GeometricalGeodesy
Thomas H. Meyer, Ph.D.
Department of Natural Resources Management &
Engineering
University of Connecticut
Surveying is…
 “Measuring distance, angles, heights,
etc., to determine the relative locations
of points on the Earth” -- NGS Glossary.
 what you do to make a map
 applied trigonometry
Technology
1. Global Positioning System
Technology
2. Optomechanical
The job is to
 collect observations of features to be
mapped and thereby determine their
coordinates
 observations are things, such as angles
and distances, a surveyor “observed”
using an instrument
 maps are drawn using the coordinates of
the features
Surveying Observables
 zenith angle
Surveying Observables, cont.
 bearings

angles in the horizontal plane between features of
interest
 slant distances
Derived Quantities
 horizontal distance
 vertical distance
bounds
metes
Suppose…
 you have a properly leveled instrument.
 you site an object in the distance…
 at a zenith angle of exactly 90 degrees.
 Is that object above, below or level to you?
How do you know the Earth is
round?
The Flat-Earth Assumption
 causes objects to appear lower and
farther away than they really are.
c  a so   c / a
cos  a /(a  h) so
cos(c / a)  a /(a  h)
a  h  a sec(c / a)
h  a sec(c / a)  a
sec x  1  x 2  5 x 4 / 24  O( x 6 ) so
h  a sec(c / a)  a  (c / a) 2  O(c / a) 4
Is this a practical problem?
 if a = 6378137 m
 then a one centimeter error occurs at
 c = 357.159 m
 Typical surveying accuracy for a 3500 m
line is around one millimeter
Ok, so far we’ve been using a
coordinate system…
 it’s Cartesian
 In what reference frame does that
coordinate system reside within?
Geodesy
“The science concerned with determining
the size and shape of the Earth.”
--NGS Glossary
Why study geodesy?
With inches to spare…
going aground in the Panama Canal is bad…
fender benders…
Barge-Bridge Collision
A horizontal geodetic
datum..
 is a system by which latitude and
longitude coordinates are assigned to
places of interest
 i.e., the foundation of spatial coordinates
 note bene: height is not included
Surveying on the Sphere
 To construct and use geodetic datums
we must be able to determine latitude
and longitude coordinates from
surveying observations.
 That requires spherical trigonometry
Spherical
Triangles
Our fundamental figure
Geodetic Coordinates
Radii of Curvature in Geodesy
 radius of curvature in the meridian
 radius of curvature in the prime vertical
 radius of curvature in the normal section
Radius of Curvature in the
Meridian
 ( ) 
2
a(1  e )
2
1  e
2
d    ( )d
1
sin 
2

3/ 2
Radius of Curvature in the
Prime Vertical
Radius of Curvature in the
Prime Vertical
 ( ) 
2
1  e
a
2
d   ( ) d
1
sin 
2

1/ 2
Normal Section
Radius of Curvature in the
Normal Section
 ( ) ( )
 ( ,  ) 
2
2
 ( ) sin   ( ) cos 
2
d   ( ,  )d
1
GeometricalGeodesy
Thomas H. Meyer, Ph.D.
Department of Natural Resources Management &
Engineering
University of Connecticut
Thomas.meyer@uconn.edu