the North American Datum of 1983 (NAD 83) & High Accuracy Reference Networks (HARNs).ⁱ 9/1/98 - Corrected problem when HPGN was selected as the Horizontal Input Format & Output Format & a conversion between NAVD88 & NGVD29 was desired. There are 2 distributions of Corpscon which may be downloaded; a standard or a minimum distribution. Please exhaust these web resources & the CORPSCON help files, before e-mailing the Point of Contact.

Orthodrome -- from MathWorld: 43% (47)
http: //mathworld. wolfram. com/ Orthodrome. html: unknown date

Orthodrome -- from MathWorld

Great Circle

Radius -- from MathWorld: 43% (48)
http: //mathworld. wolfram. com/ Radius. html: unknown date

Radius -- from MathWorld

Circles

The distance from the center of a circle to its perimeter, or from the center of a sphere to its surface. Bertrand's Problem, Circle, Circumference, Diameter, Extent, Graph Radius, Inversion Radius, Kinney's Set, Pi, Radius of Convergence, Radius of Curvature, Radius of Gyration, Radius of Torsion, Radius Vector, Sphere

Dot Product -- from MathWorld: 43% (49)
http: //mathworld. wolfram. com/ DotProduct. html: unknown date

Dot Product -- from MathWorld

The dot product can be defined for 2 vectors X & Y by where is the angle between the vectors. It follows immediately that if X is perpendicular to Y . The dot product therefore has the geometric interpretation as the length of the projection of X onto the unit vector when the 2 vectors are placed so that their tails coincide.