General Quadrilateral grid geometry
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35
Figure 13 — Example — A 7 7 two-dimensional orthogonal grid
NOTE The dimensions axes of a 2-dimensional grid are often called row and column.
A grid may be defined in terms of an external coordinate reference system. This requires additional information about the location of the grid’s origin within the external coordinate reference system, the
orientation of the grid axes, and a measure of the spacing between the grid lines. If the spacing is uniform, then there is an affine relationship between the grid and external coordinate system, and the grid Figure 14
is called a rectified grid. If, in addition, the external coordinate reference system is related to the earth by a datum, the grid is a georectified grid. The grid lines of a rectified grid need not meet at right angles; the
spacing between the grid lines is constant along each axis, but need not be the same on every axis. The essential point is that the transformation of grid coordinates to coordinates of the external coordinate
reference system is an affine transformation.
NOTE 1 The word rectified implies a transformation from an image space to another coordinate reference system.
However, grids of this form are often defined initially in an earth-based coordinate system and used as a basis for collecting data from sources other than imagery.
NOTE 2 The internal grid coordinate system is an instance of an engineering coordinate reference system as specified
by ISO 19111:2003. Its datum is a set of one or more ground control points.
Key
X, Y, Z axes to determine 3-space
V
1
, V
2
offset vectors grid origin
Figure 14 — Geometry of a rectified grid
Copyright © 2007 Open Geospatial Consortium, Inc. All Rights Reserved.
36
EXAMPLE Figure 14 shows a two-dimensional grid in the 3-space determined by the axes X, Y, and Z. The grid
origin is at O. There are two offset vectors labelled V
1
and V
2
which specify
the orientation of the grid axes and the spacing between the grid lines. The coordinates of the grid points are of the form: O + aV
1
+ bV
2
.
When the relationship between a grid and an external coordinate reference system is not adequate to specify it in terms of an origin, an orientation and spacing in that coordinate reference system, it may still be possible
to transform the grid coordinates into coordinates in the coordinate reference system. This transformation need not be in analytic form; it may be a table, relating the grid points to coordinates in the external coordinate
reference system. Such a grid is classified as a referenceable grid. If the external coordinate reference system is related to the earth by a datum, the grid is a georeferenceable grid. A referenceable grid is associated with
information that allows the location of all points in the grid to be determined in the coordinate reference system, but the location of the points is not directly available from the grid coordinates, as opposed to a rectified grid
where the location of the points in the coordinate reference system is derivable from the properties of the grid itself. The transformation produced by the information associated with a referenceable grid will produce a grid
as seen in the coordinate reference system, but the grid lines of that grid need not be straight or orthogonal, and the grid cells may be of different shapes and sizes.