Class 14: Ellipsoid of Revolution
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Transcript of Class 14: Ellipsoid of Revolution
Class 14: Ellipsoid of Revolution
GISC-33253 March 2009
Class status
Read text chapters 5 and 6 Second exam cumulative on 12 March 2009
during lab period. Homework 1 on web page due 12 March 2009, Reading assignments due 16 April 2009.
THE ELLIPSOIDMATHEMATICAL MODEL OF THE EARTH
b
a
a = Semi major axis b = Semi minor axis f = a-b = Flattening a
N
S
Geometric Parameters
a = semi-major axis length b = semi-minor axis length f = flattening = (a-b)/a e = first eccentricity = √((a2-b2)/a2) alternately
e2 = (a2-b2)/a2
e’ = second eccentricity = √((a2-b2)/b2)
Three Latitudes
• Geodetic latitude included angle formed by the intersection of the ellipsoid normal with the major (equatorial) axis.
• Geocentric latitude included angle formed by the intersection of the line extending from the point on the ellipse to the origin of axes.
• Parametric (reduced) latitude is the included angle formed by the intersection of a line extending from the projection of a point on the ellipse onto a concentric circle with radius = a
Geodetic latitude
Geocentric latitude
Parametric latitude
Unlike the sphere, the ellipsoid does not possess a constant radius of curvature.
Radius of Curvature in Prime Vertical
• N extends from the minor axis to the ellipsoid surface.
• N >= M
• It is contained in a special normal section that is oriented 90 or 270 degrees to the meridian.
a*(1-e2)*sin(lat)
Values of a*cos(lat)
Values of sqrt(1-e2*sin2(lat))
Parametric Latitude
Comparison of latitudes
Converting latitudes from geodetic
• Parametric latitude = arctan(√(1-e2)*tan(lat))
• Geocentric latitude = arctan( (1-e2)*tan(lat))
Quadrant of the Meridian
• The meridian arc length from the equator to the pole.
• Simplified formula S0 = [ a / (1 + n) ](a0phi radians)– where n = f / (2-f)
– a0 = 1 + n2/4 + n4/64
• Meter was originally defined as one ten-millionth of the Quadrant of the Meridian.
• Use the NGS tool kit to determine (using Clarke 1866) how well they did.
Geodesic
• Analogous to the great circle on the sphere in that it represents the shortest distance between two points on the surface of the ellipsoid.
• Term representing the shortest distance between any two points lying on the same surface.
– On a plane: straight line– On a sphere: great circle