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On degeneracy of lower envelopes of algebraic surfaces

We analyze degeneracy of lower envelopes of algebraic surfaces. We focus on the cases omitted in the existing complexity analysis of lower envelopes [Halperin and Sharir 1993], and re-define the degeneracy from the viewpoint of the adjacency structure and the number of connected components. We also define badness of degeneracy from such viewpoint and show how bad the degeneracy can be. This research is intended to contribute to a robust geometrical computation in the tolerant model.