Geoscience Reference
In-Depth Information
Chapter 5
Reference Ellipsoid and the Geodetic
Coordinate System
The shape of the Earth approximates a regular ellipsoid, which can therefore be
used to represent the mathematical shape of the geoid. One can also establish a one-
to-one correspondence between the points on the Earth's surface and the points on
the ellipsoid that is used as the reference surface. Based on some relevant mathe-
matical properties of the ellipsoid, this chapter discusses the methods for reducing
the elements of terrestrial triangulation and trilateration to a reference ellipsoid and
establishes the models to transform mutually between the geodetic coordinate
system, geodesic polar coordinate system, and geodetic Cartesian coordinate sys-
tem (geodetic spatial rectangular coordinate system).
5.1 Fundamentals of Spherical Trigonometry
5.1.1 Spherical Triangle
A spherical triangle is a closed figure formed on the surface of a sphere that is
bounded by three arcs of great circles. The great circle is defined to be the
intersection of a sphere with a plane containing the center of the sphere
(Fig.
5.1
). The three arcs of great circles are called the sides of the spherical
triangle, denoted by lowercase letters a, b, and c. The spherical angles formed by
the arcs of great circles are called the angles of the spherical triangle, denoted by the
uppercase letters A, B, and C.
A trihedron O-ABC is formed by connecting the vertices of the spherical triangle
ABC with the center of the sphere O (Fig.
5.1
). The radian measure of a central
angle of a circle is equivalent to the length of the arc the angle subtends, which
yields:
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