The document discusses properties of reference ellipsoids used in geodesy. It describes how an ellipsoid is a surface of revolution created by rotating an ellipse about its minor axis, with meridians of longitude being ellipses and parallels of latitude being circles. It then provides details on the mathematical definition of an ellipse as a conic section, including Cartesian equations relating the distances from two fixed points (foci) and parametric equations derived by considering intersections with auxiliary circles.

1. Geometric Geodesy
GEODESY
Chapter III
Civil Engineering and Surveying
Department [CESD]

2. Properties of Reference Ellipsoid

3. 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).

4. • 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.

5. 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.

9. 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

13. 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

14. Now the x- coordinates of A and P are the same and so the
right-hand sides of equations (5) may be equated, giving

16. 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.