## Tohoku Mathematical Journal 2012 September SECOND SERIES VOL. 64, NO. 3

 Tohoku Math. J. 64 (2012), 387-408

Title SINGULARITIES OF PARALLEL SURFACES

Author Toshizumi Fukui and Masaru Hasegawa

(Received September 6, 2010, revised December 28, 2011)
Abstract. We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and 3-dimensional $D_4^\pm$ singularities. We give criteria for these singularity types in terms of differential geometry (Theorems 3.4 and 3.5).

2000 Mathematics Subject Classification. Primary 53A05; Secondary 58K05, 58K35.

Key words and phrases. Parallel surface, Versality of distance squared functions.