Map Projections
for a Spherical Planet
Prof. David Bernstein
James Madison University
Computer Science Department
bernstdh@jmu.edu
Motivation
Some Observations:
The Earth is not flat
Maps usually are (especially on computer displays)
The Implication:
We need a way to project points on the surface of the Earth onto a map
Thinking About Map Projections
Using a Physical Model:
Easier to understand
Algebraically:
Much more flexible
Physical Models
The Idea:
Shine a light onto or through a transparent Earth and capture the shadows cast by the opaque features
The "Parameters":
The shape of the screen (called the projection surface)
The position of the projection surface
The location of the light source
Projection Surfaces
Positioning a Planar Projection Surface
Positioning a Cylindrical Projection Surface
Positioning a Conical Projection Surface
Light Sources
Polar Azimuthal Orthographic Projection
Notation
Polar Azimuthal Orthographic Projection (cont.)
Derivation of the Projection:
\(\cos \phi = d / R \Rightarrow d = R \cos \phi\)
\(\cos \lambda = p_{2}/d \Rightarrow p_{2} = d \cos \lambda = R\cos \phi \cos \lambda\)
\(\sin \lambda = p_{1}/d \Rightarrow p_{1} = d \sin \lambda = R \cos \phi \sin \lambda\)
\(\)
Derivation of the Inverse:
\(d = ||\bs{p}||\)
\(\cos \lambda = \frac{p_{2}}{||\bs{p}||} \Rightarrow \lambda = \cos^{-1}(\frac{p_{2}}{||\bs{p}||})\)
\(\cos \phi = d / R \Rightarrow \phi = \cos^{-1}(\frac{||\bs{p}||}{R})\)
Polar Azimuthal Orthographic Projection (cont.)
The World
Cylindrical Stereographic Projection
Notation
Cylindrical Stereographic Projection (cont.)
Derivation of the Projection:
Since the triangles are "similar" \(\frac{b}{R+a} = \frac{p_{2}}{R+R}\), hence \(p_{2} = (\frac{b}{a+R})2R\)
\(\sin \phi = b/R \Rightarrow b = R \sin \phi\)
\(\cos \phi = a/R \Rightarrow a = R \cos \phi\)
So, \(p_{2} = \frac{R \sin \phi}{R+R\cos \phi}2R=\frac{R \sin \phi}{R(1+\cos \phi)}2R = \frac{\sin\phi}{1+\cos\phi}2R\)
\(p_{1} = \lambda R\) since it is only determined by where the cylinder is "cut"
Derivation of the Inverse:
\(\lambda = p_{1}/R\)
\(\tan \phi = p_{2}/R \Rightarrow \phi = \tan^{-1}(p_{2}/R)\) (but you have to be careful about the domain of the \(\tan\))
Desirable Properties of Projections
The Most Common:
Conformal (i.e., angles are preserved)
Equal Area (i.e., areas are in constant proportion)
Equidistant (i.e., distances are in constant proportion)
An Important Mathematical Result:
A single projection can not be both conformal and equal area
Desirable Properties of Projections (cont.)
Nerd Humor - Bad Map Projection: South America
(Courtesy of
xkcd
)
Sinusoidal Projection
Derivation of an Equal Area Projection:
Uniformly space the parallels so \(p_{2} = \phi R\)
To be equal area given these parallels, the length of each parallel has to equal the "circumference" at that latitude (which is \(R \cos \phi\))
The projected parallel must be proportional to this circumference so it must have length \(k 2 \pi R \cos \phi\)
To be consistent at the equator (where the "circumference" is \(2 \pi R\)), we must have \(k = 0.5\)
Assuming a constant horizontal scale, we must have \(\frac{p_{1}}{\pi R \cos \phi} = \frac{\lambda}{\phi}\)
So, \(p_{1} = \lambda R \cos \phi\)
Note:
We did not use a physical model
Sinusoidal Projection (cont.)
The World
Equatorial Cylindrical Equal Area Projection
Parameters:
\(\lambda_{0}\) is the standard longitude (i.e., the horizontal center of the projection)
Projection:
\(p_{1} = R (\lambda - \lambda_{0}) \)
\(p_{2} = R \sin(\phi)\)
Inverse:
\(\lambda = \lambda_{0} + \frac{p_{1}}{R}\)
\(\phi = \sin^{-1}(p_{2} / R)\)
Equatorial Cylindrical Equal Area Projection (cont.)
The World
Equatorial Cylindrical Conformal Projection (Mercator)
Parameters:
\(\lambda_{0}\) is the standard longitude
Projection:
\(p_{1} = R (\lambda - \lambda_{0}) \)
\(p_{2} = R \ln[\tan(\pi / 4 + \phi / 2)]\)
Inverse:
\(\lambda = \lambda_{0} + \frac{p_{1}}{R}\)
\(\phi = 2 [\tan^{-1}(e^{\frac{p_{2}}{R}}) - \pi / 4]\)
Equatorial Cylindrical Conformal Projection (cont.)
The World
Conical Equal Area Projection (Albers)
Parameters:
\(\lambda_{0}\) and \(\phi_{0}\) are the longitude and latitude of the origin of the transformed coordinates
\(\phi_{1}\) and \(\phi_{2}\) are the two standard parallels (i.e., the parallels where the cone intersects the sphere)
Preliminary Calculations:
\(n = 0.5 (\sin(\phi_{1}) + \sin(\phi_{2}))\)
\(c = \cos^{2}(\phi_{1}) + 2 n \sin(\phi_{1})\)
\(\rho_{0} = \frac{\sqrt(c - 2 n \sin(\phi_{0}))}{n}\)
Projection:
\(\rho = \frac{\sqrt{c - 2 n \sin(\phi)}}{n}\)
\(\theta = n(\lambda - \lambda_{0}) \)
\(p_{1} = R \rho \sin(\theta)\)
\(p_{2} = R (\rho_{0} - \rho \cos(\theta))\)
Inverse:
\(a = \sqrt{(p_{1}/R)^{2}+(\rho_{0}-p_{2}/R)^{2}}\)
\(b = \tan^{-1}((p_{1}/R)/(\rho_{0} - p_{2}/R))\)
\(\phi = \sin^{-1}(\frac{c- a^{2} n^{2}}{2n})\)
\(\lambda = \lambda_{0} + b/n\)
Conical Equal Area Projection (cont.)
The World
There's Always More to Learn