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  • 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)



    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


  • 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



    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

    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


  • 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-



  • Source:

  • Secant cone intersects the surface of the ellipsoid NOT the earths surface.

  • Earth Center







    ab > ab

    cd < cd



  • 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


  • 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.


    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


    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