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California Coordinate System

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California Coordinate System

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California Coordinate System

Thomas Taylor, PLS

Right of Way Engineering

District 04

(510) 286-5294

Tom_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

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

- 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.

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

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

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)

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

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)

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

Example # 1

- Determine the Radial Difference: u

Example # 1

- Determine the Mapping Radius: R

Example # 1

- Determine the Plane Convergence: g

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: 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)

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: 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’’

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 + 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)

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.

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

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

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.