STATE PLANE COORDINATE
COMPUTATIONS
Lectures 14 – 15
GISC-3325
Updates and details
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Required reading assignments due 30 April 2008
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Extra credit due 23 April 2008
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Overdue lab assignments/homework will be given
credit ONLY if received by 21 April 2008.
Wednesday class: 16 April 2008 will be devoted
to RTK. Mr. Toby Stock will demonstrate, make
observations and show results. Meet him at
Blucher during lecture and lab periods.
Datum: A set of constants
specifying the coordinate system
used to calculate coordinates of
points on the Earth.
8 Constants
3 to specify the origin.
3 to specify the
orientation.
2 to specify the
dimensions of the
reference ellipsoid.
a = Semi major axis
b = Semi minor axis
f = a-b = Flattening
a
N
b
a
S
BESSEL 1841
a = 6,377,397.155 m 1/f = 299.1528128
CLARKE 1866
a = 6,378,206.4 m 1/f = 294.97869821
GEODETIC REFERENCE SYSTEM 1980 - (GRS 80)
a = 6,378,137 m
1/f = 298.257222101
WORLD GEODETIC SYSTEM 1984 - (WGS 84)
a = 6,378,137 m
1/f = 298.257223563
Image on left from Geodesy for Geomatics and GIS Professionals
by Elithorp and Findorff, OriginalWorks, 2004.
Map Projections
From UNAVCO site
hosting.soonet.ca/eliris/gpsgis/Lec2Geodesy.html
Taken from Ghilani, SPC
Conformal Mapping Projections
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Mapping a curved Earth on a flat map must
address possible distortions in angles, azimuths,
distances or area.
Map projections where angles are preserved
after projection are called “conformal”
http://www.cnr.colostate.edu/class_info/nr502/lg3/datums_coordinates/spcs.html
• SPCS 27 designed in 1930s to facilitate
the attachment of surveys to the
national system.
• Uses conformal mapping projections.
• Restricts maximum scale distortion to
less than 1 part in 10 000.
• Uses as few zones as possible to cover a
state.
• Defines boundaries of zones on countybasis.
http://www.ngs.noaa.gov/PUBS_LIB/pub_index.html
Source: http://www.cnr.colostate.edu/class_info/nr502/lg3/datums_coordinates/spcs.html
Secant cone intersects
the surface of the
ellipsoid NOT the
earth’s surface.
d’
c’
c
b
d
Ellipsoid
cd < c’d’
b’
ab > a’b’
a
a’
Grid
Earth Center
Bs: Southern standard parallel (s)
Bn: Northern standard parallel (n)
Bb: Latitude of the grid origin (0)
L0: Central meridian (0)
Nb: “false northing”
E0: “false easting”
Constants were copied from NOAA Manual NOS
NGS 5 (available on-line)
Zone constant computations
Latitude of grid origin
Mapping radius at equator.
Equations from NGS manual, SPCS of 1983 NOS NGS 5
R0: Mapping radius at latitude of true projection origin.
k0: Grid scale factor at CM.
N0:Northing value at CM intersection with central
parallel.
Convergence angle
Grid scale factor at point.
Conversion from geodetic coordinates to grid.
Formulas converted to Matlab script.
Grid to Geodetic Coordinates
http://www.ngs.noaa.gov/TOOLS/spc.shtml
Distance = √(ΔE2+ΔN2)
Azimuth =tan-1(ΔE / ΔN)
N.B. Convergence angle shown does NOT
include the arc-to-chord correction.
• STARTING COORDINATES
• AZIMUTH
• Convert Astronomic to Geodetic
• Convert Geodetic to Grid (Convergence angle)
• Apply Arc-to-Chord Correction (t-T)
• DISTANCES
• Reduction from Horizontal to Ellipsoidal
• Elevation “Sea-Level” Reduction Factor
• Grid Scale Factor
N = 3,078,495.629
E = 924,954.270
N=
-25.13
k = 0.99994523
Convergence angle
+01-12-19.0
LAPLACE Corr.
-4.04 seconds
Laplace correction
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Used to convert astronomic azimuths to geodetic azimuths.
A simple function of the geodetic latitude and the eastwest deflection of the vertical at the ground surface.
Corrections to horizontal directions are a function of the
Laplace correction and the zenith angle between stations,
and can become significant in mountainous areas.
Astronomic to Geodetic Azimuth
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=Φ–ξ
 = Λ - (η / cos )
α= A- η∙tan 
(, ) are geodetic coordinates
– (Φ, Λ) are astronomic coord.
– (ξ, η) are the Xi and Eta corrections
– (α, A) are geodetic and astronomic
azimuths respectively)
–
Grid directions (t) are based
on north being parallel to
the Central Meridian.
Remember: Geodetic and
grid north ONLY coincide
along CM.
Astronomic to Grid (via geodetic)
ag = aA + Laplace Correction – g
253d 26m 14.9s
- Observed Astro Azimuth
+ ( - 1.33s) - Laplace Correction
253d 26m 13.6s
- Geodetic Azimuth
+ 1 12m 19.0s
- Convergence Angle (g)
254d 38m 32.6s
- Grid azimuth
The convention of the sign of the convergence
angle is always from Grid to Geodetic.
Arc-to-Chord correction δ (alias t – T)
• Azimuth computed from two plane coordinate
pairs is a grid azimuth (t).
• Projected geodetic azimuth is (T).
• Geodetic azimuth is (α )
• Convergence angle (γ) is the difference between
geodetic and projected geodetic azimuths.
• Difference between t and T = “δ”, the “arc-tochord” correction, or “t-T” or “second-term”
correction.
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t = α-γ+ δ
Arc-to-Chord correction δ (alias t – T)
Where t is grid azimuth.
When should it be applied?
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Intended for during precise surveys.
Recommended for use on lines over 8 kilometers
long.
It is always concave toward the Central Parallel
of the projection.
Computed as:
δ = 0.5(sin 3-sin 0)(1- 2)
– Where 3 = (2 1 + 2)/3
–
Compute magnitude of the secondterm correction from preliminary
coordinates.
It is not significant for short sight
distances
(< 8km) but …
The effect of this correction is
cumulative!
Azimuth of line from N
Sign of N-N0
Azimuth of line from N
0 to 180
180 to 360
Positive
+
-
Negative
-
+
Angle Reductions
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Know the type of azimuth
–
–
–
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Astronomic
Geodetic
Grid
Apply appropriate corrections
Angles (difference of two directions from a
single station) do not need to consider
convergence angle.
Apply arc-to-chord correction for long sight
distances or long traverses (cumulative effect).
N1 = N + (Sg x cos ag)
E1 = E + (Sg x sin ag)
Where:
N = Starting Northing Coordinate
E = Starting Easting Coordinates
Sg = Grid Distance
ag = Grid Azimuth
Reduction of Distances
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When working with geodetic coordinates use
ellipsoidal distances.
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When working with state plane coordinates
reduce the observations to the grid (mapping
surface).
Re is the radius of
the Earth in the
azimuth of the
line.
Lm is surface
Le is ellipsoid
For most surveys the approximate radius used in NAD 27
(6,372,000 m or 20,906,000 ft) can be used for Re.
Reduce ellipsoid distance to grid
Final reduced distance
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Measured distances are first corrected for
atmospheric refraction and earth’s curvature.
Distances reduced to ellipsoid.
Distances reduced to grid by applying the
combined factor (scale factor by elevation factor).
EF at a point (numeric example)
Let R = 6372000, h = 48.98
EF = R/(R + h) = 0.999992313
if we do not have h, compute it via relationship: N + H
Reduction of distances
D
h=H+N
h
H
S
N
R=Earth Radius
6,372,161 m
20,906,000 ft.
S = D x ___R__
R+h
S=Dx
Earth Center
___R___
R+H+N
D5 is the geodetic distance.
REDUCTION TO ELLIPSOID
S = D x [R / (R + h)]
D = 1010.387 meters (Measured Horizontal Distance)
R = 6,372,162 meters (Mean Radius of the Earth)
h = H + N (H = 2 m, N = - 26 m) = - 24 meters (Ellipsoidal Height)
S = 1010.387 [6,372,162 / 6,372,162 - 24]
S = 1010.387 x 1.00000377
S = 1010.391 meters
If N is ignored:
S = 1010.387 [6,372,162 / 6,372,162 + 2]
S = 1010.387 x 0.99999969
S = 1010.387 meters -- 0.004 m or about 1: 252,600
REDUCTION TO GRID
Sg = S (Geodetic Distance) x k (Grid Scale Factor)
Sg = 1010.391 x 0.99992585
= 1010.316 meters
COMBINED FACTOR
CF = Ellipsoidal Reduction x Grid Scale
Factor (k)
= 1.00000377 x 0.99992585
= 0.99992962
CF x D = Sg
0.99992962 x 1010.387 = 1010.316 meters
STATE PLANE COORDINATE COMPUTATION
N1 = N + (Sg x cos ag)
E1 = E + (Sg x sin ag)
N1 = 4,103,643.392 + (1010.277 x Cos 253o 30’ 07.4”)
= 4,103,643.392 + (1010.277 x - 0.28398094570069)
= 4,103,643.392 + (- 286.899)
= 4,103,356.492 meters
E1 = 587,031.437 + (1010.277 x Sin 253o 30’ 07.4”)
= 587,031.437 + (1010.277 x - 0.95882992364597)
= 587,031.437 + (- 968.684)
= 586,062.753 meters
“I WANT STATE PLANE
COORDINATES RAISED TO
GROUND LEVEL”
GROUND LEVEL
COORDINATES ARE NOT
STATE PLANE
COORDINATES!!!!!
PROBLEMS WITH GROUND
LEVEL COORDINATES
• RAPID DISTORTIONS
• PROJECTS DIFFICULT TO
TIE TOGETHER
• CONFUSION OF
COORDINATE SYSTEMS
• LACK OF
DOCUMENTATION
GROUND LEVEL COORDINATES
“IF YOU DO”
TRUNCATE COORDINATE VALUES SUCH AS:
N = 13,750,260.07 ft becomes 50,260.07
E=
2,099,440.89 ft becomes 99,440.89
AND
GOOD COORDINATION BEGINS WITH
GOOD COORDINATES
GEOGRAPHY WITHOUT GEODESY IS A FELONY
The Universal Grids: Universal Transverse
Mercator (UTM) and Universal Polar
Stereographic (UPS) - TM8358.2
• Transverse Mercator Projection
• Zone width 6o Longitude World-Wide
• Northing Origin (0 meters- Northern Hemisphere)
at the Equator
• Easting Origin (500,000 meters) at Central
Meridian of Each Zone