Map Projections

Map Projections
for a Spherical Planet

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

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 locally so shapes appear "true")
- Equal Area (i.e., areas are in constant proportion)
- Note: A single projection can not be both conformal and equal area
Somewhat Less Common:
- Equidistant (i.e., distances are in constant proportion, but only from certain points or along certain lines)

Desirable Properties of Projections (cont.)

Nerd Humor - Bad Map Projection: South America
/imgs

(Courtesy of xkcd)

Sinusoidal (Sanson-Flamsteed) Projection

"Casual Derivation" of an Equal Area Projection:
- The "circumference" of a line of latitude (i.e., a parallel) is \(2 \pi R \cos \phi\)
- We want \(\frac{d p_1}{d \lambda}\) to be proportional to the "circumference", so \(\frac{d p_1}{d \lambda} = k R \cos \phi\)
- We want \(\frac{d p_2}{d \phi}\) to be proportional to the radius, so \(\frac{d p_2}{d \phi} = h R\)
- We can simplify by imposing the conditions that the scale factors are 1 (i.e., \( h = k = 1\)).
- \(\frac{d p_1}{d \lambda} = R \cos \phi \Rightarrow p_1 = R \cos \phi (\lambda - \lambda_0)\), where \(\lambda_0\) is the reference longitude that we can set to 0.
- \(\frac{d p_2}{d \phi} = R \Rightarrow p_2 = R \phi\)
Interpretation:
- \(p_2 = R \phi\) means that the parallels are evenly spaced.
- Given this, the projection preserves area because the length of each parallel equals the "circumference".

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