Coordinate
Systems
• Required readings:
• Coordinate systems:19-1 to 19-6.
• State plane coordinate systems: 20-1 to 20-5, 20-7, 20-
10, and 20-12.
• Required figures:
• Coordinate systems: 19-1, 19-2, 19-6, 19-7 and 19-8.
• State plane coordinate systems: 20-1 to 20-3, 20-10.
• Recommended, not required, readings: 19-7 to 19-11,
20-11, and 20-13.
Coordinate Systems
• Geoid and Ellipsoid, what for?
Ellipsoid Parameters
• Ellipsoid parameters (equations not required):
• semi-major axes (a), semi-minor axes (b)
• e = a 2 b 2 = first eccentricity
b2
a
• 2
N = normal length = 1 e sin 2
• Great circles and meridians
• Two main ellipsoids in North America:
• Clarke ellipsoid of 1866, on which NAD27 is based
• Geodetic Reference System of 1980 (GRS80): on
which NAD83 is based.
• For lines up to 50 km, a sphere of equal volume can be
used
Geodetic Coordinate System
• System components, coordinates
Geodetic System Coordinates
• Definitions :
– Geodetic latitude (f): the angle in the meridian
plane of the point between the equator and the normal
to the ellipsoid through that point.
– Geodetic longitude (l): the angle along the equator
between the Greenwich and the point meridians
– Height above the ellipsoid (h)
Universal Space Rectangular System
• System definition, X, Y, Z
• Advantage and disadvantage
• X, Y, Z from geodetic coordinates
Z
X = (N+h) cosf cosl
Y = (N+h) cosf sinl
Y
Z = ( N(1-e2) +h) sinf
X
State Plane Coordinate Systems
• Plane rectangular systems, why use them?
• How to construct them: Project the earth’s
surface onto a developable surface.
• Two major projections: Lambert Conformal
Conic, and Transverse Mercator.
Coneann.gif
Secants, Scales, and Distortions
• Scale is exact along the secants, smaller than
true in between.
• Distortions are larger away from the secants
Choosing a Projection
• States extending East-west: Lambert Conical
• States extending North-South: Mercator
Cylindrical.
• A single surface will provide a single zone.
Maximum zone width is 158 miles to limit
distortions to 1:10,000. States longer than 158 mi,
use more than one zone (projection).
Standard Parallels & Central
Meridians
• Standard Parallels: the secants, no distortion
along them. At 1/6 of zone width from zone edges
• Central Meridians: a meridian at the middle of
the zone, defines the direction of the Y axis.
• The Y axis points to the grid north, which is the
geodetic north only at the central meridian
• To compute the grid azimuth ( from grid north)
from geodetic azimuth ( from geodetic north):
grid azimuth = geodetic azimuth - q
Geodetic and SPCS
• Control points in
SPCS are initially
computed from
Geodetic
coordinates (direct
problem). If
NAD27 is used the
result is SPCS27. If
NAD83 is used, the
result is SPCS83.
• Define: q, R, Rb, C,
and how to get
them.
Direct and Inverse Problems
• Direct (Forward):
• given: f, l get X, Y?
• Solution: X = R sin q + C
• Y = Rb - R cos q
• Whenever q is used, it is -ve west (left) of the central
meridian.
• q = geodetic azimuth - grid azimuth
• Indirect (Inverse): Solve the above mentioned equations
to compute R, and q. Use tables to compute f, l .
• In both cases, use a computer program whenever
is available. Wolfpack can do it, see next slide.
q = geodetic azimuth - grid azimuth
Grid N Grid N
B
Geodetic N Geodetic N
Grid Az
Geodetic Az A
Geodetic Az Central Meridian of
the zone
Central Meridian of Grid Az
the zone
q : NEGATIVE at point A
West of the CM
q POSITIVE at point A
East of the CM
• Forward Computations: given (f, l) get (X, Y).
• Inverse Computations: given (X, Y) get (f, l)
Surveys Extending from one
Zone to Another
• There is always an overlap area between the
zones.
• When in the transition zone, compute the geodetic
coordinates of two points from their X, Y in first
zone (direct problem).
• Compute X, Y of the same points in the second
zone system from their geodetic coordinates
(inverse problem)
• Compute the azimuth of the line, use the azimuth
and new coordinates to proceed.
Review of Project 2
PI (V)
C
T
PC
Two vertical
BVC 1 curves can be set-
20
BVC 2 out at the same
time from either
BVC 1 or BVC 2.
The direction of
each of the
Centerlines curves
is shown to the
left. Assume that
the BM for
elevation is BVC
and IS NOT 20
Last tree
Yellow-to post
