State Plane Coordinates and Computations using themGISC-3325 - Spring 2013

Map Projections

hosting.soonet.ca/eliris/gpsgis/Lec2Geodesy.html

From UNAVCO site

Taken from Ghilani, SPC

• 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

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”

Note that the SPC in Montana is one zone only.

Source: http://www.cnr.colostate.edu/class_info/nr502/lg3/datums_coordinates/spcs.html

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

Data sheets for points near SPC zones boundaries

Some states have decided to go their own way...

Kentucky single zone slides from presentation by

SPC Expressed in Multiple Zones

When no version of the foot is specified, no values in feet units are published.

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)

http://www.ngs.noaa.gov/TOOLS/spc.shtml

• 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

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

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.

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)

Grid directions (t) are based on north being parallel to the Central Meridian.

Remember: Geodetic and grid north ONLY coincide along CM.

N.B. Convergence angle shown does NOT include the arc-to-chord correction.

Arc-to-Chord correction δ (alias t – T)

Where t is grid azimuth.

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

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

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!

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).

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.

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).

Elevation Factor at a point

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___

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● 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).

D5 is the geodetic distance.

“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

DOCUMENT DOCUMENT DOCUMENT !!

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

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

http://www.ngs.noaa.gov/PUBS_LIB/pub_index.html

http://www.cnr.colostate.edu/class_info/nr502/lg3/datums_coordinates/spcs.html

http://geodesy.noaa.gov/TOOLS/Professional_Surveyor_Articles/StatePlaneCoord.pdf