3D Theory - Referencing

This is about how we measure the position on the surface of the earth, or possibly another planet.

If we start by making an assumption that the earth is a perfect sphere, in this case we can use Spherical Polar Coordinates, as defined on this page. We can define a position on the earth using two angles:

We can assign an arbitrary x,y,z coordinate system in the local frame of the earth:

so from this diagram we can see that:

z = r sin(latitude)

and if we are on the Greenwich meridian then:

x = r cos(latitude)

but if we are not on the Greenwich meridian then this has to be modified depending on the latitude, so,

x = r cos(latitude) cos (longitude)

the y can be calculated from:

r2 = x2 + y2 + z2

therefore y = r *sqrt(1 - sin(latitude) - cos(latitude) cos (longitude))

where:

r = radius of the earth (approx. 6378135 metres)

Ellipsoid

The earth is not quite spherical, it is slightly flattened at the poles relative to the equator, this shape is known as an ellipsoid. This is a closer approximation, but not perfect, so many ellipsoids have been defined for different regions of the world.

Earth ellipsoids supported by x3d

x3d Code Ellipsoid Name a - Equatorial Radius (metres) f - Flattening
AA
Airy 1830 6377563.396 1/299.3249646
AM
Modified Airy 6377340.189 1/299.3249646
AN
Australian National 6378160 1/298.25
BN
Bessel 1841 (Namibia) 6377483.865 1/299.1528128
BR
Bessel 1841 (Ethiopia Indonesia...) 6377397.155 1/299.1528128
CC
Clarke 1866 6378206.4 1/294.9786982
CD
Clarke 1880 6378249.145 1/293.465
EA
Everest (India 1830) 6377276.345 1/300.8017
EB
Everest (Sabah & Sarawak) 6377298.556 1/300.8017
EC
Everest (India 1956) 6377301.243 1/300.8017
ED
Everest (W. Malaysia 1969) 6377295.664 1/300.8017
EE
Everest (W. Malaysia & Singapore 1948) 6377304.063 1/300.8017
EF
Everest (Pakistan) 6377309.613 1/300.8017
FA
Modified Fischer 1960 6378155 1/298.3
HE
Helmert 1906 6378200 1/298.3
HO
Hough 1960 6378270 1/297
ID
Indonesian 1974 6378160 1/298.247
IN
International 1924 6378388 1/297
KA
Krassovsky 1940 6378245 1/298.3
RF
Geodetic Reference System 1980 (GRS 80) 6378137 1/298.257222101
SA
South American 1969 6378160 1/298.25
WD
WGS 72 6378135 1/298.26
WE
WGS 84 6378137 1/298.257223563


We can assign an arbitrary x,y,z coordinate system in the local frame of the earth:

so from this diagram we can see that:

z = a * (1 -f) *sin(latitude)

and if we are on the Greenwich meridian then:

x = a cos(latitude)

but if we are not on the Greenwich meridian then this has to be modified depending on the latitude, so,

x = a cos(latitude) cos (longitude)

the y can be calculated from:

r2 = x2 + y2 + z2

therefore y = a *sqrt(1 - (1 -f) *sin(latitude) - cos(latitude) cos (longitude))

where:


metadata block
see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

cover Elements of Cartography.

Commercial Software Shop

Where I can, I have put links to Amazon for commercial software, not directly related to this site, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.

cover Palm Treo 650 GSM/GPRS Smartphone

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2015 Martin John Baker - All rights reserved - privacy policy.