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

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California Coordinate System Thomas Taylor, PLS Right of Way Engineering District 04 (510)

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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

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History

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

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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. What Map Projection to Use?

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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

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Legal Basis Public Resource Code

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R0R0 B0B0 R b E0E0 NbNb What are constants or given information within the Tables? N b is the northing of projection origin 500, meters E 0 is the easting of the central meridian 2,000, meters R 0 is the mapping radius through the projection origin What must be calculated using the constants? R is the radius of a circle, a function of latitude, and interpolated from the tables u is the radial distance from the central parallel to the station, (R 0 – R) is the convergence angle, mapping angle, mapping angle, convergence angle. What will be given? northing/easting Latitude Longitude (N,E), (X,Y), Latitude Longitude B 0 is the central parallel of the zone R b is mapping radius through grid base R u

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Geodetic to Grid Conversion Determine the Radial Difference: u B = north latitude of the station B 0 = latitude of the projection origin (tabled constant) u = radial distance from the station to the central parallel L 1, L 2, L 3, L 4 = polynomial coefficients (tabled constants)

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Geodetic to Grid Conversion Determine the Mapping Radius: R R = mapping radius of the station R 0 = mapping radius of the projection origin (tabled constant) u = radial distance from the station to the central parallel

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Geodetic to Grid Conversion Determine the Plane Convergence: g g = convergence angle L = west longitude of the station L 0 = longitude of the projection and grid origin (tabled constant) Sin(B 0 ) = sine of the latitude of the projection origin (tabled constant)

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Geodetic to Grid Conversion Determine Northing of the Station n = N 0 + u + [R(sin( g ))(tan( g /2))] or n = R b + N b – R(cos( g )) n = the northing of the station N 0 = northing of the projection origin (tabled constant) R b, N b = tabled constants

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Geodetic to Grid Conversion Determine Easting of the Station e = E 0 + R(sin( g )) e = easting of the station E 0 = easting of the projection and grid origin

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Example # 1 Compute the CCS83 Zone 6 metric coordinates of station Class-1 from its geodetic coordinates of: Latitude = 32° Longitude = 117°

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Example # 1 Determine the Radial Difference: u

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Example # 1 Determine the Radial Difference: u

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Example # 1 Determine the Mapping Radius: R

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Example # 1 Determine the Plane Convergence: g

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Example # 1 Determine Northing of the Station n = R b + N b – R(cos( g )) n = – (cos( )) n =

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Example # 1 Determine Easting of the Station e = E 0 + R(sin( g )) e = (sin( )) e =

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Problem # 1 Compute the CCS83 Zone 3 metric coordinates of station SOL1 from its geodetic coordinates of: Latitude = 38° Longitude = 122°

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Solution to Problem # 1 E B = ° u = R = g = -1° (HMS) 0r ° n = e =

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Grid to Geodetic Conversion Determine the Plane Convergence: g g = arctan[(e - E 0 )/(R b – n + N b )] g = convergence angle at the station e = easting of station E 0 = easting of the projection origin (tabled constant) R b = mapping radius of the grid base (tabled constant) n = northing of the station N b = northing of the grid base (tabled constant)

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Grid to Geodetic Conversion Determine the Longitude L = L 0 – ( g /sin(B 0 )) L = west longitude of the station L 0 = longitude of the projection origin (tabled constant) sin(B 0 ) = sine of the latitude of the projection origin (tabled constant)

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Grid to Geodetic Conversion Determine the radial difference: u u = n – N 0 – [(e – E 0 )tan( g /2)] g = convergence angle at the station e = easting of the station E 0 = easting of the projection origin (tabled constant) n = northing of the station N 0 = northing of the projection origin u = radial distance from the station to the central parallel

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Grid to Geodetic Conversion Determine latitude: B B = B 0 + G 1 u + G 2 u 2 + G 3 u 3 + G 4 u 4 B = north latitude of the station B 0 = latitude of the projection origin (tabled constant) u = radial distance from the station to the central parallel G 1, G 2, G 3, G 4 = polynomial coefficients (tabled constants)

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Example # 2 Compute the Geodetic Coordinate of station Class-2 from its CCS83 Zone 4 Metric Coordinates of: n = e =

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Example # 2 Determine the Plane Convergence: g g = arctan[(e - E 0 )/(R b – n + N b )] g = arctan[( – )/ ( – )] g = arctan(0) g = 0

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Example # 2 Determine the Longitude L = L 0 – ( g /sin(B 0 )) L = 119° – (0/sin( °)) L = 119° 00 00

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Example # 2 Determine the radial difference: u u = n – N 0 – [(e – E 0 )tan( g /2)] u = – [( – )(tan(0/2)] u =

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Example # 2 Determine latitude: B B = B 0 + G 1 u + G 2 u 2 + G 3 u 3 + G 4 u 4 B = ° E-06( ) E-15( ) E-20( ) E-28( ) 4 B = 36°

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Problem # 2 Compute the Geodetic Coordinate of station CC7 from its CCS83 Zone 3 Metric Coordinates of: n = e =

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Solution to Problem # 2 g = -1° or ° L = 122° u = B = 38°

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Convergence Angle Determining the Plane Convergence Angle and the Geodetic Azimuth or the Grid Azimuth g = arctan[(e – E 0 )/(R b – n + N b )] or g = (L 0 – L)sin(B 0 )

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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)

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

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Example # 3 Determining the Plane Convergence Angle and the Geodetic Azimuth or the Grid Azimuth g = arctan[(e – E 0 )/(R b – n + N b )] g = arctan[( – )/ ( – )] g = arctan[ ] g = 0°

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Example # 3 Determine Grid Azimuth: t or Geodetic Azimuth: a t = a – g a = t + g a = 320° ° a = 321°

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Problem # 3 Station D7 has CCS83 Zone 6 Coordinates of n = and e = , and a grid azimuth to a natural sight of 45° Compute the geodetic azimuth from D7 to the same natural sight.

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Solution to Problem # 3 g = 0° ( °) Geodetic Azimuth =

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

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

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Converting Measured Ground Distances to Grid Distances Determine Radius of Curvature of the Ellipsoid: R a R a = r 0 /k 0 R a = geometric mean radius of curvature of the ellipsoid at the projection origin r 0 = geometric mean radius of the ellipsoid at the projection origin, scaled to grid (tabled constant) k 0 = grid scale factor of the central parallel (tabled constant)

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Converting Measured Ground Distances to Grid Distances Determine the Elevation Factor: r e r e = R a /(R a + N + H) r e = elevation factor R a = radius of curvature of the ellipsoid N = geoid separation H = elevation

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Converting Measured Ground Distances to Grid Distances Determine the Point Scale Factor: k k = F 1 + F 2 u 2 + F 3 u 3 k = point scale factor u = radial difference F 1, F 2, F 3 = polynomial coefficients (tabled constants)

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Converting Measured Ground Distances to Grid Distances Determine the Combined Grid Factor: cgf cgf = r e k cgf = combined grid factor r e = elevation factor k = point scale factor

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Converting Measured Ground Distances to Grid Distances Determine Grid Distance G grid = cgf(G ground ) Note: G ground is a horizontal ground distance

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Converting Grid Distances to Horizontal Ground Distances Determine Ground Distance G ground = G grid /cgf

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

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Example # 4 Determine Radius of Curvature of the Ellipsoid: R a R a = r 0 /k 0 R a = / R a =

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Example # 4 Determine the Elevation Factor: r e r e = R a /(R a + N + H) r e = /( – ) r e =

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Example # 4 Determine the Point Scale Factor: k k = F 1 + F 2 u 2 + F 3 u 3 k = E-14(15555) E-22(15555) 3 k =

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Example # 4 Determine the Combined Grid Factor: cgf cgf = r e k cgf = ( ) cgf =

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Example # 4 Determine Grid Distance G grid = cgf(G ground ) G grid = ( ) G grid =

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Problem # 4 In CCS83 Zone 4 from station here to station there you have a measured horizontal ground distance of m. Station here and there have elevations of m and a geoid separation 0f -30.5m. Compute the horizontal grid distance from here to there. (To calculate the point scale factor assume u = 35000)

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Solution to Problem # 4 R a = r e = k = cgf = G grid = m

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Converting a Coordinate from one Zone to another Zone Firstly, convert the grid coordinate from the original zone to a GRS80 geodetic latitude and longitude using the appropriate zone constants Then, convert the geodetic latitude and longitude to the grid coordinates using the appropriate zone constants

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Problem # 5 CC7 has a metric CCS Zone 3 coordinate of n = and e = Compute a CCS Zone 2 coordinate for CC7.

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Solution to Problem # 5 n = e =

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