State Plane Coordinates and Computations using them GISC Spring 2013

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1 State Plane Coordinates and Computations using them GISC Spring 2013

2 Map Projections From UNAVCO site hosting.soonet.ca/eliris/gpsgis/lec2geodesy.html

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4 Taken from Ghilani, SPC

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6 State Plane Coordinate System 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 Uses as few zones as possible to cover a state. Defines boundaries of zones on county-basis.

7 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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16 Note that the SPC in Montana is one zone only.

17 Source:

18 N = 3,078, m E = 924, m N = m k = Convergence angle +01d12m19.0s LAPLACE Corr seconds

19 Data sheets for points near SPC zones boundaries

20 Some states have decided to go their own way... Kentucky single zone slides from presentation by

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24 SPC Expressed in Multiple Zones

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

27 B s : Southern standard parallel (f s ) B n : Northern standard parallel (f n ) B b : Latitude of the grid origin (f 0 ) L 0 : Central meridian (l 0 ) N b : false northing E 0 : false easting Constants were copied from NOAA Manual NOS NGS 5 (available on-line)

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31 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

32 Distance = (ΔE 2 +ΔN 2 ) Azimuth =tan -1 (ΔE / ΔN) N 1 = N + (S g x cos α g ) E 1 = E + (S g x sin α g ) Where: N = Starting Northing Coordinate E = Starting Easting Coordinates S g = Grid Distance α g = Grid Azimuth

33 Laplace correction Used to convert astronomic azimuths to geodetic azimuths. A simple function of the geodetic latitude and the eastwest 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.

34 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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37 Grid directions (t) are based on north being parallel to the Central Meridian. Remember: Geodetic and grid north ONLY coincide along CM.

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

39 Arc-to-Chord correction δ (alias t T) Where t is grid azimuth.

40 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-tochord correction, or t-t or second-term correction. t = α-γ+ δ

41 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 f 3 -sin f 0 )(l 1 - l 2 ) Where f 3 = (2 f 1 + f 2 )/3

42 Compute magnitude of the secondterm correction from preliminary coordinates. It is not significant for short sight distances (< 8km) but The effect of this correction is cumulative! Azimuth of line from N Azimuth of line from N Sign of N-N0 0 to to 360 Positive + - Negative - +

43 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).

44 Astronomic to Grid (via geodetic) ag = aa + Laplace Correction g 253d 26m 14.9s - Observed Astro Azimuth + ( s) - Laplace Correction 253d 26m 13.6s - Geodetic Azimuth m 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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46 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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49 Elevation Factor at a point

50 Reduction of distances H h N D S h = H + N R=Earth Radius 6,372,161 m 20,906,000 ft. S = D x R R + h Earth Center S = D x R R + H + N

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52 For most surveys the approximate radius used in NAD 27 (6,372,000 m or 20,906,000 ft) can be used for R e.

53 Reduce ellipsoid distance to grid

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55 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).

56 D5 is the geodetic distance.

57 I WANT STATE PLANE COORDINATES RAISED TO GROUND LEVEL GROUND LEVEL COORDINATES ARE NOT STATE PLANE COORDINATES!!!!!

58 PROBLEMS WITH GROUND LEVEL COORDINATES RAPID DISTORTIONS PROJECTS DIFFICULT TO TIE TOGETHER CONFUSION OF COORDINATE SYSTEMS LACK OF DOCUMENTATION

59 GROUND LEVEL COORDINATES IF YOU DO TRUNCATE COORDINATE VALUES SUCH AS: N = 13,750, ft becomes 50, E = 2,099, ft becomes 99, AND DOCUMENT DOCUMENT

60 GOOD COORDINATION BEGINS WITH GOOD COORDINATES GEOGRAPHY WITHOUT GEODESY IS A FELONY

61 The Universal Grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) - TM Transverse Mercator Projection Zone width 6 o 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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66 References Home/View/ i=893 Professional_Surveyor_Articles/ StatePlaneCoord.pdf ManualNOSNGS5.pdf LSITWorkbook/11.pdf

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