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)

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. Axis of Cone & Ellipsoid
Types of Plane Systems
Point of Origin
Plane
Apex of Cone
Ellipsoid
Axis of Cone & Ellipsoid
Tangent Plane
Local Plane
Axis of Ellipsoid
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.
Notes :

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
Notes :

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?
Nb is the northing of projection origin 500, meters u R
E0 is the easting of the central meridian 2,000, meters
R b B0
Rb is mapping radius through grid base
B0 is the central parallel of the zone
northing/easting
Latitude(F),Longitude(l)
R0 is 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

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’ ”
Longitude = 117° 00’ ”

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))

21. Example # 1 Determine Easting of the Station e = E0 + R(sin(g))
e = (sin( ))
e =

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

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

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 =
e =

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

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

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

32. Example # 2 Determine latitude: B B = B0 + G1u + G2u2 + G3u3 + G4u4
B = ° E-06( )
E-15( )2
E-20( )3
E-28( )4
B = 36° 43’ ’’

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

34. Solution to Problem # 2 g = -1° 03’ 34.026” or -1.0594517°

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 t = a – g + d
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 = and e = , and a grid azimuth to a natural sight of 320° 37’ ”. 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[( – )/
( – )]
g = arctan[ ]
g = 0° 38’ ’’

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

40. Problem # 3
Station “D7” has CCS83 Zone 6 Coordinates of n = and e = , and a grid azimuth to a natural sight of 45° 25’ ”. 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’ ”

42. 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.
Ellipsoid
Ground h H N
Radius of the Ellipsoid R
EF =
R + N + H R =
Radius of Curvature. N =
Geoidal Separation.
Geoid (MSL) H =
Mean Height above
Geoid. h =
Ellipsoidal Height
Notes :

43. Combined Grid Factor (Combined Scale Factor)
A scale factor is the Ratio of a distance on the grid projection to the corresponding distance on the ellipse. B’ A’ C A B D
Zone Limit C’
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
Notes :

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 m. Stations Me and You have elevations of m and a geoid separation 0f m. Compute the horizontal grid distance from Me to You. (To calculate the point scale factor assume u = )

51. Example # 4 Determine Radius of Curvature of the Ellipsoid: Ra
Ra = r0/k0
Ra = /
Ra =

52. Example # 4 Determine the Elevation Factor: re re = Ra/(Ra + N + H)

53. Example # 4 Determine the Point Scale Factor: k k = F1 + F2u2 + F3u3
k = E-14(15555)2
+ 5.47E-22(15555)3
k =

54. Example # 4 Determine the Combined Grid Factor: cgf cgf = re k

55. Example # 4 Determine Grid Distance Ggrid = cgf(Gground)

56. 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 m. Compute the horizontal grid distance from here to there. (To calculate the point scale factor assume u = 35000)

57. Solution to Problem # 4 Ra = 6371934.463 re = 0.999656153
k =
cgf =
Ggrid = m

58. 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

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

60. Solution to Problem # 5
n =
e =