# The Ellipsoid Faculty of Applied Engineering and Urban Planning

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

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

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

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

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.

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.

Basic Ellipsoidal Geometry
You can draw it by hand ..

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 :

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).

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

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:

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

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 = m f = 1 /

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

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.

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.

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.

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

Relationship between Latitudes

Assignment

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.

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.

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.

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.