state plane coordinates and computations using them...
TRANSCRIPT
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State Plane Coordinates and Computations using themGISC-3325 - Spring 2013
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Map Projections
hosting.soonet.ca/eliris/gpsgis/Lec2Geodesy.html
From UNAVCO site
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Taken from Ghilani, SPC
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• SPCS 27 designed in 1930s to facilitate the attachment of surveys to the national system.
• Originally suggested by a North Carolina civil engineer.
• 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 county-basis.
State Plane Coordinate System
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Conformal Mapping Projections
● 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”
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Note that the SPC in Montana is one zone only.
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Source: http://www.cnr.colostate.edu/class_info/nr502/lg3/datums_coordinates/spcs.html
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N = 3,078,495.629 m
E = 924,954.270 m
N = -25.13 m
k = 0.99994523
Convergence angle
+01d12m19.0s
LAPLACE Corr.
-4.04 seconds
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Data sheets for points near SPC zones boundaries
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Some states have decided to go their own way...
Kentucky single zone slides from presentation by
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SPC Expressed in Multiple Zones
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When no version of the foot is specified, no values in feet units are published.
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Bs: Southern standard parallel (fs)
Bn: Northern standard parallel (fn)
Bb: Latitude of the grid origin (f0)
L0: Central meridian (l0)
Nb: “false northing”
E0: “false easting”
Constants were copied from NOAA Manual NOS NGS 5 (available on-line)
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http://www.ngs.noaa.gov/TOOLS/spc.shtml
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• 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 Factor × Grid Scale Factor = Combined
Factor
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Distance = √(ΔE2+ΔN2)Azimuth =tan-1(ΔE / ΔN)
N1 = N + (Sg x cos αg)
E1 = E + (Sg x sin αg) Where: N = Starting Northing
Coordinate E = Starting Easting
Coordinates Sg = Grid Distance
αg = Grid Azimuth
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Laplace correction
● Used to convert astronomic azimuths to geodetic azimuths.● A simple function of the geodetic latitude and the east-
west 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.
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Astronomic to Geodetic Azimuth
● phi = Φ – ξ● lambda = Λ - (η / cos f)● α= A- η·tan f
– (phi,lambda ) are geodetic coordinates– (Φ, Λ) are astronomic coordinates.– (ξ, η) are the Xi and Eta corrections– (α, A) are geodetic and astronomic
azimuths (respectively)
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Grid directions (t) are based on north being parallel to the Central Meridian.
Remember: Geodetic and grid north ONLY coincide along CM.
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N.B. Convergence angle shown does NOT include the arc-to-chord correction.
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Arc-to-Chord correction δ (alias t – T)
Where t is grid azimuth.
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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-to-chord” correction, or “t-T” or “second-term” correction.
● t = α-γ+ δ
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When should it be applied?● 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 f3-sin f0)(l1- l2)– Where f3 = (2 f1 + f2)/3
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Azimuth of line from N Azimuth of line from NSign of N-N0 0 to 180 180 to 360Positive + -Negative - +
Compute magnitude of the second-term correction from preliminary coordinates.
It is not significant for short sight distances (< 8km) but …
The effect of this correction is cumulative!
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Angle Reductions● Know the type of azimuth
– 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).
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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.
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Reduction of Distances ● When working with geodetic coordinates use
ellipsoidal distances.● When working with state plane coordinates
reduce the observations to the grid (mapping surface).
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Elevation Factor at a point
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Reduction of distances
h
NH
R=Earth Radius 6,372,161 m 20,906,000 ft.
Earth Center
S
D
S = D x ___R__ R + h
h = H + N
S = D x R + H + N
___R___
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For most surveys the approximate radius used in NAD 27 (6,372,000 m or 20,906,000 ft) can be used for Re.
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Reduce ellipsoid distance to grid
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Final reduced distance● 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).
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D5 is the geodetic distance.
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“I WANT STATE PLANE COORDINATES RAISED TO GROUND LEVEL”
GROUND LEVEL COORDINATES ARE NOT STATE PLANE COORDINATES!!!!!
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PROBLEMS WITH GROUND LEVEL COORDINATES
• RAPID DISTORTIONS• PROJECTS DIFFICULT TO
TIE TOGETHER• CONFUSION OF
COORDINATE SYSTEMS• LACK OF
DOCUMENTATION
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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
DOCUMENT DOCUMENT DOCUMENT !!
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GOOD COORDINATION BEGINS WITH GOOD COORDINATES
GEOGRAPHY WITHOUT GEODESY IS A FELONY
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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
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References● http://www.surpriseaz.gov/DocumentCenter/
Home/View/3774● http://www.profsurv.com/magazine/article.aspx?
i=893● http://geodesy.noaa.gov/TOOLS/
Professional_Surveyor_Articles/StatePlaneCoord.pdf
● http://www.ngs.noaa.gov/PUBS_LIB/ManualNOSNGS5.pdf
● http://www.dot.ca.gov/hq/row/landsurveys/LSITWorkbook/11.pdf
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http://www.ngs.noaa.gov/PUBS_LIB/pub_index.html
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http://www.cnr.colostate.edu/class_info/nr502/lg3/datums_coordinates/spcs.html
http://geodesy.noaa.gov/TOOLS/Professional_Surveyor_Articles/StatePlaneCoord.pdf