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Great Circle Distances
for a Spherical Planet


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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Definitions
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  • Great Circle:
    • A section of a sphere that contains a diameter of the sphere
  • Orthodrome:
    • Shortest distance between points on a sphere (which will be a segment of a great circle)
From Longitude/Latitude to Cartesian Coordinates
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Notation

longitude-latitude-to-cartesian
From Longitude/Latitude to Cartesian Coordinates (cont.)
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  • Derivation of the Conversion:
    • \(\cos \phi = d / R \Rightarrow d = R \cos \phi\)
    • \(\cos \lambda = p_{z}/d \Rightarrow p_{z} = d \cos \lambda = R\cos \phi \cos \lambda\)
    • \(\sin \lambda = p_{x}/d \Rightarrow p_{x} = d \sin \lambda = R \cos \phi \sin \lambda\)
    • \(p_{y} = R \sin \phi\)
  • Note:
    • This is very similar to the conversion of spherical coordinates to Cartesian coordinates (but uses the latitude not the colatitude)
Finding the Great Circle Distance
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The great circle distance is the portion of the circumference described by \(\alpha\).

great-circle-distance
  • Arc Length (Using Degrees):
    • \(d = \frac{\alpha}{360} \cdot 2 \pi R\)
  • Arc Length (Using Radians):
    • \(d = \frac{\alpha}{2 \pi} \cdot 2 \pi R = \alpha R\)
Going Further: Finding the Angle from the Longitude and Latitude
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  • From the Law of Cosines:
    • \(\cos \alpha = \frac{\bs{q} \cdot \bs{p}}{||\bs{q}|| \enspace ||\bs{p}||}\)
  • Solving:
    • \(\cos \alpha = \frac{ (R \cos \phi_{q} \cos \lambda_{q}) (R \cos \phi_{p} \cos \lambda_{p}) + (R \cos \phi_{q} \sin \lambda_{q}) (R \cos \phi_{p} \sin \lambda_{p}) + (R \sin \phi_{q})(R \sin \phi_{p}) }{R R} \)
    • \(\alpha = \cos^{-1} (\cos \phi_{q} \cos \lambda_{q}) (\cos \phi_{p} \cos \lambda_{p}) + (\cos \phi_{q} \sin \lambda_{q}) (\cos \phi_{p} \sin \lambda_{p}) + (\sin \phi_{q})(\sin \phi_{p}) \)
There's Always More to Learn
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