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