LOCAL GEODETIC HORIZON COORDINATES

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Anabel Morgan
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1 LOCAL GEODETIC HOIZON COODINATES In many surveying applications it is necessary to convert geocentric Cartesian coordinates X,,Z to local geodetic horizon Cartesian coordinates E,N,U (East,North,Up). Figure 1 shows a portion of a reference ellipsoid (defined by semimajor axis a and flattening f) approximating the size and shape of the Earth. The origin of the X,,Z coordinates lies at O, the centre of the ellipsoid (assumed to be the Earth's centre of mass, hence the name Geocentric). The Z-axis is coincident with the Earth's rotational axis and the X-Z plane is the Greenwich meridian plane (the origin of longitudes λ ). The X- plane coincides with the Earth's equatorial plane (the origin of latitudes ) and the positive X-axis is in the direction of the intersection of the Greenwich meridian plane and the equatorial plane. The positive -axis is advanced 9 o east along the equator. Z Greenwich Meridian a O H λ normal D N Q h U E X Equator A Figure 1 Geocentric and Local coordinate axes and the reference ellipsoid A point on the Earth's terrestrial surface is referenced to the ellipsoid via the normal that passes through and intersects the ellipsoid at Q. The normal through intersects the equatorial plane at D and cuts the Z-axis at H. The angle between the normal and the equatorial plane is the latitude ( o to 9 o positive north, negative south). The height of the point above the ellipsoid (measured along the normal) is the ellipsoidal height h. C:\rojects\Geospatial\Geodesy\Local Horizon Coords\LOCAL HOIZON COODINATES.doc 1

2 The angle between the Greenwich meridian plane and the meridian plane of the point (the plane containing the normal and the Z-axis) is the longitude λ ( o to 18 o positive east, negative west). Geocentric Cartesian coordinates are computed from the following equations ν = a 1 e sin X = ( ν + h) coscosλ = ( ν + h) cossin λ (1) ( ν( ) ) Z = 1 e + h sin is the radius of curvature of the ellipsoid in the prime vertical plane. In Figure 1, ν = QH e = f ( f) is the square of the eccentricity of the ellipsoid. The origin of the E,N,U system lies at the point (, λ, h ). The positive U-axis is coincident with the normal to the ellipsoid passing through and in the direction of increasing ellipsoidal height. The N-U plane lies in the meridian plane passing through and the positive N-axis points in the direction of North. The E-U plane is perpendicular to the N-U plane and the positive E-axis points East. The E-N plane is often referred to as the local geodetic horizon plane. Geocentric and local Cartesian coordinates are related by the matrix equation X,, Z system and λ rotation matrices. U X X E = λ N Z Z are the geocentric Cartesian coordinates of the origin of the E,N,U is a rotation matrix derived from the product of two separate cos sin cosλ sin λ λ = λ = 1 sinλ cosλ (3) sin 1 cos () C:\rojects\Geospatial\Geodesy\Local Horizon Coords\LOCAL HOIZON COODINATES.doc

3 The first, λ (a positive right-handed rotation about the Z-axis by λ ) takes the X,,Z axes to X,, Z. The Z -axis is coincident with the Z-axis and the X plane is the Earth's equatorial plane. The X Z plane is the meridian plane passing through and the -axis is perpendicular to the meridian plane and in the direction of East. Z(Z ) λ cos λ sin λ X cos λ X sin λ cosλ sin λ = sin λ cosλ Z 1 Z λ X X The second (a rotation about the '-axis by ) takes the X,, Z axes to the X,, Z axes. The X a xis is parallel to the U-axis, the axis is parallel to the E-axis and the Z axis is parallel to the N-axis. Z X X X cos X sin Z sin Z cos Z cos sin = 1 Z sin cos Z ( ) C:\rojects\Geospatial\Geodesy\Local Horizon Coords\LOCAL HOIZON COODINATES.doc 3

4 erforming the matrix multiplication in equation (3) gives λ coscos λ cos sin λ sin = sin λ cosλ (4) sin cosλ sin sin λ cos otation matrices formed from rotations about coordinate axes are often called Euler rotation matrices in honour of the Swiss mathematician Léonard Euler ( ). They are orthogonal, satisfying the condition T 1 T = I(i.e., = ). A re-ordering of the rows of the matrix usual form E,N,U λ gives the transformation in the more E X X N = U Z Z (5) sin λ cosλ = sin cosλ sin sin λ cos cos cosλ cos sin λ sin (6) From equation (5) we can see that coordinate differences E = E E, N = Nk Ni and k U = U U i in the local geodetic horizon plane are given by E X N = (7) U Z X = Xk Xi, = k i and Z = Zk Zi are geocentric Cartesian coordinate differences. k i C:\rojects\Geospatial\Geodesy\Local Horizon Coords\LOCAL HOIZON COODINATES.doc 4

5 NOMAL SECTION AZIMUTH ON THE ELLISOID The matrix relationship given by equation (7) can be used to derive an expression for the azimuth of a normal section between two points on the reference ellipsoid. The normal section plane between points and on the Earth's terrestrial surface contains the normal at point, the intersection of the normal and the rotational axis of the ellipsoid at (see Figure 1) and. This plane will intersect the local geodetic horizon plane in a line having an angle with the north axis, which is the direction of the meridian at 1. This angle is the azimuth of the normal section plane denoted as A and will have components E and N in the local 1 1 geodetic horizon plane. From plane geometry 1 H1 1 tan A 1 E = (8) N By inspection of equations (6) and (7) we may write the equation for normal section azimuth between points and as tan A 1 1 E X sin λ + cosλ = = 1 1 N X sin 1cos λ1 sin 1sin λ1 + Z cos1 X = X X1, and = 1 Z = Z Z1 (9) C:\rojects\Geospatial\Geodesy\Local Horizon Coords\LOCAL HOIZON COODINATES.doc 5

Navigation coordinate systems

Lecture 3 Navigation coordinate systems Topic items: 1. Basic Coordinate Systems. 2. Plane Cartesian Coordinate Systems. 3. Polar Coordinate Systems. 4. Earth-Based Locational Reference Systems. 5. Reference