Observed Angles and
Spherical Excess
Learning Objectives
After completing this lecture you will be
able to:
 Illustrate concept of spherical excess
with the aid of a suitable diagram
 Apply spherical excess to geodetic
figures
Lecture Outline
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Introduction
Normal Sections
Curve of Alignment
Observed Angles
Spherical Excess
Conclusion
Introduction
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Want to read an angle and calculate
forward azimuth from known back
azimuth
But what line do we measure angles to?
If you said straight line between points,
what is this?
– Remember normal sections
Plane Sections (Normal
Sections)
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Instrument set at B
Rotation axis is
normal BN
Vertical plane
containing A = ABN.
Instrument set at A
Rotation axis is
normal AM
Vertical plane
containing B = BAM
Line A B 
Line B A
A
B
M
N
Curve of Alignment
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Locus of all points where Bearing to A = bearing to
B + 180 is called Curve of Alignment.
Marked on ground - A surveyor sets up between A
and B such that A and B are in same vertical plane
Horizontal angles are angles between curves of
alignment
– But can assume normal sections because start off same
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Spheroidal triangles are figures formed by 3 curves
of alignment joining the 3 points
A
Normal Section
A to B
B
Curve of Alignment
Normal Section
B to A
Observed Angles and Azimuth
Observed Angles
Note that a geodesic is not a line of sight and
therefore we can’t measure angles to it!
Spherical Excess
Spherical Triangle
Spherical Triangle
Spherical Excess
Practical Implication
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All geodetic figures will have spherical
excess
Apportion evenly throughout angles.
Conclusion
You can now:
 Illustrate what spherical excess with the
aid of a suitable diagram
 Apply spherical excess to geodetic
figures
Self Study
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Read relevant module in study materials
Follow numerical example
Review Questions