Lecture 11: Geometry of the Ellipse

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Lecture 11 Geometry of the Ellipse

• 25 February 2008
• GISC-3325

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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

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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

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Level Project
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Gravity values for points
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Helmert Orthometric Heights
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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!

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Parameters

• a semi-major axis length
• b semi-minor axis length
• f flattening (a-b)/a
• e first eccentricity v((a2-b2)/a2)
• e second eccentricity v((a2-b2)/b2)

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THE ELLIPSOIDMATHEMATICAL MODEL OF THE EARTH
N
b
a
S
a Semi major axis b Semi minor axis f
a-b Flattening a
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THE GEOID AND TWO ELLIPSOIDS

CLARKE 1866
GRS80-WGS84
Earth Mass Center
Approximately 236 meters
GEOID
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NAD 83 and ITRF / WGS 84

NAD 83
ITRF / WGS 84
Earth Mass Center
2.2 m (3-D) dX,dY,dZ
GEOID
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Geodetic latitude Geocentric latitude Parametric
latitude
Unlike the sphere, the ellipsoid does not possess
a constant radius of curvature.
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Radius of Curvature of the Prime Vertical
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