# STATE PLANE COORDINATE COMPUTATIONS Lectures 14 PLANE COORDINATE COMPUTATIONS Lectures 14 ......

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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 ToolWeb page

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

updatesWeb 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 earths surface.

Earth Center

a

b

a

bc

dc

d

ab > ab

cd < cd

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 = (E2+N2)Azimuth =tan-1(E / N)

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

= = - ( / cos ) = A- tan

(, ) are geodetic coordinates (, ) are astronomic coord. (, ) are the Xi and Eta corrections (, 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 (alias t T)

Azimuth computed from two plane coordinate pairs is a grid azimuth (t).

Projected geodetic azimuth is (T). Geodetic azimuth is ( )

Convergence angle () is the difference between geodetic and projected geodetic azimuths.

Difference between t and T = , the arc-to-chord correction, or t-T or second-term correction.

t = -+

Arc-to-Chord correction (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:

= 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 g) E1 = E + (Sg x sin g)

Where: N = Starting Northing Coordinate E = Starting Easting Coordinates Sg = Grid Distance g = Grid Azimuth

Reduction of Distances When working with geodetic coordinates use

ellipsoidal distances. W

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