1. General information Based on book Datums and Map Projections for Remote Sensing, GIS and Surveying 2.2.2. Coordinate systems for the sphere and ellipsoid Three-dimensional coordinates can be defined with respect to a sphere as: LATITUDE: the angle north or south from the equatorial plane LONGITUDE: the angle east or west from an identified meridian HEIGHT: a distance above the surface of the sphere

2. The definition of latitude is a natural one, in that the poles are clearly defined points on the surface of the sphere, and the equator is the circle that bisects the two. On the other hand, the definition of longitude is to some extent arbitrary, as there are no physical reasons for choosing a particular meridian as the reference. That chosen is called the prime meridian.

3. Different prime meridians have been used in history. Different countries have had their own prime meridians like: Ferro (in Canarian Islands) Oslo Paris Nowadays we use Greenwich as the international standard. The identification of the prime meridian is a part of datum definition.

4. PARALLELS AND MERIDIANS • PARALLELS: lines on the surface of the sphere parallel to the equator. These are lines of equal latitude. • MERIDIANS: lines on the surface of the sphere running from pole to pole. These are lines of equal longitude.

5. GRATICULE • Parallels and meridians intersect each other at 90°. A lattice of parallels and meridians is known as a graticule. • A similar system for depicting position can be employed on the ellipsoid. The coordinates are defined as: • latitude, • longitude and • ellipsoidal height.

6. HEIGHTS • The adjective ellipsoidal height is applied to height to distinguish it from height above the geoid. • In flat areas this formula is valid: • h = H + N, where • h is the ellipsoidal height • N is geoidal height and • H is the orthometric height (mean sea level)

7. Axis direction • Axis direction: • for latitude, angles measured northward from the equator is positive • for latitude, angles measured southward from the equator is negative

8. Axis direction • for longitude, angles measured eastward from the prime meridian are positive • for longitude, angles measured westward from the prime meridian are negative.

9. Axis directions • For height, distances measured up from the surface of the ellipsoid are positive, • for height, distances measured down from the surface of the ellipsoid are negative.

10. Axis units: • latitude and longitude are typically expressed in degrees ( degrees, minutes and seconds) • with hemisphere indicated and ellipsoidal height in meters.

11. Other units • In surveying also grads (1/400 th of a circle) is used • (basic surveying, municipalities) • In the army there is also their own circle system, • a circle is divided into 6000 parts.

12. Conversion between ellipsoidal and geocentric Cartesian coordinates • X = (N+h)cosφcosλ • Y = (N+h)cosφsinλ • Z = ((1-e^2)N+h)sinφ, where φ is the latitude λ is the longitude h is the ellipsoidal heigth e is the eccentricity N is the radius of curvature in the prime vertical

13. Rotation ellipsoid orellipsoid of revolution • Surface is formed by an ellipse that has been rotated around its shortest axis b • Semiminor axis b • Semimajor axis a • Oblate ellipsoid • Flattening f = (a-b)/b • Usually given inverse flattening

14. Ellipsoid • Other ways: • b=a(1-e^2)^½ • b=a(1-f) • c=a^2/b = a/(1-f) • n=(a-b)/(a+b) • n=f/(2-f)

15. ellipsoid • e^2=(a^2-b^2)/a^2 (first eccentricity squared) • e´^2=(a^2-b^2)/b^2 (second ecc. squared) • e^2=f(2-f) • e^2=4n/(1-n)^2 • f=(a-b)/a