Geometric algebra and singularities of ruled and developable surfaces

Any ruled surface in R³ is described as a curve of unit dual vectors in the algebra of dual quaternions (=the even Clifford algebra Cℓ⁺(0, 3, 1)). Combining this classical framework and A-classification theory of C^∞ map-germs (R², 0) → (R³, 0), we characterize local diffeomorphic types of singular ruled surfaces in terms of geometric invariants. In particular, using a theorem of G. Ishikawa, we show that local topological type of singular developable surfaces is completely determined by vanishing order of the dual torsion τ, that generalizes an old result of D. Mond for tangent developables of non-singular space curves. This work suggests that Geometric Algebra would be useful for studying singularities of geometric objects in classical Klein geometries.