# Symbols and abbreviated terms

 a semi-major axis b semi-minor axis CC Change Coordinates (package abbreviation in UML model) CD Coordinate datum (package abbreviation in UML model) CCRS Compound coordinate reference system CI Citation CRS Coordinate reference system CS Coordinate system (also package abbreviation in UML model) DQ Data quality (package abbreviation in UML model) E Easting f flattening h ellipsoidal height H gravity-related height ISO International Organization for Standardization N Northing OGC Open GIS Consortium RS Reference System (package abbreviation in UML model) SC Spatial referencing by Coordinates (package abbreviation in UML model) SI Le Système International d'Unités UML Unified Modeling Language XML eXtensible Markup Language 1D One Dimensional 2D Two Dimensional 3D Three Dimensional λ geodetic longitude φ geodetic latitude X, Y, Z Cartesian coordinates in a geocentric coordinate reference system i, j, k Cartesian coordinates in an engineering coordinate reference system, (integer or real)

# Terms and definitions

### Cartesian coordinate system

Coordinate system which gives the position of points relative to N mutually-perpendicular straight axes. In the context of geospatial coordinates the maximum value of N is three.

### Compound coordinate reference system

Coordinate system describing the position of points through two or more independent coordinate reference systems. For example, one coordinate reference system can be a two-dimensional horizontal coordinate system, and the other coordinate reference system can be a vertical gravity-related height system.

### Concatenated transformation

Sequential application of multiple transformations.

### Coordinate

One of a sequence of N numbers designating the position of a point in N-dimensional space. In a coordinate reference system, the coordinate numbers must be qualified by units.

### Coordinate conversion

Change of coordinates, based on a one-to-one relationship, from one coordinate reference system to another based on the same datum. For example between geodetic and Cartesian coordinate systems or between geodetic coordinates and projected coordinates, or change of units such as from radians to degrees or feet to metres. A conversion uses parameters which have specified values, not empirically determined values.

### Coordinate reference system

Coordinate system which is related to the real world by a datum. NOTE For geodetic and vertical datums, it will be related to the Earth.

### Coordinate system

Set of (mathematical) rules for specifying how coordinates are to be assigned to points. One coordinate system may be used in many coordinate reference systems. The geometric properties of a coordinate space determine how distances and angles between points are calculated from the coordinates. For example, in an ellipsoidal (2D) space distances are defined as curves on the surface of the ellipsoid, whereas in a Euclidean plane as used for projected CRS distance is the length of a straight line between two points. The mathematical rules that determine distances and angles are calculated from coordinates and vice versa are comprised in the concept of coordinate system.

### Coordinate transformation

Computational process of converting a position given in one coordinate reference system into the corresponding position in another coordinate reference system. A coordinate transformation can require and use the parameters of the ellipsoids associated with the source and target coordinate reference systems, in addition to the parameters explicitly associated with the transformation. The term "transformation" is used only when the parameter values associated with the transformation have been determined empirically from a measurement / calculation process. This is typically the case when a change of datum is involved.

### Covariance matrix

Matrix of elements (or cells) that contain the expected average values of the product of the error in the matrix row coordinate times the simultaneous error in the matrix column coordinate. A covariance matrix is a form of detailed error estimate data. Covariance matrices are sometimes called variance-covariance matrices. All complete covariance matrices are symmetrical, meaning that the same element values appear on both sides of the diagonal elements. Covariance matrices contain information about the absolute and/or relative accuracy of the data elements (e.g. coordinates). The absolute accuracy information is contained in the diagonal matrix elements. Relative accuracy is a function of multiple diagonal and off-diagonal elements. A complete covariance matrix for N specific points in 3D space would contain 3N rows by 3N columns. For example, for three coordinates, a covariance matrix is a 3 by 3 matrix, with the matrix rows and columns each corresponding to the three coordinates. For just two horizontal coordinates, a covariance matrix is a 2 by 2 matrix, with the matrix rows and columns each corresponding to the two horizontal coordinates. Similarly, for two image coordinates, a covariance matrix is a 2 by 2 matrix, with the matrix rows and columns each corresponding to the two image coordinates.

### Cylindrical coordinates

3-dimensional coordinates with two distance and one angular coordinate.

### Datum

Parameter or set of parameters that determine the location of the origin, the orientation and the scale of a coordinate reference system.

### Depth

Distance of a point below a chosen reference surface usually measured along the local vertical (gravity vector). Depth is sometimes measured along a line that does not follow the vector of gravity locally. An example is depth in an oil or gas well. These are generally measured along the wellbore path, which may vary significantly from the local vertical. Some sections of a wellbore path may even run horizontally or slope upwards. Nevertheless the distance along the wellbore path is referred to as "depth". See elevation, ellipsoidal height, and gravity-related height.

### Dimension

Number of ordinates needed to describe a position in a coordinate system.

### Elevation

Distance of a point from a chosen reference surface along the direction of the gravity vector from the point to that surface. See ellipsoidal height and gravity-related height. It should be noted that ellipsoidal height is defined w.r.t. an ellipsoidal model of the shape of the earth. Ellipsoidal height is measured from the point along the line perpendicular to the ellipsoid's surface. Height of a point outside the surface treated as positive; negative height is also named as depth.

### Ellipsoid

Surface formed by the rotation of an ellipse about an axis In this document the axis of rotation is always the minor axis. Sometimes the alternative word "spheroid" is used in geodetic or survey practice to express the same concept. Although mathematically speaking incorrect the more common term in geodetic or survey practice is "ellipsoid". An alternative term used in geodetic practice is "reference ellipsoid".

### Ellipsoidal coordinate system Geodetic coordinate system

Coordinate system in which position is specified by geodetic latitude, geodetic longitude and (in the three-dimensional case) ellipsoidal height, associated with one or more geographic coordinate reference systems.

### Ellipsoidal height Geodetic height

Distance of a point from the ellipsoid measured along the perpendicular from the ellipsoid to this point, positive if upwards or outside of the ellipsoid. Only used as part of a three-dimensional geodetic coordinate system and never on its own.

### Engineering coordinate reference system

A coordinate reference system that is defined for and usually used in a contextually local sense, which may be an area, significantly less than the complete surface of the earth or a moving platform and its vicinity. For example local engineering and architectural coordinates, grids, and drawings; also: vessel navigation systems and CRSs associated with orbiting spacecraft. A transformation of engineering coordinates to geodetic coordinates may or may not be possible depending on whether such operation parameters have been determined (or defined). An Engineering CRS may be defined to describe geometry that is local to the context of a moving platform, such as a car, a ship, an aircraft or a spacecraft. Transformation of such engineering coordinates to geodetic coordinates involves time dependent operation parameters and, when repeated at (regular) time-intervals, will result in a record of the "track" of the moving platform. Additionally such a transformation may be used for real-time navigation of the platform. The term "vicinity of the moving platform" may constitute an area varying from the immediate surroundings of the platform to the entire earth, the latter being the case in a number of space applications.

### Flattening

Ratio of the difference between the semi-major (a) and semi-minor axis (b) of an ellipsoid to the semi-major axis; f = (a-b)/a. Sometimes inverse flattening 1/f = a/(a-b) is given instead of flattening; 1/f is also known as reciprocal flattening.

### Geocentric coordinate reference system

3-dimensional coordinate reference system with its origin at the (approximate) centre of the Earth.

### Geodetic coordinates

Coordinates defined in a geocentric, geographic (2D or 3D) or projected coordinate reference system.

### Geodetic datum

Datum describing the relationship of a 3D or 2D coordinate system to the Earth. In most cases, the geodetic datum includes an ellipsoid definition.

### Geographic coordinate reference system

Coordinate reference system using an ellipsoidal coordinate system and based on an ellipsoid that approximates the shape of the Earth. A geographic coordinate system can be 2D or 3D. In a 3D geographic coordinate system, the third dimension is height above the ellipsoid surface.

### Geographic dataset

Dataset with a spatial content.

### Geoid

Level surface which best fits mean sea level either locally or globally. "Level surface" means an equipotential surface of the Earth's gravity field that is everywhere perpendicular to the direction of gravity.

### Gravity-related height

Height dependent on the Earth's gravity field. In particular, orthometric height or normal height, which are both approximations of the distance of a point above the geoid.

### Gregorian calendar

Calendar in general use first introduced in 1582 to correct an error in the Julian calendar. In the Gregorian calendar, common years have 365 days and leap years 366 days divided into 12 sequential months.

### Greenwich meridian

Prime meridian passing through Greenwich, United Kingdom. Most geodetic datums use the Greenwich meridian as the prime meridian.

### Ground coordinates Earth referenced coordinates Terrestrial coordinates

Coordinates of points expressed in a non-image, earth-fixed coordinate reference system. The term ground coordinates is used herein to distinguish such coordinates from image coordinates. Even when an image is collected by a near vertical camera, image coordinates are different from ground coordinates.

### Image

Record of the likeness of any features, objects, and activities. An image can be acquired through the sensing of visual or any other segment of the electromagnetic spectrum by sensors, such as thermal infrared, and high resolution radar.

### Image coordinates

Definition of position within an image, expressed in image row and column coordinates.

### Image geometry model

Mathematical model that specifies the mapping (or projection) from 3D ground position coordinates to the corresponding 2D image position coordinates. An image geometry model is alternately called an image sensor model, sensor model, imaging model, or image mathematical model. The term "sensor" is often used when the image is generated by a digital camera and is thus originally digital. The word "camera" is usually used when the image is recorded in analogue form, normally on film. Of course, film images can be later scanned or digitised and are then "digital". An image geometry model can also be used to determine the correct ground position for an image position, if used with additional data. When a single (or monoscopic) image is used, this additional data normally defines the shape and position of the visible ground (or object) surface. For example, this additional data is often a single elevation or is grid elevation data, sometimes called a Digital Terrain Model (DTM). Alternately, two stereoscopic images or multiple overlapping images can be used, that show the same ground point viewed from different directions. In this case, the two (or more) image geometry mathematical models can also be used, with the point coordinates in each individual image, to determine the corresponding 3D ground position.

### Image version

New image produced by sub-setting and/or re-sampling the pixels in an original image.

### Interface

Named set of operations that characterise the behaviour of an element. An interface standard specifies the services in terms of the functional characteristics and behaviour observed at the interface. The standard is a contract in the sense that it documents a mutual obligation between the service user and provider and assures stable definition of that obligation.

### Latitude Geodetic latitude Ellipsoidal latitude

Angle from the equatorial plane to the perpendicular to the ellipsoid through a given point, northwards treated as positive.

### Local datum Engineering datum

Datum with a local reference, used as a basis for an engineering coordinate reference system. Engineering datum excludes both geodetic and vertical datums.

### Longitude Geodetic longitude Ellipsoidal longitude

Angle from the prime meridian plane to the meridian plane of the given point, eastward treated as positive.

### Map projection

Conversion from a geodetic coordinate system to a planar surface.

### Mean sea level

Average level of the surface of the sea over all stages of tide. Mean sea level in a local context normally means mean sea level for the region as measured by tide gauge measurements at one or more points over a given period of time. Mean sea level in a global context differs from a global geoid by not more than 2 metres.

### Meridian

Intersection of an ellipsoid by a plane containing the minor axis of the ellipsoid. This term is often used for the pole-to-pole arc rather than the complete closed figure.

### Oblique Cartesian coordinate system

Coordinate system with straight axes that are not necessarily mutually perpendicular.

### Pixel

Two-dimensional picture element that is the smallest non-divisible element of a digital image. In image processing, the smallest element of a digital image that can be assigned a grey level. This term originated as a contraction for "picture element".

### Polar coordinates

2-dimensional coordinates in which position is specified by distance to the origin and the direction angle. NOTE ISO/DIS 19111 does not specify the number of dimensions and therefore implicitly permits a 3-dimensional polar coordinate system to exist. The equivalent of the latter is termed "spherical coordinate system" in this document.

### Position

Spatial reference of a point or an object.

### Prime meridian Zero meridian

Meridian from which the longitudes of other meridians are quantified.

### Projected coordinate reference system

Two-dimensional coordinate system resulting from a map projection. A projected coordinate reference system is derived from a 2D geographic coordinate reference system by applying a parameterised coordinate transformation known as a "map projection". A projected coordinate reference system commonly uses a Cartesian coordinate system.

### Reference ellipsoid

Ellipsoid used as the best local or global approximation of the surface of the geoid.

### Semi-major axis

Semi-diameter of the longest axis of a reference ellipsoid. This equates to the semi-diameter of the reference ellipsoid measured in its equatorial plane.

### Semi-minor axis

Semi-diameter of the shortest axis of a reference ellipsoid. The shortest axis coincides with the rotation axis of the reference ellipsoid and therefore contains both poles.

### Spherical coordinate system.

3-dimensional coordinate system with one distance, measured from the origin and two angular coordinates, commonly associated with a geocentric coordinate reference system. Not to be confused with an ellipsoidal coordinate system based on an ellipsoid ‘degenerated' into a sphere.

### Temporal coordinate

Distance from the origin of the interval time scale used as the basis for a temporal reference system.

### Temporal coordinate reference system

Reference system against which time is measured.

### Transformation

Change of coordinates from one coordinate reference system to another coordinate reference system based on a different datum through a one-to-one relationship. A transformation uses parameter values which may have to be derived empirically by a set of points common to both coordinate reference systems. See coordinate conversion and coordinate transformation.

### Unit Unit of measure

Defined quantity in which dimensioned parameters are expressed. In this document, the subtypes of units are length units, angular units, time units, scale units and pixel spacing units.

### UTC Coordinated Universal Time

Time scale maintained by the Bureau International des Poids et Mesures (International Bureau of Weights and Measures) and the International Earth Rotation Service (IERS) that forms the basis of a coordinated dissemination of standard frequencies and time signals.

### Vertical coordinate system

1-dimensional coordinate reference system used for elevation, height, or depth measurements.

### Vertical datum

Datum describing the relation of gravity-related heights to the Earth. In most cases the vertical datum will be related to sea level. Ellipsoidal heights are treated as related to a three-dimensional ellipsoidal coordinate system referenced to a geodetic datum. Vertical datums include sounding datums (used for hydrographic purposes), in which case the heights may be negative heights or depths.