CS 128/ES 228 - Lecture 3a1 Map projections. CS 128/ES 228 - Lecture 3a2 The dilemma Maps are flat, but the Earth is not! Producing a perfect map is like

CS 128/ES 228 - Lecture 3a 1

Map projections


The dilemma

Maps are flat, but the Earth is not!

Producing a perfect map is like peeling an orange and flattening the peel without distorting

a map drawn on its surface.


For example:

The Public Land Survey System

• As surveyors worked north along a central meridian, the sides of the sections they were creating converged

• To keep the areas of each section ~ equal, they introduced “correction lines” every 24 miles


Like this

Township SurveyKent County, MI

1885

http://en.wikipedia.org/wiki/Image:Kent-1885-twp-co.jpg


One very practical result

The jog created by these “correction lines”, where the old north-south line abruptly stopped and a new one began 50 or 60 yards east or west, became a feature of the grid, and because back roads tend to follow surveyors’ lines, they present an interesting driving hazard today. After miles of straight gravel or blacktop, the sudden appearance of a correction line catches most drivers by surprise, and frantic tire marks show where vehicles have been thrown into hasty 90-dgree turns, followed by a second skid after a short stretch running west or east when the road head north again onto the new meridian.

Andro Linklater. 2002. Measuring America. Walker & Co., NY. P. 162


Geographical (spherical) coordinates

Latitude & Longitude (“GCS” in ArcMap)

Both measured as angles from center of Earth

Reference planes: - Equator for latitude

- Prime meridian for longitude


Lat/Long. are not Cartesian coordinates

They are angles measured from the center of Earth

They can’t be used (directly) to plot locations on a plane

Understanding Map Projections. ESRI, 2000 (ArcGIS 8). P. 2


Parallels and Meridians

Parallels: lines of latitude.

Everywhere parallel

1o always ~ 111 km (69 miles)

Some variation due to ellipsoid (110.6 at equator, 111.7 at pole)

Meridians: lines of longitude.

Converge toward the poles

1o =111.3 km at 1o

= 78.5 “ at 45o

= 0 “ at 90o


Overview of the cartographic process

1. Model surface of Earth mathematically

2. Create a geographical datum

3. Project curved surface onto a flat plane

4. Assign a coordinate reference system


1. Modeling Earth’s surface

Ellipsoid: theoretical model of surface - not perfect sphere - used for horizontal measurements

Geoid: incorporates effects of gravity - departs from ellipsoid because of different rock densities in mantle - used for vertical measurements


Ellipsoids: flattened spheres

Degree of flattening given by f = (a-b)/a

(but often listed as 1/f)

Ellipsoid can be local or global


Local Ellipsoids

Fit the region of interest closely

Global fit is poor

Used for maps at national and local levels


Examples of ellipsoids

Local Ellipsoids Inverse flattening (1/f)

Clarke 1866 294.9786982

Clarke 1880 293.465

N. Am. 1983

Global EllipsoidsInternational 1924 297

GRS 80 (Geodetic Ref. Sys.) 298.257222101

WGS 84 (World Geodetic Sys.) 298.257223563


2. Then what’s a datum?

Datum: a specific ellipsoid + a set of “control points” to define the position of the ellipsoid “on the ground”

Either local or global

> 100 world wideSome of the datums stored in Garmin 76 GPS receiver


North American datums

Datums commonly used in the U.S.:

- NAD 27: Based on Clarke 1866 ellipsoid Origin: Meads Ranch, KS - NAD 83: Based on GRS 80 ellipsoid

Origin: center of mass of the Earth


Datum Smatum

NAD 27 or 83 – who cares?

One of 2 most common sources of mis-registration in GIS

(The other is getting the UTM zone wrong – more on that later)


3. Map Projections

Why use a projection?

1. A projection permits spatial data to be displayed in a Cartesian system

2. Projections simplify the calculation of distances and areas, and other spatial analyses


Properties of a map projection

Area

Shape

Projections that conserve area are called equivalent

Distance

Direction

Projections that conserve shape are called conformal


Two rules:

Rule #1: No projection can preserve all four properties. Improving one often makes another worse.

Rule #2: Data sets used in a GIS must be in the same projection. GIS software contains routines for changing projections.


Classes of projections

a. Cylindrical

b. Planar (azimuthal)

c. Conical


Cylindrical projections

Meridians & parallels intersect at 90o

Often conformal

Least distortion along line of contact (typically equator)

Ex. Mercator - the ‘standard’ school map

http://ioc.unesco.org/oceanteacher/resourcekit/Module2/GIS/Module/Module_c/module_c4.html


Transverse Mercator projection

Mercator is hopelessly poor away from the equator

Fix: rotate the projection 90° so that the line of contact is a central meridian (N-S)

Ex. Universal Transverse Mercator


Planar projections

a.k.a Azimuthal

Best for polar regions


Conical projections

Most accurate along “standard parallel”

Meridians radiate out from vertex (often a pole)

Ex. Albers Equal Area

Poor in polar regions – just omit those areas


Compromise projections

http://ioc.unesco.org/oceanteacher/resourcekit/Module2/GIS/Module/Module_c/module_c4.html

Robinson world projection Based on a set of

coordinates rather than a mathematical formula

Shape, area, and distance ok near origin and along equator

Neither conformal nor equivalent (equal area). Useful only for world maps


More compromise projections


What if you’re interested in oceans?

http://www.cnr.colostate.edu/class_info/nr502/lg1/map_projections/distortions.html


“But wait: there’s more …”

http://www.dfanning.com/tips/map_image24.html

All but upper left: http://www.geography.hunter.cuny.edu/mp/amuse.html


Buckminster Fuller’s “Dymaxion”

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CS 128/ES 228 - Lecture 3a1 Map projections. CS 128/ES 228 - Lecture 3a2 The dilemma Maps are flat, but the Earth is not! Producing a perfect map is like