Global Positioning System Reference
In-Depth Information
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A
PPENDIX A
[Fir
[34
G
ENERAL BACKGROUND
Lin
—
1
——
Sho
*PgE
This appendix provides mathematical material that is handy to have available in a
classroom situation to support key derivations or conclusions of the main chapters.
It begins with a listing of expressions from spherical trigonometry. The rotation
matrices are given along with brief definitions of positive and negative rotations.
The sections on linear algebra, linearization, and statistics contain primary reference
material for the least-squares adjustment given in Chapter 4. The subsection on the
distribution of sums of variables is particularly useful when deriving the distribution
of
v
T
Pv
.
[34
A.1 SPHERICAL TRIGONOMETRY
The sides of a spherical triangle are defined by great circles. A great circle is an
intersection of the sphere with a plane that passes through the center of the sphere.
It follows from geometric consideration of the special properties of the sphere that
great circles are normal sections and geodesic lines. Figure A.1 shows a spheri-
cal triangle with corners
(A,B,C)
, sides
(a,b,c)
, and angles
(
)
. Notice that
the sequence of the elements in the respective triplets is consistent, counterclock-
wise in this case. The sides of the spherical triangle are given in angular units.
In many applications, one of the corners of the spherical triangle represents the
North or South Pole. Documentation of the expressions listed below is readily avail-
able from the mathematical literature. Complete derivations can be found in Sigl
(1977).
α
,
β
,
γ
340
