Saturday, December 18, 2010

Math Foundations

We haven't been writing chapters in the order they will appear in our book. The most recent chapter, Math Foundations, is no exception. It will be Chapter 2, but is basically the last chapter I am writing. After this, I only have to finish the introduction and smooth out a few other sections. Kevin has a similar amount of work left, so we are on schedule!

This chapter is not on math foundations for general 3D graphics; rather, it is on useful mathematics for virtual globes, with a focus on ellipsoids. This chapter is unique in that it contains some derivations, but we are computer science practitioners, not mathematicians, so it also provides working code in a handy Ellipsoid class. Our colleague, Jim Woodburn, helped significantly with the derivations.

This chapter covers:
  • Geographic (longitude, latitude, height) and WGS84 (x, y, z) coordinates.
  • Ellipsoids
    • Oblate spheroids (Earth's shape).
    • Surface normals: geodetic vs. geocentric.
    • Conversion between geographic and WGS84 coordinates.
    • Scaling a arbitrary point in space to the ellipsoid surface.
    • Curves on ellipsoids.
I also wrote two related examples applications, one demonstrating geodetic and geocentric surface normals, and another for computing curves on an ellipsoid by slicing a plane through it. Here's a video of the later:

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