Geometric Geodesy

GEODESY

Chapter III

Civil Engineering and Surveying

Department [CESD]

Properties of Reference Ellipsoid

Figure 1 show a schematic view of the reference ellipsoid

upon which meridians (curves of constant longitude λ) and

parallels (curves of constant latitude φ) form an orthogonal

network of reference curves on the surface. This allows a

point P in space to be coordinated via a normal to the

ellipsoid passing through P. This normal intersects the

surface at Q which has coordinates of λ, φ and P is at a

height h= QP above the ellipsoid surface. We say that P

has geodetic coordinates (λ, φ, h).

• The important thing at this stage is that the ellipsoid is a surface

of revolution created by rotating an ellipse about its minor axis,

where this minor axis is assumed to be either the Earth's

rotational axis, or a line in space close to the Earth's rotational

axis.

• Meridians of longitude are curves created by intersecting the

ellipsoid with a plane containing the minor axis and these curves

are ellipses; as are all curves on the ellipsoid created by

intersecting planes. Note here that parallels of latitude (including

the equator) are circles; since the intersecting plane is

perpendicular to the rotational axis, and circles are just special

cases of ellipses.

THE ELLIPSE

• The ellipse is one of the conic sections; a name derived

from the way they were first studied, as sections of a

cone.

• A right-circular cone is a solid whose surface is obtained

by rotation a straight line, called the generator, about a

fixed axis.

1. Cartesian equations of the ellipse

An ellipse is the locus of a point Pk that moves so that the sum of

the distances r and r' from two fixed points F and F' (the foci)

separated by a distance 2a is a constant and equal to the major

axis of the ellipse, i.e.,

r + r’ = 2a

a is the semi-major

b is the semi-minor

d = OF = OF’ is the focal distance

2. Parametric Equation of Ellipse

If auxiliary circles x2 + y2 = a2, and x2 + y2 = b2 are drawn on a common

origin O of an x, y coordinate system and radial lines are drawn at angles

from the x-axis; then the ellipse is the locus of points Pk that lie at the

intersection of lines, parallel with the coordinate axes, drawn through the

intersections of the radial lines and auxiliary circles.

This definition leads to the parametric equation of the ellipse. Consider

points A (auxiliary circle) and P (ellipse) on Figure 5. Using equation (4)

and the equation for the auxiliary circle of radius a we may write

Now the x- coordinates of A and P are the same and so the

right-hand sides of equations (5) may be equated, giving

Similarly, considering points B and P; using equation of

Cartesian and the equation for the auxiliary circle of radius

of b, derive the same parametric equations of ellipse.

Using parametric equation derive the Cartesian equation of

ellipse.