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Map projections

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Map projections

CS 128/ES 228 - Lecture 3a

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.

CS 128/ES 228 - Lecture 3a

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

CS 128/ES 228 - Lecture 3a

Township Survey

Kent County, MI

1885

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

CS 128/ES 228 - Lecture 3a

http://www.texas-flyer.com/ms150/img/riders05.jpg

CS 128/ES 228 - Lecture 3a

Latitude & Longitude(“GCS” in ArcMap)

- Both measured as angles from the center of Earth
- Reference planes:
- Equator for latitude- Prime meridian (through Greenwich, England) for longitude

CS 128/ES 228 - Lecture 3a

- 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

CS 128/ES 228 - Lecture 3a

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 0o = 78.5 “ at 45o
= 0 “ at 90o

CS 128/ES 228 - Lecture 3a

- Model surface of Earth mathematically
- Create a geographical datum
- Project curved surface onto a flat plane
- Assign a coordinate reference system

(leave for next lecture)

CS 128/ES 228 - Lecture 3a

- 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

CS 128/ES 228 - Lecture 3a

- Degree of flattening given by f = (a-b)/a(but often listed as 1/f)
- Ellipsoid can be local or global

CS 128/ES 228 - Lecture 3a

- Fit the region of interest closely
- Global fit is poor
- Used for maps at national and local levels

http://exchange.manifold.net/manifold/manuals/5_userman/mfd50The_Earth_as_an_Ellipsoid.htm

CS 128/ES 228 - Lecture 3a

CS 128/ES 228 - Lecture 3a

- 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 wide

Some of the datums stored in Garmin 76 GPS receiver

CS 128/ES 228 - Lecture 3a

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

CS 128/ES 228 - Lecture 3a

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)

CS 128/ES 228 - Lecture 3a

Why use a projection?

- A projection permits spatial data to be displayed in a Cartesian system
- Projections simplify the calculation of distances and areas, and other spatial analyses

CS 128/ES 228 - Lecture 3a

Area

Shape

Projections that conserve area are called equivalent

Distance

Direction

Projections that conserveshape are called conformal

CS 128/ES 228 - Lecture 3a

Leonardo da Vinci [?], c. 1514

http://www.odt.org/hdp/

CS 128/ES 228 - Lecture 3a

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 displayed in the same projection. GIS software contains routines for changing projections.

CS 128/ES 228 - Lecture 3a

- Cylindrical
- Planar (azimuthal)
- Conical

CS 128/ES 228 - Lecture 3a

- 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

CS 128/ES 228 - Lecture 3a

- Mercator is hopelessly distorted away from the equator
- Fix: rotate 90° so that the line of contact is a central meridian (N-S)
- Ex. Universal Transverse Mercator (UTM)

CS 128/ES 228 - Lecture 3a

- a.k.a Azimuthal
- Best for polar regions

CS 128/ES 228 - Lecture 3a

- Most accurate along “standard parallel”
- Meridians radiate out from vertex (often a pole)
- Poor in polar regions – just omit those areas
- Ex. Albers Equal Area. Used in most USGS topographic maps

CS 128/ES 228 - Lecture 3a

- Robinson world projection
- Based on a set ofcoordinates 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

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

CS 128/ES 228 - Lecture 3a

CS 128/ES 228 - Lecture 3a

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

CS 128/ES 228 - Lecture 3a

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

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

CS 128/ES 228 - Lecture 3a

CS 128/ES 228 - Lecture 3a