  The linear-time ham-sandwich cut algorithm of Lo, Matou\v{s}ek, and Steiger
for bi-chromatic finite point sets in the plane works by appropriately
selecting crossings of the lines in the dual line arrangement with a set of
well-chosen vertical lines. We consider the setting where we are not given the
coordinates of the point set, but only the orientation of each point triple
(the order type) and give a deterministic linear-time algorithm for the
mentioned sub-algorithm. This yields a linear-time ham-sandwich cut algorithm
even in our restricted setting. We also show that our methods are applicable to
abstract order types.
