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California Coordinate System

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  1. California Coordinate System Capital Project Skill Development Class (CPSD) G100497

  2. California Coordinate System Thomas Taylor, PLS Right of Way Engineering District 04 (510) 286-5294 Tom_Taylor@dot.ca.gov

  3. Course Outline • History • Legal Basis • The Conversion Triangle • Geodetic to Grid Conversion • Grid to Geodetic Conversion • Convergence Angle • Reducing Measured Distances to Grid Distances • Zone to Zone Transformations

  4. History

  5. Types of Plane Systems Point of Origin Plane Apex of Cone Ellipsoid Axis of Cone & Ellipsoid Axis of Ellipsoid Tangent Plane Local Plane Line of intersection Axis of Cylinder Ellipsoid Ellipsoid Intersecting Cylinder Transverse Mercator Intersecting Cone 2 Parallel Lambert

  6. What Map Projection to Use? • A number of Conformal Map Projections are used in the United States. • Universal Transverse Mercator. • Transverse Mercator. • Oblique Transverse Mercator. • Lambert Conformal Conical. • The Transverse Mercator is used for states (or zones in states) that are long in a North-South direction. • The Lambert is used for states (or zones in states) that are long in an East-West direction. • The Oblique Mercator is used in one zone in Alaska where neither the TM or Lambert were appropriate.

  7. Characteristics of the Lambert Projection • The secant cone intersects the surface of the ellipsoid at two places. • The lines joining these points of intersection are known as standard parallels. By specifying these parallels it defines the cone. • Scale is always the same along an East-West line. • By defining the central meridian, the cone becomes orientated with respect to the ellipsoid

  8. Legal Basis • Public Resource Code

  9. What will be given? g , q , mapping angle, convergence angle. (N,E), (X,Y), Latitude(F), Longitude(l) R0 What are constants or given information within the Tables? Nbis the northing of projection origin 500,000.000 meters u R E0 is the easting of the central meridian 2,000,000.000 meters R b B0 Rbis mapping radius through grid base B0 is the central parallel of the zone northing/easting Latitude(F),Longitude(l) R0is the mapping radius through the projection origin What must be calculated using the constants? Nb R is the radius of a circle, a function of latitude, and interpolated from the tables E0 u is the radial distance from the central parallel to the station, (R0 – R) g , q is the convergence angle, mapping angle

  10. Geodetic to Grid Conversion • Determine the Radial Difference: u B = north latitude of the station B0 = latitude of the projection origin (tabled constant) u = radial distance from the station to the central parallel L1, L2, L3, L4 = polynomial coefficients (tabled constants)

  11. Geodetic to Grid Conversion • Determine the Mapping Radius: R R = mapping radius of the station R0 = mapping radius of the projection origin (tabled constant) u = radial distance from the station to the central parallel

  12. Geodetic to Grid Conversion • Determine the Plane Convergence: g g = convergence angle L = west longitude of the station L0 = longitude of the projection and grid origin (tabled constant) Sin(B0) = sine of the latitude of the projection origin (tabled constant)

  13. Geodetic to Grid Conversion • Determine Northing of the Station n = N0 + u + [R(sin(g))(tan(g/2))] or n = Rb + Nb – R(cos(g)) n = the northing of the station N0 = northing of the projection origin (tabled constant) Rb, Nb = tabled constants

  14. Geodetic to Grid Conversion • Determine Easting of the Station e = E0 + R(sin(g)) e = easting of the station E0 = easting of the projection and grid origin

  15. Example # 1 Compute the CCS83 Zone 6 metric coordinates of station “Class-1” from its geodetic coordinates of: Latitude = 32° 54’ 16.987” Longitude = 117° 00’ 01.001”

  16. Example # 1 • Determine the Radial Difference: u

  17. Example # 1 • Determine the Radial Difference: u

  18. Example # 1 • Determine the Mapping Radius: R

  19. Example # 1 • Determine the Plane Convergence: g

  20. Example # 1 • Determine Northing of the Station n = Rb + Nb – R(cos(g)) n = 9836091.7896 + 500000.000 – 9754239.92234(cos(-0.4122909785)) n = 582104.404

  21. Example # 1 • Determine Easting of the Station e = E0 + R(sin(g)) e = 2000000.000 + 9754239.92234(sin(-0.4122909785)) e = 1929810.704

  22. Problem # 1 Compute the CCS83 Zone 3 metric coordinates of station “SOL1” from its geodetic coordinates of: Latitude = 38° 03’ 59.234” Longitude = 122° 13’ 28.397”

  23. Solution to Problem # 1 EB = 0.315384453° u = 35003.7159064 R = 8211926.65249 g = -1° 03’ 20.97955” (HMS) 0r -1.05582765° n = 675242.779 e = 1848681.899

  24. Grid to Geodetic Conversion • Determine the Plane Convergence: g g = arctan[(e - E0)/(Rb – n + Nb)] g = convergence angle at the station e = easting of station E0 = easting of the projection origin (tabled constant) Rb = mapping radius of the grid base (tabled constant) n = northing of the station Nb = northing of the grid base (tabled constant)

  25. Grid to Geodetic Conversion • Determine the Longitude L = L0 – (g/sin(B0)) L = west longitude of the station L0 = longitude of the projection origin (tabled constant) sin(B0) = sine of the latitude of the projection origin (tabled constant)

  26. Grid to Geodetic Conversion • Determine the radial difference: u u = n – N0 – [(e – E0)tan(g/2)] g = convergence angle at the station e = easting of the station E0 = easting of the projection origin (tabled constant) n = northing of the station N0 = northing of the projection origin u = radial distance from the station to the central parallel

  27. Grid to Geodetic Conversion • Determine latitude: B B = B0 + G1u + G2u2 + G3u3 + G4u4 B = north latitude of the station B0 = latitude of the projection origin (tabled constant) u = radial distance from the station to the central parallel G1, G2, G3, G4 = polynomial coefficients (tabled constants)

  28. Example # 2 Compute the Geodetic Coordinate of station “Class-2” from its CCS83 Zone 4 Metric Coordinates of: n = 654048.453 e = 2000000.000

  29. Example # 2 • Determine the Plane Convergence: g g = arctan[(e - E0)/(Rb – n + Nb)] g = arctan[(2000000.000 – 2000000.000)/ (8733227.3793 – 654048.453 + 500000.000)] g = arctan(0) g = 0

  30. Example # 2 • Determine the Longitude L = L0 – (g/sin(B0)) L = 119° 00’ 00’’ – (0/sin(36.6258593071°)) L = 119° 00’ 00’’

  31. Example # 2 • Determine the radial difference: u u = n – N0 – [(e – E0)tan(g/2)] u = 654048.453 – 643420.4858 - [(2000000.000 – 2000000.000)(tan(0/2)] u = 10627.967

  32. Example # 2 • Determine latitude: B B = B0 + G1u + G2u2 + G3u3 + G4u4 B = 36.6258593071° + 9.011926076E-06(10627.967) + -6.83121E-15(10627.967)2 + -3.72043E-20(10627.967)3 + -9.4223E-28(10627.967)4 B = 36° 43’ 17.893’’

  33. Problem # 2 Compute the Geodetic Coordinate of station “CC7” from its CCS83 Zone 3 Metric Coordinates of: n = 674010.835 e = 1848139.628

  34. Solution to Problem # 2 g = -1° 03’ 34.026” or -1.0594517° L = 122° 13’ 49.706” u = 33761.9722245 B = 38° 03’ 18.958”

  35. Convergence Angle • Determining the Plane Convergence Angle and the Geodetic Azimuth or the Grid Azimuth g = arctan[(e – E0)/(Rb – n + Nb)] or g = (L0 – L)sin(B0)

  36. Convergence Angle • Determine Grid Azimuth: t or Geodetic Azimuth: a t = a – g + d t = grid azimuth a = geodetic azimuth g = convergence angle (mapping angle) d = arc to chord correction, known as the second order term (ignore this term for lines less than 5 miles long)

  37. Example # 3 Station “Class-3” has CCS83 Zone 1 Coordinates of n = 593305.300 and e = 2082990.092, and a grid azimuth to a natural sight of 320° 37’ 22.890”. Compute the geodetic azimuth from Class-3 to the same natural sight.

  38. Example # 3 • Determining the Plane Convergence Angle and the Geodetic Azimuth or the Grid Azimuth g = arctan[(e – E0)/(Rb – n + Nb)] g = arctan[(2082990.092 – 2000000.000)/ (7556554.6408 – 593305.300 + 500000.000)] g = arctan[0.0111198338] g = 0° 38’ 13.536’’

  39. Example # 3 • Determine Grid Azimuth: t or Geodetic Azimuth: a t = a – g a = t + g a = 320° 37’ 22.890’’ + 0° 38’ 13.536’’ a = 321° 15’ 36.426’’

  40. Problem # 3 Station “D7” has CCS83 Zone 6 Coordinates of n = 489321.123 and e = 2160002.987, and a grid azimuth to a natural sight of 45° 25’ 00.000”. Compute the geodetic azimuth from D7 to the same natural sight.

  41. Solution to Problem # 3 g = 0° 55” 51.361’ (0.9309335°) Geodetic Azimuth = 46 20’ 51.361”

  42. Ground h H Ellipsoid N Radius of the Ellipsoid Combined Grid Factor (Combined Scale Factor) • Elevation Factors • Before a Ground Distance can be reduced to the Grid, it must first be reduced to the ellipsoid of reference. R EF = R + N + H R = Radius of Curvature. N = Geoidal Separation. • Geoid (MSL) H = Mean Height above Geoid. h = Ellipsoidal Height

  43. Combined Grid Factor (Combined Scale Factor) • A scale factor is the Ratio of a distance on the grid projection to thecorresponding distance on the ellipse. B’ A’ C A B D C’ Zone Limit Zone Limit D’ Scale Decreases Scale Increases Scale Increases - Grid Distance A-B is smaller than Geodetic Distance A’-B’. - Grid Distance C-D is larger than Geodetic Distance C’-D’. Scale Decreases

  44. Converting Measured Ground Distances to Grid Distances • Determine Radius of Curvature of the Ellipsoid: Ra Ra = r0/k0 Ra = geometric mean radius of curvature of the ellipsoid at the projection origin r0 = geometric mean radius of the ellipsoid at the projection origin, scaled to grid (tabled constant) k0 = grid scale factor of the central parallel (tabled constant)

  45. Converting Measured Ground Distances to Grid Distances • Determine the Elevation Factor: re re = Ra/(Ra + N + H) re = elevation factor Ra = radius of curvature of the ellipsoid N = geoid separation H = elevation

  46. Converting Measured Ground Distances to Grid Distances • Determine the Point Scale Factor: k k = F1 + F2u2 + F3u3 k = point scale factor u = radial difference F1, F2, F3 = polynomial coefficients (tabled constants)

  47. Converting Measured Ground Distances to Grid Distances • Determine the Combined Grid Factor: cgf cgf = re k cgf = combined grid factor re = elevation factor k = point scale factor

  48. Converting Measured Ground Distances to Grid Distances • Determine Grid Distance Ggrid = cgf(Gground) Note: Gground is a horizontal ground distance

  49. Converting Grid Distances to Horizontal Ground Distances • Determine Ground Distance Gground = Ggrid/cgf

  50. Example # 4 In CCS83 Zone 1 from station “Me” to station “You” you have a measured horizontal ground distance of 909.909m. Stations Me and You have elevations of 3333.333m and a geoid separation 0f -30.5m. Compute the horizontal grid distance from Me to You. (To calculate the point scale factor assume u = 15555.000)