California Coordinate System

Capital Project Skill Development Class (CPSD)G100497

California Coordinate System

Thomas Taylor, PLSRight of Way EngineeringDistrict 04

(510) 286-5294Tom_Taylor@dot.ca.gov

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

History

Types of Plane SystemsPlane

Ellipsoid

Tangent PlaneLocal Plane

Point of Origin

Intersecting CylinderTransverse Mercator

Axis of Ellipsoid

Ellipsoid

Axis of Cylinder

Line of intersection

Apex of Cone

Intersecting Cone2 Parallel Lambert

Axis of Cone & Ellipsoid

Ellipsoid

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?

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

Legal Basis

Public Resource Code

R0

B0

R b

E0

Nb

What are constants or given information within the Tables?

Nb is the northing of projection origin 500,000.000 meters

E0 is the easting of the central meridian 2,000,000.000 meters

R0 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, (R0 – R)

is the convergence angle, mapping angle

, mapping angle, convergence angle.

What will be given?

northing/easting

LatitudeLongitude

(N,E), (X,Y), Latitude Longitude

B0 is the central parallel of the zone

Rb is mapping radius through grid base

Ru

Geodetic to Grid Conversion

Determine the Radial Difference: u

)B(L)B(L)(L B)(L u 44

3321

0B-BB

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)

Geodetic to Grid Conversion

Determine the Mapping Radius: R

u - RR 0

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

Geodetic to Grid Conversion

Determine the Plane Convergence: g

)L)sin(B-(L 00g = 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)

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

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

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”

Example # 1

Determine the Radial Difference: u

34-0.4292043B

44733.3339229-904718611.32B

44733.3339229 - '16.987' 54' 32B

Example # 1

Determine the Radial Difference: u

1441-47599.846u

4)0.429204330.016171(-

).4292043345.65087(-0

).4292043348.94188(-0

4334)4(-0.42920110905.327 u

4

3

2

Example # 1

Determine the Mapping Radius: R

2349754239.92R

61441)(-47599.84-0762.9706640R

Example # 1

Determine the Plane Convergence: g

'24752' .44 '24-

or

785-0.4122909

575763))(0.549517056117.000278-25.116(

575763))(0.549517'01.001' 00' 15'-117 (116

)L)sin(B-(L 00

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

Example # 1

Determine Easting of the Station

e = E0 + R(sin(g))

e = 2000000.000 + 9754239.92234(sin(-0.4122909785))

e = 1929810.704

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”

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

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)

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)

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

Grid to Geodetic Conversion

Determine latitude: BB = 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)

Example # 2

Compute the Geodetic Coordinate of station “Class-2” from its CCS83 Zone 4 Metric Coordinates of:

n = 654048.453

e = 2000000.000

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

Example # 2

Determine the Longitude

L = L0 – (g/sin(B0))

L = 119° 00’ 00’’ – (0/sin(36.6258593071°))

L = 119° 00’ 00’’

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

Example # 2

Determine latitude: BB = 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’’

Problem # 2

Compute the Geodetic Coordinate of station “CC7” from its CCS83 Zone 3 Metric Coordinates of:

n = 674010.835

e = 1848139.628

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”

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)

Convergence Angle

Determine Grid Azimuth: t or Geodetic Azimuth: a

t = a – g + dt = 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)

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.

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’’

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’’

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.

Solution to Problem # 3

g = 0° 55” 51.361’ (0.9309335°)

Geodetic Azimuth = 46 20’ 51.361”

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

Groundh H

NR

adius of the Ellipsoid

REF =

R + N + H

R = Radius of Curvature.N = Geoidal Separation.H = Mean Height above

Geoid.h = Ellipsoidal Height

Combined Grid Factor (Combined Scale Factor)

Scale Decreases

Zon

e Li

mit

Zon

e Li

mit

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

A B D

CA’

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)

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)

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

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)

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

Converting Measured Ground Distances to Grid Distances

Determine Grid Distance

Ggrid = cgf(Gground)

Note: Gground is a horizontal ground distance

Converting Grid Distances to Horizontal Ground Distances

Determine Ground Distance

Gground = Ggrid/cgf

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)

Example # 4 Determine Radius of Curvature of

the Ellipsoid: Ra

Ra = r0/k0

Ra = 6374328/0.999894636561

Ra = 6374999.69189

Example # 4

Determine the Elevation Factor: re

re = Ra/(Ra + N + H)

re = 6374999.69189/(6374999.69189 – 30.5 + 3333.333)

re = 0.9994821768

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

Example # 4

Determine the Combined Grid Factor: cgf

cgf = re k

cgf = 0.9994821768(0.9998976162)

cgf = 0.999379846

Example # 4

Determine Grid Distance

Ggrid = cgf(Gground)

Ggrid = 0.999379846(909.909)

Ggrid = 909.3447

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)

Solution to Problem # 4

Ra = 6371934.463

re = 0.999656153

k = 0.999955870

cgf = 0.999612038

Ggrid = 1234.088m

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

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.

Solution to Problem # 5

n = 543163.942

e = 1979770.624