What are the 'lines' in \(PC \)?

We have two choices:

  1. We can define a line to be a path traced out as the rolling ball rolls along a geodesic without slipping or twisting.

    This notion of line depends on the ratio \(R\) of the radii of the fixed and the rolling balls.

  2. We can define a line using the split octonions.

    (Details later.)



Theorem. These notions agree if and only if \(R = 1/3 \) or \(R = 3\).