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Map Projections
for a Spherical Planet


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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Motivation
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  • 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
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  • Using a Physical Model:
    • Easier to understand
  • Algebraically:
    • Much more flexible
Physical Models
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  • 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
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projection-surfaces
Positioning a Planar Projection Surface
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projections_planar
Positioning a Cylindrical Projection Surface
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projections_cylindrical
Positioning a Conical Projection Surface
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projections_conical
Light Sources
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projections_light-sources
Polar Azimuthal Orthographic Projection
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Notation

projection_derivation_polar-azimuthal-orthographic
Polar Azimuthal Orthographic Projection (cont.)
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  • 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.)
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The World

projection_polar-azimuthal-orthographic_world
Cylindrical Stereographic Projection
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Notation

projection_derivation_cylindrical-stereographic
Cylindrical Stereographic Projection (cont.)
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  • 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
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  • 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.)
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Nerd Humor - Bad Map Projection: South America
/imgs
(Courtesy of xkcd)
Sinusoidal Projection
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  • 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.)
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The World

projection_sinusoidal_world
Equatorial Cylindrical Equal Area Projection
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  • 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.)
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The World

projection_equatorial-cylindrical-equalarea_world
Equatorial Cylindrical Conformal Projection (Mercator)
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  • 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.)
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The World

projection_equatorial-mercator_world
Conical Equal Area Projection (Albers)
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  • 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.)
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The World

projection_albers_world
There's Always More to Learn
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