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Class 14: Ellipsoid of Revolution

GISC-3325 3 March 2009. Class 14: Ellipsoid of Revolution. 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 ELLIPSOID MATHEMATICAL MODEL OF THE EARTH. N.

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Class 14: Ellipsoid of Revolution

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  1. GISC-3325 3 March 2009 Class 14: Ellipsoid of Revolution

  2. 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.

  3. THE ELLIPSOIDMATHEMATICAL MODEL OF THE EARTH N b a S a = Semi major axis b = Semi minor axis f = a-b = Flattening a

  4. 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)

  5. 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

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

  7. 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.

  8. a*(1-e2)*sin(lat)

  9. Values of a*cos(lat)

  10. Values of sqrt(1-e2*sin2(lat))

  11. Parametric Latitude

  12. Comparison of latitudes

  13. Converting latitudes from geodetic • Parametric latitude = arctan(√(1-e2)*tan(lat)) • Geocentric latitude = arctan( (1-e2)*tan(lat))

  14. 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.

  15. 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

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