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Systems

• Required readings:• Coordinate systems:19-1 to 19-6.• State plane coordinate systems: 20-1 to 20-5, 20-7, 20-

10, and 20-12.• Required figures:

• Coordinate systems: 19-1, 19-2, 19-6, 19-7 and 19-8.• State plane coordinate systems: 20-1 to 20-3, 20-10.• Recommended, not required, readings: 19-7 to 19-11,

20-11, and 20-13.

Coordinate Systems

• Geoid and Ellipsoid, what for?

Ellipsoid Parameters• Ellipsoid parameters (equations not required):

• semi-major axes (a), semi-minor axes (b)• e = = first eccentricity

• N = normal length = • Great circles and meridians• Two main ellipsoids in North America:

• Clarke ellipsoid of 1866, on which NAD27 is based• Geodetic Reference System of 1980 (GRS80): on

which NAD83 is based.• For lines up to 50 km, a sphere of equal volume can be

used

a

e12 2 sin

2 2

2

a b

b

Geodetic Coordinate System• System components, coordinates

Geodetic System Coordinates• Definitions :

– Geodetic latitude (f): the angle in the meridian plane of the point between the equator and the normal to the ellipsoid through that point.

– Geodetic longitude (l): the angle along the equator between the Greenwich and the point meridians

– Height above the ellipsoid (h)

Universal Space Rectangular System• System definition, X, Y, Z• Advantage and disadvantage• X, Y, Z from geodetic coordinates

X = (N+h) cosf coslY = (N+h) cosf sinlZ = ( N(1-e2) +h) sinf

X

Z

Y

State Plane Coordinate Systems• Plane rectangular systems, why use them?• How to construct them: Project the earth’s

surface onto a developable surface. • Two major projections: Lambert Conformal

Conic, and Transverse Mercator.

Coneann.gif

Secants, Scales, and Distortions• Scale is exact along the secants, smaller than

true in between.• Distortions are larger away from the secants

Choosing a Projection

• States extending East-west: Lambert Conical• States extending North-South: Mercator

Cylindrical.• A single surface will provide a single zone.

Maximum zone width is 158 miles to limit distortions to 1:10,000. States longer than 158 mi, use more than one zone (projection).

Standard Parallels & Central Meridians

• Standard Parallels: the secants, no distortion along them. At 1/6 of zone width from zone edges

• Central Meridians: a meridian at the middle of the zone, defines the direction of the Y axis.

• The Y axis points to the grid north, which is the geodetic north only at the central meridian

• To compute the grid azimuth ( from grid north) from geodetic azimuth ( from geodetic north):

grid azimuth = geodetic azimuth - q

Geodetic and SPCS• Control points in

SPCS are initially computed from Geodetic coordinates (direct problem). If NAD27 is used the result is SPCS27. If NAD83 is used, the result is SPCS83.

• Define: q, R, Rb, C, and how to get them.

Direct and Inverse Problems• Direct (Forward):

• given: f, l get X, Y?• Solution: X = R sin q + C• Y = Rb - R cos q• Whenever q is used, it is -ve west (left) of the central

meridian.• q = geodetic azimuth - grid azimuth• Indirect (Inverse): Solve the above mentioned equations

to compute R, and q. Use tables to compute f, l .• In both cases, use a computer program whenever

is available. Wolfpack can do it, see next slide.

Grid N

Grid Az

Geodetic Az

Geodetic NGrid NB

A

Geodetic N

Geodetic Az

Grid Az

q = geodetic azimuth - grid azimuth

qPOSITIVE at point AEast of the CM

: NEGATIVE at point AWest of the CM

Central Meridian of the zone

Central Meridian of the zone

q

• Forward Computations: given (f, ) l get (X, Y).

• Inverse Computations: given (X, Y) get (f, ) l

Surveys Extending from one Zone to Another

• There is always an overlap area between the zones.

• When in the transition zone, compute the geodetic coordinates of two points from their X, Y in first zone (direct problem).

• Compute X, Y of the same points in the second zone system from their geodetic coordinates (inverse problem)

• Compute the azimuth of the line, use the azimuth and new coordinates to proceed.

Review of Project 2

PI (V)

PC

TC

20BVC 1

BVC 2

Last tree

Yellow-to post

Two vertical curves can be set-out at the same time from either BVC 1 or BVC 2.The direction of each of the Centerlines curves is shown to the left. Assume that the BM for elevation is BVC and IS NOT 20