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Systems

• Required readings:• Coordinate systems:19-1 to 19-6.

• State plane coordinate systems: 20-1 to 20-5 to 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 (): the angle in the meridian plane of the point between the equator and the normal to the ellipsoid through that point.

– Geodetic longitude (): 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) cos cosY = (N+h) cos sinZ = ( N(1-e2) +h) sin

State Plane Coordinate Systems• Plane rectangular systems, why use them• Can we just use a single 2D Cartesian coordinate

system, (X, Y), origin Southwest of California?• The answer is that the distortions because of that flat

surface will be big, we have to use systems that cover smaller areas to limit the distortions. They are the “state plane coordinate systems” in the US

• How to construct them: Project the earth’s surface onto a developable surface.

• Two major projections: Lambert Conformal Conic, and Transverse Mercator.

Secants no distortions

Secants no distortions

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

true in between.

• Distortions are larger as you move away from the secants, we limit the width to limit distortions

Zone of limited distortion (1:10,000), 158 miles

Zones of limited distortion (1:10,000), 158 miles

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 -

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: , R, Rb, C, and how to get them.

Direct and Inverse Problems• Direct (Forward):

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

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

compute R, and . Use tables to compute , .

• In both cases, use a computer program whenever is available. Wolfpack can do it, see next slide.

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

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

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.

Central meridian