Table of Contents
|
POSC Specifications Version 2.2 |
Epicentre Usage Guide Projections and Projected Coordinate Systems |
| | |
Table of Contents |
Only formulas for computation on the ellipsoid are considered in Epicentre. Projection formulas for the spherical earth are simpler but the spherical figure is inadequate to represent positional data with great accuracy at large map scales for the real earth. Projections of the sphere are only suitable for illustrative maps at scale of 1:1 million or less where precise positional definition is not critical.
The formulas which follow are largely taken from Map Projections - A Working Manual by J.P. Snyder, published by the U.S. Geological Survey as Professional Paper No.1395. As well as providing an extensive overview of most map projections in current general use, and the formulas for their construction for both the spherical and ellipsoidal earth, this excellent publication provides computational hints and details of the accuracies attainable by the formulas. It is strongly recommended that all those who have to deal with map projections for medium and large scale mapping should follow its guidance.
There are a number of different formulas available in the literature for coordinate transformations other than those quoted by Snyder. Some are closed formulas; others, for ease of calculation, may depend on series expansions and their precision will generally depend on the number of terms used for computation. Generally those formula which follow in this chapter will provide results which are accurate to within a decimetre, which is normally adequate for exploration mapping purposes. Coordinate expression and computations for engineering operations are usually consistently performed in grid terms.
The importance of one further variable should be noted. This is the unit of linear measurement used in the definition of projected coordinate systems. For projected coordinate systems the unit of measurement is restricted to this unit. For non-metric projected coordinate systems the ellipsoid semimajor axis needs to be converted to the projected coordinate system linear unit before use in the formulas below.
In the formulas for computing coordinate transformation in the projected coordinate systems which follow, the basic ellipsoidal parameters are represented by symbols and derived as follows:
|
a |
ellipsoidal semi-major axis |
|
b |
ellipsoidal semi-minor axis |
|
f |
flattening of the ellipsoid where 1/f = a/(a - b) |
|
e |
eccentricity of the ellipsoid where e2 = 2f - f2 |
|
e' |
second eccentricity where e'2 = e2 / (1-e2) |
|
r |
radius of curvature of the meridian at latitude j, where r = a (1-e2)/(1-e2sin2j)3/2 |
|
n |
radius of curvature on the prime vertical (i.e. perpendicular to the meridian) at latitude j, where n = a/(1-e2sin2j)1/2 |
|
j |
latitude of the point to be transformed, positive if North and negative if South of the Equator |
|
l |
longitude of the point to be transformed, positive if East and negative if West of the prime meridian |
|
j0 |
latitude of the natural origin |
|
l0 |
longitude of the natural origin (with respect to the prime meridian) |
|
j F |
latitude of the false origin |
|
l F |
longitude of the false origin (with respect to the prime meridian) |
|
j1 |
latitude of the first standard parallel |
|
j2 |
latitude of the second standard parallel |
|
k0 |
scale factor at the natural origin |
|
E |
Easting measured from the grid origin |
|
N |
Northing measured from the grid origin |
|
FE |
false easting, the Eastings value assigned to the natural origin |
|
FN |
false northing, the Northings value assigned to the natural origin |
|
EF |
Eastings value assigned to the false origin |
|
NF |
Northings value assigned to the false origin |
|
EC |
Eastings value assigned to the projection center |
|
NC |
Northings value assigned to the projection center |
(Note that the origin of most projected coordinate systems is given false coordinates to avoid negative coordinates. In the formulas which follow these values, (FE and FN or EF and NF or EC and NC) are included where appropriate so that the projected coordinates of points result directly from the quoted formulas).
| | |
Table of Contents |