6Basic 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.
7Basic 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.
8Basic Ellipsoidal Geometry You can draw it by hand ..
9Basic 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 :
10Basic 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 thatE is called the linear eccentricity of the ellipse (and of the ellipsoid).
11Basic Ellipsoidal Geometry From these geometrical considerations it is easy to prove that the equation of the ellipse is given byAssignment: Prove ellipse equation
12Basic 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:
13Basic Ellipsoidal Geometry We also have the following useful relationships among these parameters
14Basic 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 bya = mf = 1 /
18LatitudesThere are many different latitudes, we are concerned here with only three of them:The geodetic latitudeThe geocentric latitudeThe reduced latitude
19LatitudesThe 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.
20LatitudesThe 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.
21Latitudeslines 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.
22Geodetic latitudeGeocentric latitudeParametric latitudeUnlike the sphere, the ellipsoid does not possess a constant radius of curvature.
25Radii of CurvatureConsider 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.
26Radii of CurvatureThe 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.
27Radii of CurvatureNote that the curvature has units of inverse- distance. The reciprocal of the curvature is called the radius of curvature, ρWe 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.
28Radii of CurvatureNote that the curvature has units of inverse- distance. The reciprocal of the curvature is called the radius of curvature, ρWe 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.