Lecture 11: Geometry of the Ellipse

25 February 2008

GISC-3325

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

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

Level Project

Gravity values for points

Helmert Orthometric Heights

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!

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)

THE ELLIPSOIDMATHEMATICAL MODEL OF THE EARTH

b

a

a = Semi major axis b = Semi minor axis f = a-b = Flattening a

N

S

THE GEOID AND TWO ELLIPSOIDS

GRS80-WGS84CLARKE 1866

GEOID

Earth Mass Center

Approximately 236 meters

NAD 83 and ITRF / WGS 84

ITRF / WGS 84NAD 83

Earth Mass Center

2.2 m (3-D) dX,dY,dZ

GEOID

Geodetic latitude

Geocentric latitude

Parametric latitude

Unlike the sphere, the ellipsoid does not possess a constant radius of curvature.

Radius of Curvature of the Prime Vertical