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# Distance between coordinates

I placed some cones on a sports playing field and estimated their locations using latitude/longitude coordinates. An example pair of estimated latitude/longitude coordinates is 54.96975, -1.51407.

On average, my estimated latitude was 0.00009 away from the actual latitude. On average, my estimated longitude was 0.0001 from the actual longitude.

Am I able to say, on average, what distance my estimated coordinates were from the actual coordinates?

The process should be:

2. Project the known points.
3. Measure the distances.

Your comments indicate that you've decided you were "about 10m out," but why not just measure it properly and have the "right" answer (according to the projection you choose)?

(This answer was prepared by Robert G. Chamberlain of Caltech (JPL):
[email protected] and reviewed on the comp.infosystems.gis
newsgroup in Oct 1996.)

If the distance is less than about 20 km (12 mi) and the locations of the
two points in Cartesian coordinates are X1,Y1 and X2,Y2 then the

will result in an error of
less than 30 meters (100 ft) for latitudes less than 70 degrees
less than 20 meters ( 66 ft) for latitudes less than 50 degrees
less than 9 meters ( 30 ft) for latitudes less than 30 degrees
(These error statements reflect both the convergence of
the meridians and the curvature of the parallels.)

The flat-Earth distance d will be expressed in the same units as
the coordinates.

If the locations are not already in Cartesian coordinates, the
computational cost of converting from spherical coordinates and
then using the flat-Earth model may exceed that of using the
more accurate spherical model.

Otherwise, presuming a spherical Earth with radius R (see below), and the
locations of the two points in spherical coordinates (longitude and
latitude) are lon1,lat1 and lon2,lat2 then the

Haversine Formula (from R.W. Sinnott, "Virtues of the Haversine",
Sky and Telescope, vol. 68, no. 2, 1984, p. 159):

dlon = lon2 - lon1
dlat = lat2 - lat1
a = sin^2(dlat/2) + cos(lat1) * cos(lat2) * sin^2(dlon/2)
c = 2 * arcsin(min(1,sqrt(a)))
d = R * c

will give mathematically and computationally exact results. The
intermediate result c is the great circle distance in radians.
The great circle distance d will be in the same units as R.

The min() function protects against possible roundoff errors that
could sabotage computation of the arcsine if the two points are
very nearly antipodal (that is, on opposide sides of the Earth).
Under these conditions, the Haversine Formula is ill-conditioned
(see the discussion below), but the error, perhaps as large as
2 km (1 mi), is in the context of a distance near 20,000 km
(12,000 mi).

Most computers require the arguments of trignometric functions to
be expressed in radians. To convert lon1,lat1 and lon2,lat2 from
degrees, minutes, and seconds to radians, first convert them to
decimal degrees. To convert decimal degrees to radians, multiply
the number of degrees by pi/180 = 0.017453293 radians/degree.

Inverse trigonometric functions return results expressed in
radians. To express c in decimal degrees, multiply the number of
multiply the number of RADIANS by R to get d.)

The problem of determining the great circle distance on a sphere
has been around for hundreds of years, as have both the Law of
Cosines solution (given below but not recommended) and the
Haversine Formula. Sinnott gets the credit here because he was
quoted by Snyder (cited below). Perhaps someone will provide the
truly seminal reference so the proper attribution can be given?

The Pythagorean flat-Earth approximation assumes that meridians are
parallel, that the parallels of latitude are negligibly different from
great circles, and that great circles are negligibly different from
straight lines. Close to the poles, the parallels of latitude are not only
shorter than great circles, but indispensably curved. Taking this into
account leads to the use of polar coordinates and the planar law of cosines
for computing short distances near the poles: The

Polar Coordinate Flat-Earth Formula

a = pi/2 - lat1
b = pi/2 - lat2
c = sqrt(a^2 + b^2 - 2 * a * b * cos(lon2 - lon1)
d = R * c

will give smaller maximum errors than the Pythagorean Theorem for
higher latitudes and greater distances. (The maximum errors, which
depend upon azimuth in addition to separation distance, are equal
at 80 degrees latitude when the separation is 33 km (20 mi),
82 degrees at 18 km (11 mi), 84 degrees at 9 km (5.4 mi).) But
even at 88 degrees the polar error can be as large as 20 meters
(66 ft) when the distance between the points is 20 km (12 mi).

The latitudes lat1 and lat2 must be expressed in radians (see
above) pi/2 = 1.5707963. Again, the intermediate result c is the
distance in radians and the distance d is in the same units as R.

An UNRELIABLE way to calculate distance on a spherical Earth is the

Law of Cosines for Spherical Trigonometry
** NOT RECOMMENDED **

a = sin(lat1) * sin(lat2)
b = cos(lat1) * cos(lat2) * cos(lon2 - lon1)
c = arccos(a + b)
d = R * c

Although this formula is mathematically exact, it is unreliable
for small distances because the inverse cosine is ill-conditioned.
Sinnott (in the article cited above) offers the following table
to illustrate the point:
cos (5 degrees) = 0.996194698
cos (1 degree) = 0.999847695
cos (1 minute) = 0.9999999577
cos (1 second) = 0.9999999999882
cos (0.05 sec) = 0.999999999999971
A computer carrying seven significant figures cannot distinguish
the cosines of any distances smaller than about one minute of arc.

The function min(1,(a + b)) could replace (a + b) as the argument
for the inverse cosine for the same reason as in Sinnott's Formula,
but doing so would "polish a cannonball".

## Systems & Utilities

Tower Construction Notifications System (TCNS) and Electronic Section-106 System (E-106).
The Tower Construction Notification System (TCNS) allows companies to voluntarily submit notifications of proposed tower constructions to the FCC. The FCC provides this information to federally-recognized Indian Tribes, Native Hawaiian Organizations (NHOs), and State Historic Preservation Officers (SHPOs), and allows them to respond directly to the companies if they have concerns about a proposed construction.

The Section 106 System is used in completing the review process for proposed construction of towers and other communications facilities under Section 106 of the National Historic Preservation Act (NHPA).

### Utilities

#### Utilities

The AM Tower Locator is a tool that allows you to determine whether the construction of a proposed tower requires you to notify AM stations prior to construction.

This notification process is required by FCC rules.

The Line A and Line C Program determines whether an entered coordinate is SOUTH of Line A or WEST of Line C. Line A is an imaginary line within the US, approximately paralleling the US-Canadian border. To the north of Line A, FCC coordination with Canadian authorities is generally required in the assignment of frequencies.

Line C is an imaginary line in Alaska approximately paralleling the Alaskan-Canadian border. To the east of Line C, FCC coordination with Canadian authorities is generally required in the assignment of frequencies.

Geographic coordinates provided to the Commission via the Universal Licensing System must be referenced to the North American Datum of 1983 (NAD83). If the source from which you obtain the coordinates is referenced to another datum (e.g., NAD27, PRD40) you must convert the coordinates to NAD83.

The FCC uses the procedures outlined below when converting licensing data to NAD83 coordinates when a radio service is converted to the Universal Licensing System (ULS). In most cases, this procedure uses the NADCON software developed by the National Geological Survey. For certain Pacific island areas, the FCC uses a specified shift from the applicable local datum. For other Pacific island areas where a conversion is not yet available, coordinates should continue to be referenced to the applicable local datum.

The Population program provides access to the population 200k and 600k databases.

The program uses these databases to list cities with 200,000 people within 75 miles of the entered coordinates. The program also lists the cities with 600,000 people within 87 miles of the entered coordinates.

The program verifies compliance with Rule Sections 90.261, 90.20, 90.17, 90.35, 90.63, 90.65, 90.67, 90.73, 90.75, 90.79, and 90.93.

TOWAIR Determination can be used to determine whether or not an antenna strcuture registration with the FCC is necessary.

The US Borders program determines the distance to the Canadian and Mexican borders and determines what region the user-specified coordinates reside as defined in Rule Section 90.619. Rule Section 90.619 defines Canadian regions for 800 and 900 MHz land mobile radio stations. This rule also defines which frequencies may or may not be assigned in regions near the Canadian and Mexican borders.

This program provides you the distance to Chicago. Rule 90.617 defines a unique channel plan for the Chicago area that the FCC defines as stations with a 70-mile radius of 41º 52' 28"N and 87º 38' 22"W.

This program alerts you if the entered coordinates are in proximity to a defined peak as defined in Rule Section 90.621. Rule section 90.621 defines mountain peaks that should be provided special protection criteria.

## Uses of GIS

GIS is used for many different topics. Some of its uses include human geography fields, politics, natural sciences, urban planning, and economics. Within these fields, GIS can be used for a very large range of topics and problems. It can be used to study precipitation patterns, soils, population density and distribution, diseases, natural resource management, natural hazards and disasters, transportation and communication networks, and any other topic or problem that has a location and spatial component, especially interactions between space. Α] GIS can also be used to study topics across time and between topics, such as how crop health in a specific area of cropland has changed over time or the relationship between wildlife populations and urban growth.

## What is the Distance between Two Points?

For any two points there is exactly one line segment connecting them. The distance between two points is the length of the line segment connecting them. Note that the distance between two points is always positive. Segments that have equal length are called congruent segments.

Distance between 2 Points
(xA, yA) and (xB, yB)Distance
(1, 2) and (3, 4)2.8284
(1, 3) and (-2, 9)6.7082
(1, 2) and (5, 5)5
(1, 2) and (7, 6)7.2111
(1, 1) and (7, -7)10
(13, 2) and (7, 10)10
(1, 3) and (5, 0)5
(1, 3) and (5, 6)5
(9, 6) and (2, 2)8.0623
(5, 7) and (7, 7)2
(8, 2) and (3, 8)7.8102
(8, -3) and (4, -7)5.6569
(8, 2) and (6, 1)2.2361
(-6, 8) and (-3, 9)3.1623
(7, 11) and (-1, 5)10
(-6, 5) and (-3, 1)5
(-6, 7) and (-1, 1)7.8102
(5, -4) and (0, 8)13
(5, -8) and (-3, 1)12.0416
(-5, 4) and (2, 6)7.2801
(4, 7) and ( 2, 2)5.3852
(4, 2) and ( 8, 5)5
(4, 6) and (3, 7)1.4142
(-3, 7) and (8, 6)11.0454
(-3, 4) and (5, 4)8
(-3, 2) and (5, 8)10
(-3, 4) and (1, 6)4.4721
(-2, 4) and (3, 9)7.0711
(-2, 4) and (4, 7)6.7082
(-2, 5) and (5, 2)7.6158
(-12, 1) and (12, -1)24.0832
(-1, 5) and (0, 4)1.4142
(-1, 4) and (4, 1)5.831
(0, 1) and (4, 4)5
(0, 5) and (12, 3)12.1655
(0, 1) and (6, 3.5)6.5
(0, 8) and (4, 5)5
(0, 0) and (3, 4)5
(0, 0) and (1, 1)1.4142
(0, 1) and (4, 4)5
(0, 5) and (12, 3)12.1655
(2, 3) and (5, 7)5
(2, 5) and (-4, 7)6.3246
(2, 3) and (1, 7)4.1231
(2, 8) and (5, 3)5.831
(3, 2) and (-1, 4)4.4721
(3, 12) and (14, 2)14.8661
(3, 7) and (6, 5)3.6056
(3, 4) and (0, 0)5

### How to Calculate Distance between 2 Points?

The length of a segment is usually denoted by using the endpoints without an overline. For instance, the ` ext` is denoted by `overline` or sometimes `moverline`. A ruler is commonly used to find the the distance between two points. If we place the `0` mark at the left endpoint, and the mark on which the other endpoint falls on is the distance between two points. In general, we do not need to measure from the 0 mark. By the ruler postulate, the distance between two points is the absolute value between the numbers shown on the ruler. On the other hand, if two points `A and B` are on the x-axis, i.e. the coordinates of `A and B` are `(x_A,0)` and `(x_B,0)` respectively, then the distance between two points `AB = |x_B −x_A|`. The same method can be applied to find the distance between two points on the y-axis. The formula for the distance between two points in two-dimensional Cartesian coordinate plane is based on the Pythagorean Theorem. So, the Pythagorean theorem is used for measuring the distance between any two points `A(x_A,y_A)` and `B(x_B,y_B)`

### Real World Problems Using Length between Two Points

If we compare the lengths of two or more line segments, we use the formula for the distance between two points. We usually use the distance formula for finding the length of sides of polygons if we know coordinates of their vertices. In this case, we can explore the nature of polygons. It can also help us for finding the area and perimeter of polygon.

Length between two points calculator is used in almost all fields of mathematics. For example, the distance between two complex numbers `z_1 = a + ib` and `z_2 = c + id` in the complex plane is the distance between points `(a,b) and (c,d)`, i.e.

### Distance between Two Points Practice Problems

Practice Problem 1:
Starting at the same point, Michael and Ann walked. Michael walked 5 miles north and 2 miles west, while Ann walked 7 miles east and 2 miles south. How far apart are them?

Practice Problem 2:
Find the distance between points `E and F.`

## Calculate the Distance from One Point to Another

What if you are given two coordinates for latitude and longitude and you need to know how far it is between the two locations? You could use what is known as a haversine formula to calculate the distance — but unless you are a whiz at trigonometry, it is not easy. Luckily, in today's digital world, computers can do the math for us.

• Most interactive map applications will allow you to input GPS coordinates of latitude and longitude and tell you the distance between the two points.
• There are a number of latitude/longitude distance calculators available online. The National Hurricane Center has one that is very easy to use.

Keep in mind that you can also find the precise latitude and longitude of a location using a map application. In Google Maps, for example, you can simply click on a location and a pop-up window will give latitude and longitude data to a millionth of a degree. Similarly, if you right-click on a location in MapQuest you will get the latitude and longitude data.

Advanced computer technology has placed new tools in the hands of geographers to not only create maps much more efficiently, but to analyze spatial data in map form as well. A geographic information system is a computer-based technology that enters, analyzes, manipulates, and displays geographic information. It is a marriage between computer-based cartography and database management.

A simple way of visualizing a geographic information system is to think of a set of overhead transparencies. On each transparency is a map of a particular set of data. Examine Figure 1.25. The bottom transparency is the most important as it has the coordinate system (latitude and longitude) upon which we can align or register the other layers of information. The second layer is a map of industrial sites, the third shopping centers and so on. By layering the information one on top of the other, a geographer can show the relationship and degree of connectivity between various land uses and transportation routes. Transportation geographers can then plan new routes between population centers found on the census tract map layer and business locations. Geographic Information Systems are being employed to study a number of geographic issues like flood hazard mapping, earthquake hazard studies, economic market area analysis, etc.

Figure 1.26 Earthquakes 1568 - 1996 and population density 2000, the National Atlas. (Courtesy USGS)

Figure 1.26 is a map constructed using a GIS from the online National Atlas of the United States. Layers of data, earthquakes 1568 - 1996 and population density 2000, are turned on and off with digital buttons. The map product from the GIS permits us to visualize those population centers most threatened by earthquake activity.

Video: GIS Specialists at Work

For Citation: Ritter, Michael E. The Physical Environment: an Introduction to Physical Geography.
Date visited. https://www.thephysicalenvironment.com/

Lisa Pitts ([email protected])

Help keep this site available by donating through PayPal.

## Coordinates Finder

A) The coordinates finder can help you find the latitude and longitude of a country, place or other location.

The coordinates along with the city, state, county, country and other relevant information about the location is returned by the form.

Coordinates are returned in DD (Decimal Degrees), DMS (Degree-Minute-Second) and UTM (Universal Transverse Mercator).

In addition to getting the coordinates by entering in an address, you can also perform reverse geocoding requests.

With reverse geocoding, you can figure out the address, city, country etc., by entering in the latitude and longitude of a location.

### Location to Coordinates or Coordinates to Location

Geographic coordinates help you locate any place on earth using numbers. These numbers are the longitude and latitude of the location. Longitude is perpendicular and latitude is parallel to the equator.

So whether you're looking to find a location using coordinates that you already know, or whether you'd like to figure out the coordinates of a location that you know of, the finder can help.

Additionally, if you need to see the place, you may do so on the labeled map.

GeoDistance [ loc 1 , loc 2 ] gives the distance between locations loc 1 and loc 2 as measured along the geodesic joining them on the surface of the reference ellipsoid. Heights are ignored. The result is returned as a Quantity object with dimensions of length. The unit used can be chosen with the option UnitSystem , which has \$UnitSystem as default value. Latitudes and longitudes can be given as numbers in degrees, as DMS strings, or as Quantity angles. Position objects in GeoDistance [ loc 1 , loc 2 ] can be given as GeoPosition , GeoPositionXYZ , GeoPositionENU , or GeoGridPosition objects. In GeoDistance [ loc 1 , loc 2 ] , the loc i can be Entity objects with domains such as "City" , "Country" , and "AdministrativeDivision" . For entities corresponding to extended geographic regions, GeoDistance by default computes the minimum distance between any points in the regions. GeoDistance [ loc 1 , loc 2 ] by default uses the reference ellipsoid associated with the datum for loc 1 . GeoDistance automatically threads over lists of locations or GeoPosition arrays, so that GeoDistance [ loc , locs ] returns a list of distances, and GeoDistance [ locs 1 , locs 2 ] returns a matrix of distances. Results are given as QuantityArray objects. GeoDistance and GeoDirection , or their combination in GeoDisplacement , solve the geodetic inverse problem. GeoDistance has option DistanceFunction , with the following settings:
•  "Boundary" minimum distance between any points in regions "Center" distance between centers of regions "SignedBoundary" distance to boundary , negative for interior points
GeoDistance by default uses the setting DistanceFunction  "Boundary" .

## What is a coordinate system?

A coordinate system is a method for identifying the location of a point on the earth. Most coordinate systems use two numbers, a coordinate, to identify the location of a point. Each of these numbers indicates the distance between the point and some fixed reference point, called the origin. The first number, known as the X value, indicates how far left or right the point is from the origin. The second number, known as the Y value, indicates how far above or below the point is from the origin. The origin has a coordinate of 0, 0.

Longitude and latitude are a special kind of coordinate system, called a spherical coordinate system, since they identify points on a sphere or globe. However, there are hundreds of other coordinate systems used in different places around the world to identify locations on the earth. All of these coordinate systems place a grid of vertical and horizontal lines over a flat map of a portion of the earth.

A complete definition of a coordinate system requires the following:

• The projection that is used to draw the earth on a flat map
• The location of the origin
• The units that are used to measure the distance from the origin

### GIS Mapping Software

Maptitude Mapping Software gives you all of the tools, maps, and data you need to analyze and understand how geography affects you and your business. Maptitude supports dozens of coordinate systems allowing you to work with data from almost any source.