Map Projections and Coordinate Systems · 2020. 6. 28. · Specifying Projections • The type of developable surface (e.g., cone) • The size/shape of the Earth (ellipsoid, datum),

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Map Projections and Coordinate Systems

Map Projections

A systematic

rendering

3-D spherical

to

2-D Cartesian

system.

Many Projections: Minnesota examplehttp://rocky.dot.state.mn.us/geod/projections.htm

Projections may be categorized by:

•The projection surface

•Distortion properties

•Surface orientation

•The location of projection source

Cone – Conic

Cylinder -

Cylindrical

Plane -

Azimuthul

The projection

surface:

Projection Surfaces – “developable”

Developable Surface- Cone

Developable Surface - Cylinder

Projections are like political parties, they distort everything

Distortion properties:

preserve local shape - conformal (orthomorphic)

preserve scale – equidistant

preserve direction – azimuthal

preserve great circles – gnomic

preserve circular shapes - stereographic

Distortion Varies Across Map

Distortion varies by developable

surface

CylindricalConic

Projections Categorized by

Orientation:

Equatorial - intersecting equator

Transverse - at right angle to

equator

Gnomonic -

center of globe

Stereographic -

at the antipode

Orthographic -

at infinity

Source:http://www.fes.uwaterloo.ca/crs/geog165/mapproj.htm

Categorized by the Location of Projection

Source

Specifying Projections

• The type of developable surface (e.g., cone)

• The size/shape of the Earth (ellipsoid,

datum), and size of the surface

• Where the surface intersects the ellipsoid

• The location of the map projection origin on

the surface, and the coordinate system units

Defining a Projection – LCC(Lambert Conformal Conic)

• The LCC requires

we specify an

upper and lower

parallel

• A spheroid

• A central meridian

• A projection origin

centralmeridian

origin

Many LCC Projections: Minnesota examplehttp://rocky.dot.state.mn.us/geod/projections.htm

Lecture Question:

Ellipsoid, Projection

Are the related? How

Lecture Question:

What is a developable surface?

What are the common shapes?

Lecture Question:

What is a datum transformation?

How does it differ from a map projection?

“Standard” Projections

• Governments (and other organizations)

define “standard” projections to use

• Projections preserve specific geometric

properties, over a limited area

•Imposes uniformity, facilitates data exchange,

provides quality control, establishes limits on

geometric distortion.

State Plane Coordinate System Zones

State Plane Coordinate System

• Each state partitioned into zones

• Each zone has a different projection specified

• Distortion in surface measurement less than 1 part in 10,000 within a zone

CaliforniaState PlaneZones

State Plane Coordinate Systems

• Uses Lambert conformal conic (LCC) and Transverse Mercator (TM, cylindrical)

• LCC when long dimension East-West

• TM when long dimension N-S

• May be mixed, as many zones used as needed

State Plane Coordinate System

e.g., Minnesota Central State Plane Zone

Projection type: Lambert Conformal Conic

Spheroid: GRS80

Central Meridian: -94.25

Reference Latitude: 45.00

1st Stand. Parallel: 45.6166666

2nd Stand. Parallel: 47.05

False Easting: 800000

False Northing: 100000

Universal Transverse Mercator –

UTM System

UTM Zone

Details

Each Zone is 6 degrees wide

Zone location defined by a

central meridian

Origin at the Equator, 500,000m

west of the zone central Meridian

Coordinates are always positive

(offset for south Zones)

Coordinates discontinuous across

zone boundaries

Universal Transverse Mercator

Projection – UTM Zones for the U.S.

UTM

Measurements of

distance, shape &

area with .04% or

less distortion.*

Grid allows a slight

tilt from True North.

“UTM grid

declination”*

From USGS 1:24k map sheet

Coordinate Systems Notation

Latitude/Longitude

Degrees Minutes Seconds 45° 3' 38" N

Degrees Minutes (decimal) 45° 3.6363' N

Degrees (decimal) 45.0606° N

State Plane (feet) 2,951,384.24 N

UTM (meters) 4,996,473.72 N

UTM Coordinates (meters)

Common “Topo” format283000m E “Easting” (East of the zone meridian;

remember false easting

concept)3904000m N “Northing” (North of the Equator)

Standard

UTM 10S 0545980E 4185742N

National Projections

Map Projections vs. Datum

Transformations

• A map projections is a systematic

rendering from 3-D to 2-D

• Datum transformations are from one

datum to another, 3-D to 3-D

• Changing from one projection to

another may require both

Simple Projection: Example

f = 30, l = 45

Example:

X = R (30*p/180- 0)

= 6,378 (1.7320)

= 11,047

Y = R ln(tan(45+45/2))

= 6,378 ln(1.73205)

= 6,378 * 0.5493

= 3,503

45

From one Projection to Another

Map Projections Summary

• Projections specify a two-dimensional

coordinate system from a 3-D globe

• All projections cause some distortion

• Errors are controlled by choosing the proper

projection type, limiting the area applied

• There are standard projections

• Projections differ by datum – know your

parameters

Non-Coordinate Systems

Public Land Survey System

(PLSS)

Have you ever looked out the windows while flying

on an airplane over central and western US?



Historical Surveying

Metes and Bounds

Used in many of the Eastern and

Southeastern States.

Also used for land titles and transactions.

Descriptions have:

A point of beginning,definite corners, lengths and

directions of the property sides, names of adjoining

property owners and area.*

* Source:http://feature.geography.wisc.edu/sco/geodetic/plss.html

Historical Surveying

Metes and Bounds

from Mercer County, Kentucky Deed Book 7, p.417

13 February 1810. Rice BEADLES of Lincoln Co., Ky sells 140 acres

and 36 poles of land in Mercer County on Doctors Fork to John

CRAIN for 200 pounds.

Land bounded as follows...: [Ref. 1]Beginning at the mouth of a branch at an ash stump

thence up the creek S 20 poles to 2 beach

thence east 41 poles to a small walnut in Arnett's line

thence north 50 east 80 poles to a linn hickory dogwood in said line

thence north 38 poles to an ash

thence west 296 poles with Potts's line till it intersects with Tolly's line

thence south 30 west 80 poles to a whiteoak and sugar thence east 223 poles to

beginning....

Source: http://www.outfitters.com/genealogy/land/crain.html

Metes and Bounds

Source: http://www.outfitters.com/genealogy/land/crain.html

Source:http://www.ca.blm.gov/cadastral/meridian.html

The non-Coordinate System: The Public Land Survey System

Public Land Survey System PLSS(digitized Section, township and Range information

Begun by Thomas Jefferson(To pay war debts to Revolutionary War Soldiers

Land allotments rather than hard currency)

Used in much of the Central and Western US

Established for inventory and transfer of property

Grid system

6 square mile Sections within

East West oriented Ranges

North South oriented Townships


PLSS – a method of defining parcels by successive subdivision

Measurement Systems

How much a man behind an ox

could plow in one day

http://www.glorecords.blm.gov/images/page2_large.gif

6 miles by 6

Miles

Each Section 1

mile by 1 mile

Measured in

Chains (66 feet)

Easy way to figure

acres before

calculators

3 chains by 7 chains =

21 sq chains

divide by 10 to get acres

2.1 acres

Example of a PLSS Section Corner Monument





Source:http://feature.geography.wisc.edu/sco/geodetic/geodetic_graphics/Township_Big.jpg

36 sections in

a Township,

each section

approximately

a mile square.

(80 chains x 80 chains

= 6400 sq chains or

640 acres)

Note:

zig-zag or

boustrophedon

numbering

scheme

Source:http://feature.geography.wisc.edu/sco/geodetic/geodetic_graphics/Township_Big.jpg

Sections may

be further

subdivided

640 acres

1 mile by 1 mile

160 acres3

20

acre

s

40 acres

“Aliquot division”

Format:

Section,Township, Range, Meridian, County

For Example:

Northeast Quarter of the Northeast Quarter

of Section 5 in Township 101, Range 35 W

of the Fifth Principal Meridian, Jackson

County, Minnesota

NE 1/4, NE 1/4, S5, T101, R35W, 5th PM,

Jackson County, Minnesota


Lake City

7.5 minute

USGS

Topographical

Map

1:24,000 scale

REVIEW

Format:

Section,Township, Range, Meridian, County

For Example:

Northeast Quarter of the Northeast Quarter

of Section 5 in Township 101, Range 35 W

of the Fifth Principal Meridian, Jackson

County, Minnesota

NE 1/4, NE 1/4, S5, T101, R35W, 5th PM,

Jackson County, Minnesota

REVIEW

Lake City

7.5 minute

USGS

Topographical

Map

1:24,000 scale

(detail examples

of this map

follow)

Lake City

7.5 minute

USGS

Topographical

Map

1:24,000 scale

Next Slides are examples of:

Public Land Survey System PLSS(digitized Section, township and Range information

Grid system

6 square mile Sections within

East West oriented Ranges

North South oriented Townships

Documents

Map Projections and Coordinate Systems · 2020. 6. 28. · Specifying Projections • The type of developable surface (e.g., cone) • The size/shape of the Earth (ellipsoid, datum),