STATE PLANE COORDINATE COMPUTATIONSLectures 14 – 15

GISC-3325

Updates and details Required reading assignments were due last

Thursday. Extra credit still available Review class syllabus to see how final grades are

assigned: labs, reading assignments and homework make up a significant part of the grade.

Plane Coordinate Trends Two contrary trends have emerged in the

implementation of SPCS.− Some states, e.g. MT and recently KY, have adopted one

zone.− Others, e.g. ME, have wanted to add zones.

Why?− One zone simplifies working with projects spanning

multiple zones; adding zones makes scale factor closer to one making grid distances close to grid.

Also low-distortion projections.

Low Distortion Projections – what are we talking about?• A mapping projection that minimizes the difference

between distances depicted in a GIS when compared to the real-world distances “at ground”.

• “Standard” mapping projections are “at sea level” (ellipsoid), elevation increases the distortion– Flagstaff, AZ (ellipsoid ht ~ 7000 ft)• SPC Distortion = ~ 1:2,300 or -2.3 ft per mile

– Phoenix, AZ (ellipsoid ht ~ 1000 ft.)• SPC Distortion = ~ 1:6,800 or -0.8 ft per mile

• “Standard” mapping projections usually do not have Central Meridian and Latitude origin near project, which increases distortion variability and convergence angle.

Cartoon: Distortion due to change in Earth curvature (1 of 2)

Linear distortion due to Earth curvature

Grid length greaterthan ellipsoidal length

(distortion > 0)

Ellipsoidsurface

Projectionsurface(secant)

Maximum projection zone width for balanced positive

and negative distortion

Grid length less thanellipsoidal length

(distortion < 0)

LDPs – Who wants them and why?• Engineers & Surveyors use them daily• The value of a GIS increases directly as a function of

its accurate portrayal of items of interest– Local govt. GIS managers are realizing the

benefits of incorporating as-builts and COGO– Better decision support from the GIS

• There is virtually no “cost” to using them– “On-the-fly” reprojection is a reality

• Standard Projections are not good enough for local GIS – UTM distortion is 1:2,500 (2.1 ft per mile)– SPC distortion is 1:10,000 (0.5 ft per mile)– But in both cases distortion at ground usually much

greater

LDP Definition Tool1. User specifies area of interest 2. LDP Tool:– Determines projection parameters– Utilizes USGS National Elevation Dataset and NGS Geoid

Model to:• Determine a representative ellipsoid height• Generate a distortion contour plot

– Displays distortion plot to user3. User accepts, or modifies parameters and iterates4. Upon completion:– a final graphic is provided along with metadata files – Offer to “register” the projection

Grid distance less than"ground" distance(distortion < 0)

Linear distortion due to ground height above ellipsoid

Horizontal distance betweenpoints on the ground

(at average height)

Ground surfacein project area

Localprojectionsurface

Ellipsoidsurface

Grid distancegreater than

"ground" distance(distortion > 0)

Typical published "secant" projection

surface (e.g., State Plane, UTM)

Distortion < 0for almost all cases

Projection “Registry”• A single, national source for the projection

parameters of participating local governments– Registration accomplished via• LDP Tool•Web page

• Emergency Responders access the Registry through two means:– Subscription – push technology gives them instant

updates–Web page – 24 hour, publicly accessible web site

Image on left from Geodesy for Geomatics and GIS Professionals by Elithorp and Findorff, OriginalWorks, 2004.

Map Projections

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

From UNAVCO site

Taken from Ghilani, SPC

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”

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 county-

basis.

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.

Earth Center

a’

b’

a

bc

dc’

d’

ab > a’b’

cd < c’d’

Grid

Ellipsoid

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

Equations from NGS manual, SPCS of 1983 NOS NGS 5

Latitude of grid origin

Mapping radius at equator.

R0: Mapping radius at latitude of true projection origin.

k0: Grid scale factor at CM.

N0:Northing value at CM intersection with central parallel.

Conversion from geodetic coordinates to grid.

Convergence angle

Grid scale factor at point.

Formulas converted to Matlab script.

Grid to Geodetic Coordinates

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

Combined Factor While the SPCS83 tool will compute the scale

factor (SF), we must account for the height of the point with respect to the ellipse.

The elevation factor (EF) = R / (R + h) The combined factor (CF) = SF * EF The CF is used to convert ground distances to

grid. Inverting allows conversion of grid to ground.

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

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

= 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-to-chord” correction, or “t-T” or “second-term” correction.

t = α-γ+ δ

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

Where t is grid azimuth.

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 3-sin 0)(1- 2)− Where 3 = (2 1 + 2)/3

Azimuth of line from N Azimuth of line from N

Sign 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).

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

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).

Lm is surface

Le is ellipsoid

Re is the radius of the Earth in the azimuth of the line.

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).

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

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___

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 αg)

E1 = E + (Sg x sin αg)

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