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Faculty of Applied Engineering and Urban Planning Civil Engineering Department Introduction to Geodesy and Geomatics The Ellipsoid 2 nd Semester 2008/2009

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Computational and Geometric Geodesy Basic Ellipsoidal Geometry Ellipsoidal Coordinates Elementary Differential Geodesy Direct / Inverse Problem Transformation Between Geodetic and Cartesian Coordinates

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THE GEOID AND TWO ELLIPSOIDS GRS80-WGS84 CLARKE 1866 GEOID Earth Mass Center Approximately 236 meters

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NAD 83 and ITRF / WGS 84 ITRF / WGS 84 NAD 83 Earth Mass Center 2.2 m (3-D) dX,dY,dZ GEOID

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Basic Ellipsoidal Geometry The ellipsoid is formed by rotating an ellipse about its minor axis, which for present purposes we assume to be parallel to the Earth’s spin axis.

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Basic Ellipsoidal Geometry The basic geometric construction of an ellipse is as follows: for any two points, F1 and F2, called focal points, the ellipse is the locus (path) of points, P, such that the sum of the distances PF 1 + PF2 is a constant.

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Basic Ellipsoidal Geometry You can draw it by hand..

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Basic Ellipsoidal Geometry Introducing a coordinate system (x, z ) with origin halfway on the line F1 F2 and z –axis perpendicular to F1 F2, we see that if P is on the x -axis, then that constant is equal to twice the distance from P to the origin; this is the length of the semi-major axis; call it a :

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Basic Ellipsoidal Geometry Moving the point, P, to the z -axis, and letting the distance from the origin point to either focal point (F1 or F2 ) be E, we also find that E is called the linear eccentricity of the ellipse (and of the ellipsoid).

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Basic Ellipsoidal Geometry From these geometrical considerations it is easy to prove that the equation of the ellipse is given by Assignment: Prove ellipse equation

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Basic Ellipsoidal Geometry We see that the ellipse, and hence the ellipsoid, is defined by two essential parameters: a shape parameter and a size (or scale) parameter. The following are also used; in particular, the flattening: the first eccentricity: the second eccentricity:

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Basic Ellipsoidal Geometry We also have the following useful relationships among these parameters

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Basic Ellipsoidal Geometry When specifying a particular ellipsoid, we will, in general, denote it by the pair of parameters, a,f. Geodetic Reference System of 1980 (GRS80) and has parameter values given by a = 6378137 m f = 1 / 298.257222101

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Latitudes There are many different latitudes, we are concerned here with only three of them: The geodetic latitude The geocentric latitude The reduced latitude

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Latitudes The geocentric latitude of P is the angle, ψ, at the origin and in the meridian plane from the equator to the radial line through P. Note, however, that the geocentric latitude is independent of any defined ellipsoid and is identical to the complement of the polar angle defined earlier for the spherical coordinates.

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Latitudes The most common latitude used in geodesy, appropriately called the geodetic (geographic) latitude. This is the angle, φ, in the meridian plane from the equator to the line through P that is also perpendicular to the basic ellipsoid a, f. also called the normal latitude.

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Latitudes lines of reduced or parametric latitude, β, form circles whose radii are the same as the radii of circles formed by the corresponding lines of latitude on a sphere with radius equal to the equatorial radius of the spheroid.

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Geodetic latitude Geocentric latitude Parametric latitude Unlike the sphere, the ellipsoid does not possess a constant radius of curvature.

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Relationship between Latitudes

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Assignment

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Radii of Curvature Consider a curve on a surface, for example a meridian arc or a parallel circle on the ellipsoid, or any other arbitrary curve. The meridian arc and the parallel circle are examples of plane curves, curves that are contained in a plane that intersects the surface. The amount by which the tangent to the curve changes in direction as one moves along the curve indicates the curvature of the curve.

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Radii of Curvature The curvature, χ, of a plane curve is the absolute rate of change of the slope angle of the tangent line to the curve with respect to arc length along the curve.

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Radii of Curvature Note that the curvature has units of inverse- distance. The reciprocal of the curvature is called the radius of curvature, ρ We may think of the radius of curvature at a point of an arbitrary curve as being the radius of the circle tangent to the curve at that point and having the same curvature.

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Radii of Curvature Note that the curvature has units of inverse- distance. The reciprocal of the curvature is called the radius of curvature, ρ We may think of the radius of curvature at a point of an arbitrary curve as being the radius of the circle tangent to the curve at that point and having the same curvature.

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Radii of Curvature

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GRS80

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Normal curvature

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