Satellite Posi+oning Lecture 6b: Map Projec+ons Jan Johansson [email protected] Chalmers University of Technology, 2013 Satellite Posi+oning Map projections
What is a map projection?
How are they constructed?
We will derive the Mercator map projection
Summary
1. Geocentric
Satellite Posi+oning Three ways to
describe your position
on the earth?
Cartesian
2. Geodetic
3. in the map
projection
Satellite Posi+oning How to describe your
position on the
earth?
Latitude
Longitude
Meridians
and
Parallels
Pictures from:
Snyder, John P. 1987. Map projections: a working manual. USGS Professional
Paper 1395. Washington, DC: United States Government Printing Office
Satellite Posi+oning What is a map projection?
“any systematic arrangements of
meridians and parallells portraying
the curved surface of the earth
upon a plane”
from Maling
Most pictures from:
Snyder, John P. 1987. Map projections: a working manual. USGS Professional
Paper 1395. Washington, DC: United States Government Printing Office
Satellite Posi+oning Properties of a map
projection?
The Indicatrix may tell
you!
Or how is an infinitesimal
small circle in reality,
depicted on the projection
plane?
As an ellips!
For conform projections,
the ellipsoid is an circle
with changed size, h=k,
For equal area
projections, the ellipsoid
will have size=1, h*k=1.
Pictures from:
Snyder, John P. 1987. Map projections: a working manual. USGS Professional
Paper 1395. Washington, DC: United States Government Printing Office
Satellite Posi+oning Gerardus Mercator
(1512-1594)
Used a normal
cylindrical conformal
projection for a large
world map in 1569.
Pictures from:
Snyder, John P. 1987. Map projections: a working
manual. USGS Professional Paper 1395. Washington,
DC: United States Government Printing Office
Satellite Posi+oning Derivation of Mercator map projection
(assuming a spherical earth)
MercatorP
map
projection
Satellite osi+oning Pictures from:
Snyder, John P. 1987. Map projections: a working
manual. USGS Professional Paper 1395. Washington,
DC: United States Government Printing Office
Satellite Posi+oning Lambert cylindrical equal area map projection
Length of equator same as for the Mercator map projection
Pictures from:
Snyder, John P. 1987. Map projections: a working manual. USGS Professional
Paper 1395. Washington, DC: United States Government Printing Office
Summary
Satellite Posi+oning • What is a map projection?
• Types of map projections?
Perspective
non-perspective
cylindrical
conic
azimuthal
• Properties
the indicatrix
h=k
conformal or orthomorphic
h·k=1
equal-area
• The Mercator projection
cylindrical
conformal (orthomorphic)
non-perspective
Pictures from:
Snyder, John P. 1987. Map projections: a working manual. USGS Professional
Paper 1395. Washington, DC: United States Government Printing Office
Comment
Satellite Posi+oning The mathematical derivation for the Mercator map
projection was done under the assumption of a
spherical earth.
For an ellipsoid the principles are equal, however
the mathematical derivation becomes much more
complicated.