Using coordinate systems

A coordinate system (CS) defines a set of axes that span the coordinate space (Iliffe, 2003). A CS defines the attribute of axes, such as number of axes (dimensions: 1D, 2D, or 3D), direction, names, units, and even the order of axes. The most common systems used in GIS are as follows (Allan A.L., 2007):

• Cartesian coordinates (X, Y, Z)
• Geodesic coordinates (geodetic latitude and longitude)
• Spherical coordinates (geocentric latitude and longitude)
• A hybrid 2D coordinate system and 1D coordinate system on a map projection (for example, E=Easting, N=Northing, and H= orthometric height)
• A hybrid 2D coordinate system and 1D coordinate system for the ellipsoid and sphere (for example: φ, λ, and h= ellipsoidal height)

  Note

  For theory aspects regarding coordinate systems, datum, and map projections, please refer to: Jonathan Iliffe and Roger Lott; Datums and Map projections for remote sensing, GIS, and surveying 1st Edition; Whittles Publishing, 2003.

Mathematic cartography uses the ellipsoid of revolution (spheroid), sphere, and plane as "reference surfaces" to define coordinate systems.

The ellipsoidal coordinate system

The first approximation of the shape of the Earth is that it is a rotational ellipsoid or a reference ellipsoid. The following figure shows the graticule of parallels and meridians at 10⁰ intervals:

The surface of revolution is obtained by rotating the meridian ellipse around its minor (short) axis. This ellipsoid is called an oblate ellipsoid. The major and minor axes of a reference ellipsoid do not vary greatly. This is because its shape approximates the shape of a sphere, so the terms "ellipsoid" and "spheroid" are often used interchangeably by Esri's software.

The size and shape of the ellipsoid can be defined by at least two geometric parameters, where one should be a semi-axis. For example, the most common ellipsoid, called GRS 1980, has the following parameters: its semi-major axis is 6,378,137 m and its flattening is 1/298.25722 21008 827 (Hooijberg, M. Practical Geodesy, 1997).

The ellipsoidal coordinate system or geodetic coordinate system can be two-dimensional (geodetic latitude, ? or φ and geodetic longitude, λ) or three-dimensional (?, λ, and ellipsoidal height h), as shown in the following figure:

The latitude and longitude are expressed as sexagesimal degrees in minutes and seconds (DMS), or they are expressed as sexagesimal decimal degrees (DD), especially for the digital storage and computation, as shown in the following screenshot:

This example shows the geographic coordinate for Paris in DMS and DD. A short example of converting from 2⁰19'53" E DMS to DD is explained, as follows:

1. Divide the minute value by the number of minutes in a degree (60):

   19 minutes= 19/60 = 0.316666 degrees

2. Divide the seconds value by the number of seconds in a degree (3600):

   53 seconds=53/3600=0.014722 degrees

3. Add up the degrees: 2⁰ + 0.31 66 66⁰ + 0.01 47 22⁰ = 2.33 13 88 DD ~ 2.3314⁰ DD

The ellipsoidal height is expressed in meters.

The Equatorial plane splits the ellipsoid into two hemispheres: the Northern hemisphere (N) where latitude has values between [0⁰, +90⁰], and the Southern hemisphere (S) where latitude has values between [0⁰, -90⁰].

The Greenwich meridian plane (prime meridian) splits the ellipsoid into two hemispheres: the Eastern hemisphere (E. Greenwich) where longitude has values between [0⁰, +180⁰], and the Western hemisphere (W. Greenwich) where latitude has values between [0⁰, -180⁰].

The spherical coordinate system

A more simplified approximation of the shape of the Earth is the sphere. The spherical coordinate system can be two-dimensional (latitude φ and longitude λ) or three-dimensional (latitude φ, longitude λ, and height above sphere, h), as shown in the following figure:

A sphere has a constant curve on any latitude, in every direction, which allows generalizations in map-projection formulae. In the ArcMap coordinate system list, you will find the term "authalic", which is used for some coordinate reference systems. An authalic sphere is a sphere with a surface area equal to the surface area of the given ellipsoid. For example, the GRS 1980 Authalic Sphere has the following parameters: the semi-major and semi-minor axes are 6,371,007 m and the flattening is 0.

The spherical polar coordinate system

The spherical polar coordinate system is an oblique or transverse coordinate system that is defined by a chosen pole on a sphere called Q that has the latitude on the sphere, 0⁰ ≤ φ <90⁰. The spherical polar coordinates are azimuth (A) and zenith distance (Z), as shown in the following figure:

Azimuth (A) is the angle from the arc of a great circle that crosses through the pole Q, point P (geodesic line QP'), and the meridian of pole Q. The angle is measured clockwise and has values between [0⁰, 360⁰].

Zenith distance (Z) is the central angle that subtends the arc of the great circle from pole Q to a given point on sphere P'. The subtended angle Z has values between [0⁰, 180⁰].

Verticals (A is constant) are arcs of a great circle between the poles Q and Q'. Almucantars (Z is constant) are small circles with variable radius, and they are perpendicular to the verticals. The graticule is made up of verticals and almucantars.

The spherical-polar coordinate system is an additional coordinate system that is used by the traverse and oblique map projections.

The three-dimensional (3D) Cartesian coordinate system

The 3D Cartesian coordinates (X, Y, Z) rotate and are attached to the model of the Earth. The origin of coordinate system is the center of the ellipsoid or sphere. The plane XOY is in the Equatorial plane. The OX-axis is a line from the center that runs through the intersection of the Greenwich meridian with Equatorial plane. The OY-axis is perpendicular to the OX-axis. The OZ-axis coincides with the minor axis of the ellipsoid (polar axis), as you can see in the following figure:

In most of these cases, the 3D Cartesian coordinate system coincides with the center of mass of the Earth called geocenter. In this case, the OX-axis is toward 90⁰ E longitude, and the OZ-axis is a line from the geocenter through the Conventional Terrestrial Pole (which coincides with the Earth's axis of rotation). This Cartesian coordinate system is known as Terrestrial Reference System (TRS). The Cartesian coordinates are expressed in meters.