California Coordinate System

1
California Coordinate System

• Capital Project Skill Development Class (CPSD)
• G100497

2
California Coordinate System

• Thomas Taylor, PLS
• Right of Way Engineering
• District 04
• [phone]
• Tom_Taylor_at_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?
Nb is 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
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
(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
arctan0.0111198338 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
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 the corresponding 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)
51
Example 4

• Determine Radius of Curvature of the Ellipsoid Ra

Ra r0/k0 Ra 6374328/0.999894636561 Ra
6374999.69189
52
Example 4

• Determine the Elevation Factor re

re Ra/(Ra N H) re 6374999.69189/(6374999.6
9189 30.5 3333.333) re 0.9994821768
53
Example 4

• Determine the Point Scale Factor k

k F1 F2u2 F3u3 k 0.999894636561
1.23062E-14(15555)2 5.47E-22(15555)3 k
0.9998976162
54
Example 4

• Determine the Combined Grid Factor cgf

cgf re k cgf 0.9994821768(0.9998976162) cgf
0.999379846
55
Example 4

• Determine Grid Distance

Ggrid cgf(Gground) Ggrid 0.999379846(909.909)
Ggrid 909.3447
56
Problem 4
In CCS83 Zone 4 from station here to station
there you have a measured horizontal ground
distance of 1234.567m. Station here and there
have elevations of 2222.222m 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)
57
Solution to Problem 4
Ra 6371934.463 re 0.999656153 k
0.999955870 cgf 0.999612038 Ggrid 1234.088m
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
674010.835 and e 1848139.628. Compute a CCS
Zone 2 coordinate for CC7.
60
Solution to Problem 5
n 543163.942 e 1979770.624