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Geometry of the Ellipse: Earth Modeling and Parameters

Explore the ellipsoid model of the Earth, including its mathematical representation, parameters like semi-major and minor axis, flattening, and eccentricity. Learn about geoid variations, geodetic and geocentric latitude, and differences between Clarke 1866 and GRS80-WGS84 models. Understand the curvature of ellipsoids and their significance in geospatial calculations.

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Geometry of the Ellipse: Earth Modeling and Parameters

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  1. Lecture 11: Geometry of the Ellipse 25 February 2008 GISC-3325

  2. Class Update • Next exam 12 March 2008 • Labs 1-4 due today! • Homework 2 due 3 March 2008 • Will have exams graded by next Monday • Will post solutions to class web page

  3. Note on orthometric heights • Orthometric height differences are provided by leveling ONLY when there is parallelism between equipotential surfaces. • Over short distances this may be the case. • To account for non-parallelism we use geopotential numbers in computations. • In general, geopotential surfaces are NOT parallel in a N-S direction but are E-W

  4. Level Project

  5. Gravity values for points

  6. Helmert Orthometric Heights

  7. Geometry of the Ellipsoid • Ellipsoid of revolution is formed by rotating a meridian ellipse about its minor axis thereby forming a 3-D solid, the ellipsoid. • Modern models are chosen on the basis of their fit to the geoid. • Not always the case!

  8. Parameters • a = semi-major axis length • b = semi-minor axis length • f = flattening = (a-b)/a • e = first eccentricity = √((a2-b2)/a2) • e’ = second eccentricity = √((a2-b2)/b2)

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

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

  11. NAD 83 and ITRF / WGS 84 NAD83 ITRF / WGS 84 Earth Mass Center 2.2 m (3-D) dX,dY,dZ GEOID

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

  13. Radius of Curvature of the Prime Vertical

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