This paper describes how to calculate the Legendre transformation of the beak-to-beak form of the parabolic umbilic. The result is a multivalued surface in three dimensions, with multiplicity of up to five, which contains cusped edges of regression and lines of self-intersection. The calculation is self-contained and is elementary in method but intricate in detail. The ideas in it will be available in other such calculations of similar complexity, and for any required context. Our underlying motivation is the study of atmospheric fronts in semi-geostrophic theory. We show how a front may be regarded as a self-intersection line terminating at a swallowtail point on the Legendre transform surface. A physical stability argument requires excision of certain parts of the surface to leave a jointly convex and single-valued surface having a gradient discontinuity along the front. A numerical illustration is presented.