# Determining Horizontal Coordinate System data (azimuth, altitude) for geographic locations

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I would like to determine, specifically, azimuth and elevation data for any given location on the surface of the earth relative to another location on the surface of the earth (for my purposes, the latter will be London, UK).

Data and APIs are readily available for certain celestial objects, but I haven't had any luck finding data as required.

Can anyone enlighten me as to a method for calculating such data?

Many thanks.

This can be done using the Python package Skyfield fairly easily.

Here is a shell of a script; I used some dictionaries to hold data, you may want to do something else. I've stored a lot of goodies but only printed azimuth, altitude and distance in kilometers.

By default Skyfield's.altaz()method calculates atmospheric refraction for anything higher than -1 degrees (1 degree below the horizon geometrically). You can turn that off by replacing it with.altaz(pressure_mbar=0).

OUTPUT:

place altitude azimuth distance (km) Caracas -33.67 258.26 7075.28 Edinburgh -2.39 338.98 533.65 Canberra -76.25 64.93 12377.25

Here's the Python script:

import numpy as np import matplotlib.pyplot as plt from skyfield.api import Topos, Loader load = Loader('~/Documents/fishing/SkyData') # avoids multiple copies of large files data = load('de421.bsp') ts = load.timescale() Earth = data['earth'] London = Earth + Topos(latitude_degrees = 51.51, longitude_degrees = -0.1275, elevation_m = 11.) places = {'Caracas':(10.48, -66.90, 900), 'Edinburgh':(55.95, -3.19, 47.), 'Canberra':(-35.29, 149.13, 578.)} now = ts.now() answer_dict = dict() for name, (lat, lon, elev) in places.items(): place = Earth + Topos(latitude_degrees = lat, longitude_degrees = lon, elevation_m = elev) vector_km = place.at(now).position.km - London.at(now).position.km distance_km = np.sqrt((vector_km**2).sum()) dic = dict() answer_dict[name] = dic dic['lat'] = lat dic['lon'] = lon dic['elev'] = elev dic['vector_km'] = vector_km dic['vector_distance_km'] = distance_km alt, az, d = London.at(now).observe(place).apparent().altaz() dic['altaz'] = alt._degrees, az._degrees dic['dist'] = d.km for name, dic in answer_dict.items(): print(name, dic['altaz'], dic['dist'])

## Horizontal coordinate system

The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane. This conveniently divides the sky into the upper hemisphere that you can see, and the lower hemisphere that you cannot (because the Earth is in the way). The pole of the upper hemisphere is called the zenith . The pole of the lower hemisphere is called the nadir .

The horizontal coordinates are:
* altitude (Alt), sometimes referred to as elevation, that is the angle between the object and the observer's local horizon.
* azimuth (Az), that is the angle of the object around the horizon, usually measured from the north point towards the east. In former times, it was common to refer to azimuth from the south, as it was then zero at the same time the hour angle of a star was zero. This assumes, however, that the star (upper) culminates in the south, which is only true for most stars in the Northern Hemisphere .

The horizontal coordinate system is sometimes also called the az/el [http://www2.keck.hawaii.edu/inst/KSDs/40/html/ksd40-55.4.html] or Alt/Az coordinate system.

General observations

The horizontal coordinate system is fixed to the Earth, not the stars. Therefore, the altitude and azimuth of an object changes with time, as the object appears to drift across the sky. In addition, because the horizontal system is defined by your local horizon, the same object viewed from different locations on Earth at the same time will have different values of altitude and azimuth.

Horizontal coordinates are very useful for determining the rise and set times of an object in the sky. When an object's altitude is 0°, it is on the horizon, if at that moment its altitude is increasing, it is rising, if its altitude is decreasing it is setting. However all objects on the celestial sphere are subject to the diurnal motion , which is always from east to west, so the inherent cumbersome determination whether altitude is increasing or decreasing can be easily found by considering the azimuth of the celestial object instead (referenced to North as 0°):
*if the azimuth is between 0° and 180° (north—east—south), it is rising.
*if the azimuth is between 180° and 360° (south—west—north), it is setting.There are the following special cases:
*At the north pole all directions are south, and at the south pole all directions are north, so the azimuth is undefined in both locations. A star (or any object with fixed equatorial coordinates ) has constant altitude, and therefore never rises or sets when viewed from either pole. The Sun , Moon , and planets can rise or set over the span of a year when viewed from the poles because their right ascension s and declination s are constantly changing.
*At the equator objects on the celestial poles stay at fixed points on the horizon.

Note that the above considerations are strictly speaking true for the "geometric horizon" only: the horizon as it would appear for an observer on sea level on a perfect smooth Earth without atmosphere. In practice the "apparent horizon", which you see, has a negative altitude, which absolute value gets larger when you come higher, due to the curvature of the Earth. In addition the atmospheric refraction adds another 0.5° to that value.

Transformation of coordinates

It is possible to convert from the equatorial coordinate system to the horizontal coordinate system and back, once the observer's geographic latitude phi is known (+90° on the north pole, 0° on the equator, -90° on the south pole).

We will use A for the azimuth, a for the altitude.

We will use delta for the declination , H for the hour angle expressed in degrees (If in hours, it needs to be multiplied by 15).

equatorial to horizontal

sin a = sin phi cdot sin delta + cos phi cdot cos delta cdot cos H

cos A cdot cos a = cos phi cdot sin delta - sin phi cdot cos delta cdot cos H

sin A cdot cos a = - cos delta cdot sin H

One may be tempted to 'simplify' the last two equations by dividing out the cos a leaving one expression in an A only. But the tangent cannot distinguish between (for example) an azimuth of 45° and 225°. These two values are very different: they are opposite directions, NE and SW respectively. One can do this only when the quadrant in which the azimuth lies is already known.

If the calculation is done with an electronic pocket calculator, it is best not to use the functions "arcsin" and "arccos" when possible, because of their limited 180° only range, and also because of the low accuracy the former gets around ±90° and the latter around 0° and 180°. Most scientific calculators have a "rectangular to polar" (R&rarrP) and "polar to rectangular" (P&rarrR) function, which avoids that problem and gives us an extra sanity check as well.

The algorithm then becomes as follows.
*Calculate the terms right of the = sign of the 3 equations given above
*Apply a R&rarrP conversion taking the cos A cdot cos a as the X value and the sin A cdot cos a as the Y value
*The angle part of the answer is the azimuth, an angle over the full range of 0° to 360° (or -180° to +180° etc)
*Apply a second R&rarrP conversion taking the radius part of the last answer as the X and the sin a of the first equation as the Y value
*The angle part of the answer is the altitude, an angle between -90° and +90°
*The radius part of the answer must be 1 exactly, or you have made an error.

horizontal to equatorial

sin delta = sin phi cdot sin a + cos phi cdot cos a cdot cos A

cos delta cdot cos H = cos phi cdot sin a - sin phi cdot cos a cdot cos A

cos delta cdot sin H = - sin A cdot cos a

The same quadrant considerations from the first set of formulas also hold for this set.

The position of the Sun

There are several ways to compute the apparent position of the Sun in horizontal coordinates.

Complete and accurate algorithms to obtain precise values can be found in Jean Meeus 's book "Astronomical Algorithms".

Instead a simple approximate algorithm is the following:

Given:
* the date of the year and the time of the day
* the observer's latitude , longitude and time zone

* The Sun declination of the corresponding day of the year, which is given by the following formula: delta = -23.45^circ cdot cos left ( frac<360^circ> <365>cdot left ( N + 10 ight ) ight )

where N is the number of days spent since January 1 .

* The true hour angle that is the angle which the earth should rotate to take the observer's location directly under the sun.
** Let "hh":"mm" be the time the observer reads on the clock.
** Merge the hours and the minutes in one variable T = "hh" + "mm"/60 measured in hours.
** "hh":"mm" is the official time of the time zone, but it is different from the true local time of the observer's location. T has to be corrected adding the quantity + (Longitude/15 - Time Zone), which is measured in hours and represents the difference of time between the true local time of the observer's location and the official time of the time zone.
** If it is summer and Daylight Saving Time is used, you have to subtract one hour in order to get Standard Time .
** The value of the Equation of Time in that day has to be added. Since T is measured in hours, the Equation of Time must be divided by 60 before being added.
** The hour angle can be now computed. In fact the angle which the earth should rotate to take the observer's location directly under the sun is given by the following expression: H = (12 - T ) * 15. Since T is measured in hours and the speed of rotation of the earth 15 degrees per hour, H is measured in degrees. If you need H measured in radians you just have to multiply by the factor 2π/360.

* Use the Transformation of Coordinates to compute the apparent position of the Sun in horizontal coordinates.

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## RA / DEC¶

The most common way to give sky coordinates is as right ascension (RA) and declination (DEC) in the equatorial coordinate system.

Actually there are several equatorial coordinate systems in use, the most common ones being FK4, FK5 and ICRS. If you’re interested to learn more about these and other astronomical coordinate systems, look into the Explanatory Supplement to the Astronomical Almanac.

But in practice it’s pretty simple: when someone gives or talks about RA / DEC coordinates, they mean either ICRS or FK5 J2000 coordinates. The difference between those two is at the sub-arcsecond level for the whole sky, i.e. irrelevant for gamma-ray astronomy.

We recommend you by default assume RA / DEC is in the ICRS frame, which is the default in astropy.coordinates.SkyCoord and also the current standard celestial reference system adopted by the IAU (see Wikipedia - ICRS).

## Descriptions of Free Geodetic Software Developed by NGS

Programs and utilities to perform least squares adjustment on horizontal, vertical angle, and/or GPS observations. Data checking programs are included.

Suite of programs that are used in conjunction with PC program ADJUST.

This program is used to determine the scale and constant corrections for electronic distance measuring instruments by making measurements over previously determined base lines. The formulas used in the program are found in NOAA Technical Memorandum NOS NGS-10, "Use of Calibration Base Lines."

Specially designed software to compute geoid heights for the Caribbean Sea. Boundaries of area are defined as latitude 9-28N and longitude 86-58W. Suggest you read details before obtaining software.

Tests the consistency and compatibility of the Blue Book B file (GPS project and station occupation data) and G file (GPS vector data transfer file).

Program COMPVECS finds differences among redundant vectors in a G-file. It needs a horizontal bluebook with *80* position data.

Version 6.x, is a MS-Windows-based program which allows the user to convert coordinates between Geographic, State Plane and Universal Transverse Mercator (UTM) systems on the North American Datum of 1927 (NAD 27), the North American Datum of 1983 (NAD 83) and High Accuracy Reference Networks (HARNs).

Reformats GPS project information to fit the requirements of the National Geodetic Survey data base. The file created, which is called the B-file, contains project information, station information, and survey measurements. The CR8BB software functions independently of the type of GPS receivers used in a project.
Version 5.4 implements changes reflected in the July, 2000, updates to the Blue Book. These changes include the use of *72* rather than *71* records and the use of Annex M rather than Annex J. Version 5.5 provides additional sorting capabilities, renumbering functions, default values, and makes minor corrections to the handling of imported SERFILs.
Version 5.5.2 provides additional corrections.
Version 6.1 Further extends sorting, renumbering and SERFILs and makes field checking less overbearing. Also, sized better for 800x600 resolution and provides for the new ARP columns in the *27* record. This version supersedes all previous versions.

Extracts data from GPS Blue Book G file to create a station serial number file (serfil) for GPS observations.

Specially designed software to compute deflections of the vertical and Laplace corrections for the Caribbean Sea. Boundaries of area are defined as latitude 9-28N and longitude 86-58W. Suggest you read details before obtaining software.

Computes deflections of the vertical and Laplace corrections for the conterminous United States, Alaska, Puerto Rico, Virgin Islands, and Hawaii.
An on-line interactive version is also available from the Geodetic Tool Kit.

Computes deflections of the vertical and Laplace corrections for Mexico. Suggest you read details before obtaining software.

DSWIN is windows based software for Data Sheet view and extraction. It displays a list of county names as found on your CD-ROM. Click on a county and a list of stations appears. Click on a station from the list and a data sheet appears. You may save the data sheet to a file or print it. The search feature allows for filtering the station list by: Point Radius, Min/Max Box, Station Name, or PID. You may also filter by type of control, such as 1st order bench marks only

DSFILES is windows based software for manipulating Data Sheet files. Using DSFILES you can split a Data Sheet file or join several files into one. You can also merge one or more files into a single delimited, one record per station file for importing into a database or GIS package.

DSUPDATE is windows based software for updating Data Sheet files. Using DSUPDATE you can merge/replace one Data Sheet file into another so that newly retrieved Data Sheets will replace older ones, thereby keeping your Data Sheet file up-to-date.

Performs field check computations, such as tape standardization, eccentric reduction, datum transformation, triangle (plane, spherical, geodetic), special (three-point fix, intersection, resection), geodetic (traverse, inverse, direct) computations, solar observation for azimuth, and State plane coordinate system computations on the NAD 27 and NAD 83 datums.

Computes gravimetric, geocentric geoid undulations relative to the ITRF97/GRS-80 reference ellipsoid in the conterminous United States. Primarily for scientific use.
An on-line interactive version is also available from the Geodetic Tool Kit.

The National Geodetic Survey has released updated models for transforming heights between ellipsoidal coordinates and physical height systems that relate to water flow. These models cover regions including the conterminous United States (CONUS), Alaska, Hawaii, VI, PR, Guam and the Commonwealth of the Northern Mariana Islands, and American Samoa.

Designed to work with standard dsdatasheet files, this program allows the user to download and update data via the web create horizontal and vertical geodetic control listings view, print, and export datasheets and digital photos create databases of geodetic control, and much more. Complete built-in help is provided, including instructions for recovering, positioning and photographing geodetic control points. Another tool from Malcolm Archer-Shee.

USGG2003 is a gravimetric geoid model, and is an update of the G99SSS model. USGG2003 covers the coterminous U.S., Alaska, Hawaii, Puerto Rico and the American Virgin Islands.
An on-line interactive version is also available from the Geodetic Tool Kit.

USGG2009 is a gravimetric geoid model, and is an update of the G99SSS model. USGG2009 covers the coterminous U.S., Alaska, Hawaii, American Samoa, Guam and Northern Mariana Islands, Puerto Rico and the American Virgin Islands.
An on-line interactive version is also available from the Geodetic Tool Kit.

Converts NAD 27 State plane coordinates to NAD 27 geographic positions (latitudes and longitudes) and conversely. Includes defining constants for all NAD 27 plane coordinate zones.

This horizontal time-dependent positioning software program allows users to predict horizontal displacements and/or velocities at locations throughout the United States and its territories. This software also enables users to update geodetic coordinates and/or observations from one date to another.
An on-line interactive version is also available from the Geodetic Tool Kit.

Interpolates binary GPS orbit files with a record length of 52.

Comprises four programs - INVERSE which computes the geodetic azimuth and distance between two points, given their geographic positions FORWARD which computes the geographic position of a point, given the geodetic azimuth and distance from a point with known geographic position and the three-dimensional versions of these programs -- INVERS3D and FORWRD3D -- which include the height component.
These programs are based on the April, 1975 Survey Review paper by T. Vincenty.
An on-line interactive version is also available from the Geodetic Tool Kit.

Determines the loop misclosures of GPS base lines using the delta x, delta y, delta z vector components computed from a group of observing sessions.

Estmates the leveled height difference between two bench marks by removing the orthometric correction from the differences of published heights.

Computes geoid height values for Mexico. Suggest you read details before obtaining software.

Computes and verifies classical horizontal field observations (angles and distances), and formats these data to conform to Blue Book specifications. MTEN4 does not adjust field observations, but performs certain field check computations such as triangle computations, distance reductions, and trigonometric height computations. MTEN4 allows four-digit station numbers and new position codes set forth in the Blue Book.

Program na2vbbk performs file format conversion from a Leica/Wild NA2000/3000 series digital level output file format to a NGS vertical bluebook format. Minimal data checks are performed.
NOTE: Program na2vbbk does NOT verify that field observations were within FGCS tolerance limits nor does it compute the partial refraction correction. It should only be used when upper and lower thermistor probe temperatures are OBSERVED at each level instrument setup.

Transforms geographic coordinates between the NAD 27, Old Hawaiian, Puerto Rico, or Alaska Island datums and NAD 83 values. Can be used for America Samoa as well, but PLEASE READ THE SAMOA README FILE FIRST. Recommended for converting coordinate data for mapping, low-accuracy surveying, or navigation.
An on-line interactive version is also available from the Geodetic Tool Kit.
NOAA's Coastal Services Center offers a windows utility that does NADCON conversions using shapefiles as input and output.

Simplifies entering vertical observation data records into Blue Book format. The program formats the data onto a computer disk which can then be sent to NGS for further processing and incorporation of the data into the National Geodetic Reference System.

Creates horizontal control point records in Blue Book format with geodetic positions in geographic coordinates, state plane coordinates, or Universal Transverse Mercator coordinates.

Converts NAD 83 state plane coordinates to NAD 83 geographic positions and conversely. Includes defining constants for NAD 83 coordinate zones. State plane coordinates are entered or computed to 1 mm accuracy, while the latitudes and longitudes entered or computed correspond to approximately 0.3 mm accuracy.
An on-line interactive version is also available from the Geodetic Tool Kit.

Adds a user-specified shift in seconds to each input latitude and longitude.

Converts geographic coordinates (latitudes and longitudes) on the Clarke 1866, GRS 80/WGS 84, International, WGS 72, or any user-defined reference ellipsoid to Universal Transverse Mercator (UTM) coordinates, and vice-versa.

This program facilitates the process of edtiting, formatting and checking digital leveling observation data and creates abstracts, bok files, and VERTOBS datasets for submission to the National Geodetic Survey (NGS). The program includes many built-in functions such as predicting temperature differences, refaction corrections, rod corrections and plotting. Also included are routines for editing *.lvl files and VERTOBS files.

Converts between USNG, UTM, and geodetic latitude/longitude. The U.S. National Grid (USNG) System is an alpha-numeric reference system that overlays the UTM coordinate system. It is a Federal Geographic Data Committee (FGDC) standard developed to improve public safety, commerce, as well as aid the casual GPS user. The USNG provides an easy to use geoaddress system for identifying and determining locations with the help of a USNG gridded map and/or a USNG enabled GPS system.

VDatum is under development as part of the joint NOAA/USGS Bathymetric-Topographic Demonstration Project, and is a tool for the conversion of elevation data among 28 different vertical datums. Vertical datums can be based on Mean Sea Level (such as NAVD88), tidally derived surfaces (nautical charts) or three dimensional space systems (GPS, for example).

Computes the modeled difference in orthometric height between the North American Vertical Datum of 1988 (NAVD 88) and the National Geodetic Vertical Datum of 1929 (NGVD 29) for a given location specified by latitude and longitude. This conversion is sufficient for many mapping purposes. An on-line interactive version is also available from the Geodetic Tool Kit.

NGS supported Description Entry Software.

This PC Windows program provides methods for converting between Geodetic Latitude-Longitude-Ellipsoid_ht and XYZ on the GRS80 Ellipsoid.
An on-line interactive version is also available from the Geodetic Tool Kit.

## Geodetic Datums: NAD 27, NAD 83 and WGS84

When you need to accurately enter latitude and longitude coordinates in a GIS, the first step is to give it a datum. A geodetic datum uniquely defines all locations on Earth with coordinates.

Because where would you be on Earth without having reference to it?

Because the Earth is curved and in GIS we deal with flat map projections, we need to accommodate both the curved and flat views of the world. In surveying and geodesy, we accurately define these properties with geodetic datums.

We begin modelling the Earth with a sphere or ellipsoid. Over time, surveyors have gathered a massive collection of surface measurements to more reliably estimate the ellipsoid.

When you combine these measurements, we arrive at a geodetic datum. Datums precisely specify each location on Earth’s surface in latitude and longitude. For example, NAD27, NAD83 and WGS84 are geodetic datums.

### A Mammoth Collection of Survey Benchmarks

In order to create a geodetic datum, surveyors undertook a mammoth collection of monument locations in the late 1800s. Surveyors installed brass or aluminum disks at each reference location.

Each monument location was connected using mathematical techniques like triangulation.

From the unified network of survey monument, the result of triangulation was the North American Datum of 1927 (NAD 27). After, geodesists developed the more accurate NAD 83, which we still use today. NAD 27 and NAD 83 provide a frame of reference for latitude and longitude locations on Earth.

Surveyors now rely almost exclusively on the Global Positioning System (GPS) to identify locations on the Earth and incorporate them into existing geodetic datums.

For example, NAD27, NAD83, and WGS84 are the most common geodetic datums in North America.

#### North American Datum 1927 (NAD27)?

NAD27 stands for the North American Datum of 1927. NAD27 is the adjustment of long-baseline surveys. Overall, it established a network of standardized horizontal positions in North America. Most historical USGS topographic maps and projects by the US Army Corps of Engineers used NAD27 as a reference system.

A horizontal datum provides a frame of reference as a basis for placing specific locations at specific points on the spheroid. Geodesists use a horizontal datum as the model to translate a spheroid/ellipsoid into locations on Earth with latitude and longitude lines. Geodetic datums form the basis of coordinates of all horizontal positions on Earth. All coordinates on Earth are referenced to a horizontal datum. The North American Datum of 1927 (NAD27) is one of the main three geodetic datums used in North America.

NAD27 uses all horizontal geodetic surveys collected at this time using a least-square adjustment. This datum uses the Clarke Ellipsoid of 1866 with a fixed latitude and longitude at Meade’s Ranch, Kansas. (39°13’26.686″ north latitude, 98°32’30.506″ west longitude)

Kansas was selected as a common reference point because it was near the center of the contiguous United States. The latitudes and longitudes of every other point in North America were based on its direction, angle, and distance away from Meade’s Ranch. Any point with a latitude and longitude away from this reference point could be measured on the Clarke Ellipsoid of 1866.

Surveyors gathered approximately 26,000 stations in the United States and Canada. At each station, surveyors collected latitudes and longitude coordinates. NOAA’s National Geodetic Survey used these survey stations and triangulation to form the NAD27 datum.

As time went on, the number of stations also grew. For example, surveyors benchmarked approximately 250,000 stations. This set of horizontal positions formed the basis for the North American Datum of 1983 (NAD83). In 1983, the NAD27 datum was eventually replaced with NAD83.

#### North American Datum 1983 (NAD83)?

The North American Datum of 1983 (NAD 83) is the most current datum being used in North America. It provides latitude and longitude and some height information using the reference ellipsoid GRS80. Geodetic datums like the North American Datum 1983 (NAD83) form the basis of coordinates of all horizontal positions for Canada and the United States.

The North American Datum of 1983 (NAD 83) is a unified horizontal or geometric datum and successor to NAD27 providing a spatial reference for Canada and the United States.

NAD83 corrects some of the distortions from NAD27 over distance by using a more dense set of positions from terrestrial and Doppler satellite data. NAD83 is a geocentric datum (referenced to the center of Earth’s mass) offset by about 2 meters. Even today, geodesists are continually improving horizontal geodetic datums.

#### WGS84: Unifying a Global Ellipsoid Model with GPS

It wasn’t until the mainstream use of Global Positioning Systems (GPS) until a unified global ellipsoid model was developed. The radio waves transmitted by GPS satellites enable extremely precise Earth measurements across continents and oceans. Global ellipsoid models have been created because of the enhancement of computing capabilities and GPS technology.

This has led to the development of global ellipsoid models such as WGS72, GRS80, and WGS84 (current). The World Geodetic System (WGS84) is the reference coordinate system used by the Global Positioning System.

Never before have we’ve been able to estimate the ellipsoid with such precision because of the global set of measurements provided by GPS. It’s made of a reference ellipsoid, a standard coordinate system, altitude data, and a geoid. Similar to NAD 83, it uses the Earth’s center mass as the coordinate origin. The error is believed to be less than 2 centimeters to the center mass.

NAD83 corrects some of the distortions from NAD27 over distance by using a more dense set of positions from terrestrial and Doppler satellite data. Approximately 250,000 stations were used to develop the NAD83 datum. This compares to only 26,000 used in the NAD27 datum.

One of the primary differences is that NAD83 uses an Earth-centered reference, rather than a fixed station in NAD27. All coordinates were referenced to Kansas Meade’s Ranch (39°13’26.686″ north latitude, 98°32’30.506″ west longitude) for NAD27 datum. The National Geodetic Survey relied heavily on the use of the Doppler satellite to locate the Earth’s center of mass. However, NAD83 is not geocentric with an offset of about two meters.

North American Datum of 1983 is based on the reference ellipsoid GRS80 which is physically larger than NAD27’s Clarke ellipsoid. The GRS80 reference ellipsoid has a semi-major axis of 6,378,137.0 meters and a semi-minor axis of 6,356,752.3 meters. This compares to the Clarke ellipsoid with a semi-major axis of 6,378,206.4 m and a semi-minor axis of 6,356,583.8 meters.

### The Varying Historical Accuracy of the Ellipsoid

And since the beginning of the 19th century, the dimensions of the ellipsoid have been calculated at least 20 different times with considerably different accuracy.

The early attempts at measuring the ellipsoid used small amounts of data and did not represent the true shape of the Earth. In 1880, the Clarke ellipsoid was adopted as a basis for its triangulation computations. The first geodetic datum adopted for the United States was based on the Clarke ellipsoid with its starting point in Kansas known as Meades Ranch.

### One Datum with Many Versions and Abbreviations

Since 1986, geodesists have made several updates to NAD83. Actually, because of these changes, there is more than one version of NAD83. For example, the National Geodetic Survey has adjusted the NAD83 datum four times since the original geodetic datum estimation in 1986.

• NAD83 (1986): This version was intended to be geocentric and used the GRS80 ellipsoid.
• NAD83 (1991, HARN, HPGN): High Accuracy Reference Network (HARN) and High Precision Geodetic Network reworked geodetic datums from 1986-1997
• NAD83 (CORS96): Continually Operating Reference Stations (CORS) are composed of permanently operating Global Positioning System (GPS) receivers
• NAD83 (CSRS, CACS): Canadian Spatial Reference System and Canadian Active Control System with GPS processing.
• NAD83 (NSRS 2007, 2011): National Spatial Reference System and current survey standard using multi-year adjusted locations based on GNSS from the CORS.

### The Importance of Datum Transformations

The coordinates for benchmark datum points are typically different between geodetic datums. For example, the latitude and longitude location in a NAD27 datum differs from that same benchmark in NAD83 or WGS84. This difference is known as a datum shift.

Depending on where you are in North America, NAD27 and NAD83 may differ in tens of meters for horizontal accuracy. The average correction between NAD27 and NAD83 is an average of 0.349″ northward and 1.822″ eastward.

It’s important to note that the physical location has not changed. To be clear, most monuments have not moved. Datum shifts happen because survey measurements improve. Also, it happens when there are more of them and methods of geodesy change. This results in more accurate geodetic datums over time. The horizontal datums that form the basis of coordinates of all horizontal positions in North America improve.

Because maps were created in different geodetic datums throughout history, datum transformations are often necessary. Especially, this is true when using historical data. For example, USGS topographic maps generally were published using a NAD27 datum. You would need to apply a datum transformation when working with NAD83 data.

### When Do We Need Datum Transformations?

A coordinate transformation is a conversion from a non-projected coordinate system to a coordinate system. A coordinate transformation is done through a series of mathematical equations.

The geodetic datum is an integral part of projections. All coordinates are referenced to a datum. A datum describes the shape of the Earth in mathematical terms. A datum defines the radius, inverse flattening, semi-major axis, and semi-minor axis for an ellipsoid. The North American datum of 1983 (NAD 83) is the United States horizontal or geometric datum. It provides latitude and longitude and some height information.

Unfortunately, NAD 83 is not the only datum you’ll encounter. Before the current datum was defined, many maps were created using different starting points. And even today, people continue to change geodetic datums in an effort to make them more accurate. A common problem is when different coordinate locations are stored in different reference systems. When combining data from different users or eras, it is important to transform all information into common geodetic datums.

Projected coordinate systems are based on geographic coordinates, which are in turn referenced to a datum. For example, State Plane coordinate systems can be referenced to either NAD83 and NAD27 geodetic datums.

The NAD27 datum was based on the Clarke Ellipsoid of 1866:
Semi-major axis: 6,378,206.4 m
Semi-minor axis: 6,356,583.8 m
Inverse flattening: 294.98

The NAD83 datum was based on the Geodetic Reference System (GRS80) Ellipsoid:
Semi-major axis: 6,378,137.0 m
Semi-minor axis: 6,356,752.3 m
Inverse flattening: 298.26

When you transform NAD83 and NAD27 geographic coordinates to projected State Plane coordinates, it is the same projection method. However, because the geodetic datums were different, the resulting projected coordinates will also be different. In this case, a datum transformation is necessary.

For any type of work where coordinates need to be consistent with each other, you must use the same geodetic datum. If you are marking property or land boundaries or building roads or planning for coastal inundation scenarios, you must know about and use the correct geodetic datums.

## How It Works

The first thing the algorithm does is convert both points, Point A and Point B, to Cartesian (x, y, z) coordinates. This is complicated by the fact that the Earth is not a perfect sphere. Due to rotation, the Earth bulges at the equator, forming a shape known as an oblate spheroid. The distance between the poles is 6357 km, but the distance of a diameter passing through the equator is 6378 km. Consider this diagram, in which the bulging is exaggerated for the purposes of illustration:

Here, X is a point on the Earth’s surface, N is the north pole, S is the south pole, and C is the center of the Earth. The equator is represented by the horizontal line passing through C. Imagine you are standing at the point X. The dotted line tangent to the Earth there indicates your local horizon.

Perpendicular to your horizon is another dotted line that passes into the Earth. Where that dotted line touches the plane of the Earth’s equator, you see an angle marked φ (Greek letter phi). This angle is called the geodetic latitude. This is the kind of latitude reported by GPS.

Indeed, geodetic latitude is the classic kind of latitude used since ancient times. Roughly speaking, it indicates a latitude measured by astronomy: it is the angle of the North Celestial Pole (a point in the sky near the pole star Polaris) above your local horizon. If you are standing on the Earth’s north pole, Polaris will be straight up, or 90 degrees above the horizon. Likewise, if you are standing on the equator, Polaris will be on the horizon, or 0 degrees above the horizon. Thus geodetic latitude can be measured directly by any sailor with a clear nighttime sky.

However, to calculate the Cartesian coordinates of X with respect to the center C of the Earth, the calculator needs geocentric latitude, indicated by θ (Greek letter theta) in the diagram. This is the angle between the equator and X as seen from C. The conversion is based on the World Geodetic System standard WGS 84. In the calculator this is implemented in the function GeocentricLatitude , shown here:

Likewise, the distance from the center of the Earth to a given point on its surface is variable. The function EarthRadiusInMeters calculates this distance as a function of geodetic latitude, also using WGS 84:

As an interesting aside, geocentric and geodetic latitudes are the same when you are on the equator: both are 0 degrees. Both kinds of latitude are also the same if you are on either of the Earth’s poles: +90 or -90 degrees. But they disagree by different amounts at other locations on the Earth.

The calculator uses the function LocationToPoint to calculate the Cartesian coordinates of a point given its geodetic latitude, longitude, and elevation. This function also calculates the normal vector at the given point, meaning the upward direction perpendicular to the observer’s horizon, to correctly adjust for elevation above mean sea level. The resulting vector components are all expressed in meters.

To complete the calculations, the algorithm uses a trick. It rotates the coordinate system so that the observation point (Point A) is at a pretend equator and prime meridian. In other words, imagine rotating a globe you are holding so that your observation site is now on the equator and directly facing you. The coordinates now are ideally situated for calculating azimuth and altitude angles as seen by the observer at Point A:

• The x-axis points straight up (out from the center of the globe).
• The y-axis points due east.
• The z-axis points due north.

Here is the rotation function RotateGlobe :

After RotateGlobe rotates the coordinate system, you can draw a line from Point A on the rotated globe toward Point B on that same rotated globe, through three-dimensional space. This line will usually pass through the Earth, assuming both A and B are on or near the Earth’s surface. Using a combination of trigonometry functions and vector dot products, the calculator determines the azimuth angle and altitude angles. It uses the Pythagorean formula to determine the straight-line distance between the points.

All of the code is available in the links below.

## Measurement types available with the Measure tool

The Choose Measurement Type drop-down list provides a selection of measurement types to use for distance measurement. Measurement types available include Planar , Geodesic , Loxodrome , and Great Elliptic .

#### Measurement type of the measure tool

Planar measurement use 2D Cartesian mathematics to calculate lengths and areas. This option is only available when measuring in a projected coordinate system and the 2D plane of that coordinate system will be used as the basis for the measurements. All area measurements calculated with the measure tool are planar.

The shortest line between any two points on the earth's surface on a spheroid (ellipsoid). One use for a geodesic line is when you want to determine the shortest distance between two cities for an airplane's flight path. This is also known as a great circle line if based on a sphere rather than an ellipsoid.

A loxodrome is not the shortest distance between two points but instead defines the line of constant bearing, or azimuth. Great circle routes are often broken into a series of loxodromes, which simplifies navigation. This is also known as a rhumb line.

The line on a spheroid (ellipsoid) defined by the intersection at the surface by a plane that passes through the center of the spheroid and the start and endpoints of a segment. This is also known as a great circle when a sphere is used. The great elliptic type allows you to create lines only.

When measuring in a data frame with a projected coordinate system, the default measurement type will be Planar . This means that 2D Cartesian mathematics are used to calculate lengths. Planar measurements reflect the projection of geographic data onto the 2D surface (in other words, they will not take into account the curvature of the earth). Geodesic , Loxodrome , and Great Elliptic measurement types may be chosen as an alternative if desired.

When measuring in a data frame with a geographic coordinate system, the default measurement type is Geodesic . Planar line measurements and all area measurements will be unavailable when measuring in a geographic coordinate system. Loxodrome and Great Elliptic measurement types may be chosen as an alternative if desired.

## Coordinate systems

Data is defined in both horizontal and vertical coordinate systems. Horizontal coordinate systems locate data across the surface of the earth, and vertical coordinate systems locate the relative height or depth of data.

Horizontal coordinate systems can be of three types: geographic, projected, or local. You can determine which type of coordinate system your data uses by examining the layer's properties. Geographic coordinate systems (GCS) typically have units in decimal degrees, measuring degrees of longitude (x-coordinates) and degrees of latitude (y-coordinates). The location of data is expressed as positive or negative numbers: positive x- and y-values for north of the equator and east of the prime meridian, and negative values for south of the equator and west of the prime meridian.

Spatial data can also be expressed using projected coordinate systems (PCS). Linear measurements are used for the coordinates rather than angular degrees. Finally, some data may be expressed in a local coordinate system with a false origin (0, 0 or other values) in an arbitrary location that can be anywhere on earth. Local coordinate systems are often used for large-scale (small area) mapping. The false origin may or may not be aligned to a known real-world coordinate, but for the purpose of data capture, bearings and distances can be measured using the local coordinate system rather than global coordinates. Local coordinate systems are usually expressed in feet or meters.

A geographic coordinate system measured in angular units is compared to a projected coordinate system measured in linear units.

Vertical coordinate systems are either gravity-based or ellipsoidal. Gravity-based vertical coordinate systems reference a mean sea level calculation. Ellipsoidal coordinate systems reference a mathematically derived spheroidal or ellipsoidal volumetric surface.

## Scaling and Pixel Size

• None—This applies a linear adjustment by modifying the z factor according to the cell size, thereby accounting for altitude changes (scale) as the viewer zooms in and out. This is ideal for single raster datasets covering a local area. This is not recommended for worldwide datasets as it will produce a fairly flat (gray) image when zoomed out.
• Adjusted—This applies a nonlinear adjustment using the default Pixel Size Power and Pixel Size Factor values, which accounts for the altitude changes (scale) as the viewer zooms in and out. These values are recommended when using the worldwide dataset.

The z factor is adjusted using the following equation:

## 5.5 Land Surveying and Conventional Techniques for Measuring Positions on the Earth’s Surface

Ease, accuracy, and worldwide availability have made ‘GPS’ a household term. Yet, none of the power or capabilities of GPS would have been possible without traditional surveyors paving the way. The techniques and tools of conventional surveying are still in use and, as you will see, are based on the very same concepts that underpin even the most advanced satellite-based positioning.

Geographic positions are specified relative to a fixed reference. Positions on the globe, for instance, may be specified in terms of angles relative to the center of the Earth, the equator, and the prime meridian.

Land surveyors measure horizontal positions in geographic or plane coordinate systems relative to previously surveyed positions called control points, most of which are indicated physically in the world with a metal “benchmark” that fixes the location and, as shown here, may also indicate elevation about mean sea level (Figure 5.10). In 1988 NGS established four orders of control point accuracy, ranging in maximum base error from 3mm to 5cm. In the U.S., the National Geodetic Survey (NGS) maintains a National Spatial Reference System (NSRS) that consists of approximately 300,000 horizontal and 600,000 vertical control stations (Doyle,1994).

Doyle (1994) points out that horizontal and vertical reference systems coincide by less than ten percent. This is because:

. horizontal stations were often located on high mountains or hilltops to decrease the need to construct observation towers usually required to provide line-of-sight for triangulation, traverse and trilateration measurements. Vertical control points however, were established by the technique of spirit leveling which is more suited to being conducted along gradual slopes such as roads and railways that seldom scale mountain tops. (Doyle, 2002, p. 1)

You might wonder how a control network gets started. If positions are measured relative to other positions, what is the first position measured relative to? The answer is: the stars. Before reliable timepieces were available, astronomers were able to determine longitude only by careful observation of recurring celestial events, such as eclipses of the moons of Jupiter. Nowadays, geodesists produce extremely precise positional data by analyzing radio waves emitted by distant stars. Once a control network is established, however, surveyors produce positions using instruments that measure angles and distances between locations on the Earth's surface.

### 5.5.1 Measuring Angles and Distances

You probably have seen surveyors working outside, e.g., when highways are being realigned or new housing developments are being constructed. Often one surveyor operates equipment on a tripod while another holds up a rod some distance away. What the surveyors and their equipment are doing is carefully measuring angles and distances, from which positions and elevation can be calculated. We will briefly discuss this equipment and their methodology. Let us first take a look at angles and how they apply to surveying.

Although a standard compass can give you a rough estimate of angles, the Earth’s magnetic field is not constant and the magnetic poles, which slowly move over time, do not perfectly align with the planet’s axis of rotation as a result of the latter, true (geographic) north and magnetic north are different. Moreover, some rocks can become magnetized and introduce subtle local anomalies when using compass. For these reasons, land surveyors rely on transits (or their more modern equivalents, called theodolites) to measure angles. A transit (Figure 5.11) consists of a telescope for sighting distant target objects, two measurement wheels that work like protractors for reading horizontal and vertical angles, and bubble levels to ensure that the angles are true. A theodolite is essentially the same instrument, except that it is somewhat more complex and capable of higher precision. In modern theodolites, some mechanical parts are replaced with electronics.

When surveyors measure angles, the resultant calculations are typically reported as either azimuths or bearings, as seen in Figure 5.12. A bearing is an angle less than 90° within a quadrant defined by the cardinal directions. An azimuth is an angle between 0° and 360° measured clockwise from North. "South 45° East" and "135°" are the same direction expressed as a bearing and as an azimuth.

### 5.5.2 Measuring Distances

To measure distances, land surveyors once used 100-foot long metal tapes that are graduated in hundredths of a foot. An example of this technique is shown in Figure 5.13. Distances along slopes were measured in short horizontal segments. Skilled surveyors could achieve accuracies of up to one part in 10,000 (1 centimeter error for every 100 meters distance). Sources of error included flaws in the tape itself, such as kinks variations in tape length due to extremes in temperature and human errors such as inconsistent pull, allowing the tape to stray from the horizontal plane, and incorrect readings.

Since the 1980s, electronic distance measurement(EDM) devices have allowed surveyors to measure distances more accurately and more efficiently than they can with tapes. To measure the horizontal distance between two points, one surveyor uses an EDM instrument to shoot an energy wave toward a reflector held by the second surveyor. The EDM records the elapsed time between the wave's emission and its return from the reflector. It then calculates distance as a function of the elapsed time (not unlike what we’ve learned about GPS!). Typical short-range EDMs can be used to measure distances as great as 5 kilometers at accuracies up to one part in 20,000, twice as accurate as taping.

Instruments called total stations (Figure 5.14) combine electronic distance measurement and the angle measuring capabilities of theodolites in one unit. Next we consider how these instruments are used to measure horizontal positions in relation to established control networks.

### 5.5.3 Combining Angles and Distances to Determine Positions

Surveyors have developed distinct methods, based on separate control networks, for measuring horizontal and vertical positions. In this context, a horizontal position is the location of a point relative to two axes: the equator and the prime meridian on the globe, or to the x and y axes in a plane coordinate system.

We will now introduce two techniques that surveyors use to create and extend control networks (triangulation and trilateration) and two other techniques used to measure positions relative to control points (open and closed traverses).

Surveyors typically measure positions in series. Starting at control points, they measure angles and distances to new locations, and use trigonometry to calculate positions in a plane coordinate system. Measuring a series of positions in this way is known as "running a traverse." A traverse that begins and ends at different locations, in which at least one end point is initially unknown, is called an open traverse. A traverse that begins and ends at the same point, or at two different but known points, is called a closed traverse. "Closed" here does not mean geometrically closed (as in a polygon) but mathematically closed (defined as: of or relating to an interval containing both its endpoints). By "closing" a route between one known location and another known location, the surveyor can determine errors in the traverse.

Measurement errors in a closed traverse that connects at the point where it started can be quantified by summing the interior angles of the polygon formed by the traverse. The accuracy of a single angle measurement cannot be known, but since the sum of the interior angles of a polygon is always (n-2) × 180, it's possible to evaluate the traverse as a whole, and to distribute the accumulated errors among all the interior angles. Errors produced in an open traverse, one that does not end where it started, cannot be assessed or corrected. The only way to assess the accuracy of an open traverse is to measure distances and angles repeatedly, forward and backward, and to average the results of calculations. Because repeated measurements are costly, other surveying techniques that enable surveyors to calculate and account for measurement error are preferred over open traverses for most applications.

### 5.5.4 Triangulation

Closed traverses yield adequate accuracy for property boundary surveys, provided that an established control point is nearby. Surveyors conduct control surveys to extend and add point density to horizontal control networks. Before survey-grade satellite positioning was available, the most common technique for conducting control surveys was triangulation (Figure 5.16).

1. Using a total station equipped with an electronic distance measurement device, the control survey team commences by measuring the azimuth alpha, and the baseline distance AB.
2. These two measurements enable the survey team to calculate position B as in an open traverse.
3. The surveyors next measure the interior angles CAB, ABC, and BCA at point A, B, and C. Knowing the interior angles and the baseline length, the trigonometric "law of sines" can then be used to calculate the lengths of any other side. Knowing these dimensions, surveyors can fix the position of point C.
4. Having measured three interior angles and the length of one side of triangle ABC, the control survey team can calculate the length of side BC. This calculated length then serves as a baseline for triangle BDC. Triangulation is thus used to extend control networks, point by point and triangle by triangle.

### 5.5.5 Trilateration

An alternative to triangulation is trilateration, which uses distances alone to determine positions. By eschewing angle measurements, trilateration is easier to perform, requires fewer tools, and is therefore less expensive. Having read this chapter so far, you have already been introduced to a practical application of trilateration, since it is the technique behind satellite ranging used in GPS.

You have seen an example of trilateration in Figure 5.8 in the form of 3-dimensional spheres extending from orbiting satellites. Demo 1 below steps through this process in two dimensions.

#### Try This: Step through the process of 2-dimensional trilateration.

Once a distance from a control point is established, a person can calculate a distance by open traverse, or rely on a known distance if one exists. A single control point and known distance confines the possible locations of an unknown point to the edge of the circle surrounding the control point at that distance there are infinitively many possibilities along this circle for the unknown location. The addition of a second control point introduces another circle with a radius equal to its distance from the unknown point. With two control points and distance circles, the number of possible points for the unknown location is reduced to exactly two. A third and final control point can be used to identify which of the remaining possibilities is the true location.

Trilateration is noticeably simpler than triangulation and is a very valuable skill to possess. Even with very rough estimates, one can determine a general location with reasonable success.

#### Practice Quiz

Registered Penn State students should return now take the self-assessment quiz Land Surveying.

You may take practice quizzes as many times as you wish. They are not scored and do not affect your grade in any way.

### Earth Survey Plugin: NGS Benchmarks, PLSS Data, USGS Quad Data And More

A while back, I posted about a free web app from Metzger and Willard that shows National Geodetic Survey control points (benchmarks) near a specific area, and lets you view data for those landmarks. I’ve just noticed that they’ve created a newer web app called the Earth Survey Plugin, running in a Google Earth browser plugin that not only has the same capability, but also adds a bunch of additional features:

• An NGS Survey Marker capability that works very similarly to the previous app, but now offers the ability to export the data into a static KML file
• A PLSS point geocoder function that either gives you the section data for the point in the middle of the display:

… or lets you enter the PLSS parameters, and find the center point associate with them:

These can also be saved as a KML file.

• USGS topo quad index orange dots for 1:24K, purple for 1:100K, cyan for 1:250K. Clicking on a dot brings up a pop-up balloon with the name of a quad, and a direct link to the GeoPDF for that quad at the USGS store. Note that GeoPDF quads are not currently available for quads in US National Forests, and that at this time, some states (e.g. AZ, CA) don’t have full topographic information on their GeoPDF quads.

### National Geodetic Survey Online Toolkit

The National Geodetic Survey has a set of links online conversion utilities for performing some basic geodetic conversions, including:

• High-accuracy state plane coordinate system (SPCS) conversions
• UTM/USNG/geographic coordinate conversions
• Magnetic declination
• Surface gravity prediction
• Azimuth/distance to a second position (or second position from azimuth/distance data)
• Position from a dual-frequency GPS data file

Plus many more. Some of these are also available as PC programs, but these tend to be old-school DOS-based conversion utilities, not always the most user friendly.

### View NGS Benchmarks, UTM Zones, PLSS Meridians And More In A Web Interface

The Surveying.Org website plots a number of useful survey-related data features in a Google Maps interface select one or multiple data features to display with a checkbox.

National Geodetic Survey Benchmark locations

Takes a few seconds for them to pop up. The different symbols correspond to various classes of accuracy for the benchmarks, both horizontal and vertical. Click on a benchmark icon …

… and get at popup with the name/designation of the benchmark. Clicking on the datasheet link brings up a full datasheet with coordinates, quality information, and more.

NGS Vertcon info

State Plane Coordinate System (SPCS) Zones

A few of the icons/popups are slightly misplotted, like the second one in the lower left, but they’re close enough to their respective SPCS zones to let you figure it out.

This is the only dataset that’s good for outside the US UTM zones are displayed for the entire world.

Meridian locations for Public Land Survey System (PLSS) designations (Township/Range/Section)

Also slightly misplotted in a few cases, as with the Navajo Meridian above, but close enough to figure out.

Area/length measurements

Finally, you can put the web app into either line length or area mode, and then click on the map to define vertices for a measurement. Above, an area is defined …

… and the area given in various units. Click and drag on a vertex to move it Ctrl-click on a vertex to delete it. Doesn’t seem to be a way to add a vertex to a line segment, though.

### Lookup EPSG Coordinate System Code From PRJ/WKT File

PRJ files are often included with GIS vector or raster data, and define the coordinate system associated with the data using the Well-Known Text (WKT) format. The Prj2EPSG utility site lets you upload or paste PRJ file data, and looks up the standard EPSG code designation associated with it (or gives you a list of what it thinks are the best matches). Utility seems to work well with many standard coordinate systems:

It’s not perfect, though. I threw an oddball Lambert Conformal Conic projection at it, from a USGS GeoPDF of a Utah 1:24K topo map, and it was a bit stumped:

Estonia and India )? To be fair, this particular coordinate system gives a number of other programs heartburn as well, and it was the only outright failure in 10 sample files the other 9 came up with one candidate answer, and it was always the correct one.

### Handy Set Of Online Calculators

APSalin, a company specializing in “Software development and data management for radio and television broadcast engineering”, offers a free set of handy online calculators and reference materials:

• Converters to/from geographic/geodetic – Mercator – UTM – Cartesian coordinates
• NAD27 to NAD 83 converter (latitude/longitude only)
• Latitude/longitude DMS to decimal & vice versa
• Great circle distance/bearing between two points
• Destination (enter position-bearing-distance, get coordinates of destination)
• Reference tables showing what distances on map represent for various map scales, and what various distances in real life correspond to on a map at a specific scale. Tables are given for both metric and imperial units if you ever need an argument for doing everything in metric, this is a good place to start.
• Reference tables for DEM resolutions in arc-seconds converted to meters and feet
• Ellipsoid parameters for various datums

### Compare Geographic Boundaries With Move Outlines

I can remember as a kid (way too many years ago) being impressed with a map of the lower 48 United States that had the outline of Alaska superimposed on top of it. The Alaska outline virtually covered the entire map, and there was a comment to the effect that,”Alaska is almost as large as the lower 48 states”. It wasn’t until years later that I realized the creator of that map had just traced an outline of Alaska off of a Mercator projection map and laid it on top of the US map without compensating for the change in scale.

If only the map maker had had access to Mapfrappe’s Move Outlines site, he might not have made that mistake. Draw an outline of a geographic area in one Google Maps window:

And have the outline be superimposed on top of another Google Maps window, scaled correctly to compensate for changes in the Mercator scale at different latitudes:

Alaska’s still pretty dang big, but this shows it at its true scale, roughly one-third the size of the lower 48.

Another classic example of this is Greenland, which looks humungous on a standard Mercator projection:

In true area, though, it’s roughly the same size as Mexico big, but not gargantuan:

The site has some pre-drawn comparisons, like the Great Lakes against the Black Sea:

### Multiple Coordinate Systems In Google Maps, Reverse Geocoding, And More With The Worldwide Coordinate Converter

Clement Ronzon emails about his new website, The Worldwide Coordinate Converter (TWCC for short). Drag the globe-shaped icon to the desired location in a Google Maps interface, and get a pop-up balloon with the geographic coordinates for that spot, elevation in meters, and the nearest reverse-geocoded address:

At right is a two-part coordinates box, with latitude/longitude/WGS84 always in the top part, and a user-selectable coordinate system at the bottom:

You can also enter coordinates into the appropriate boxes in either the top or bottom section, click Convert, and have them converted to the other coordinate system automatically (and plotted on the map).

### Look Up EPSG Codes From Well-Known-Text (WKT) / prj Files With Prj2EPSG

OpenGeo announces Prj2EPSG, a free online service that can look up the EPSG (European Petroleum Study Group) code for a coordinate system based on:

yields the EPSG code 4326.

• A .prj file, often found with shapefiles or other geographic data files to define the coordinate system select the prj file to upload, and get back the original prj data in the box and the EPSG code:

• Or just type in the EPSG code, and get back a data page with the coordinate system defined, and the WKT listed.

Click on any of the code links in the search results to get a full data page: