In this chapter we apply the rather technical ideas of chapter 6 to some very concrete geometrical problems. Although the ideas are technical (and their proofs even more so), applying them is, fortunately, relatively easy. We shall concentrate on elucidating the local structure of bifurcation and discriminant sets (such as envelopes), though this is by no means the only application.

Here is the pattern. We start with a family F : ℝ × ℝr ↦ ℝ, where r = 2 or 3, whose bifurcation or discriminant set interests us. If x0 is a point of one of these sets, with corresponding value t0 (so ∂F/∂t = ∂2t/∂t2 = 0 at (t0, x0) or F = ∂F/∂t = 0 at (t0, x0), respectively), then we work out the type Ak of the singularity which f = Fx0 has at t0, by counting the number of derivatives of f which vanish at t0. We then decide whether F versally unfolds f at t0 by finding ∂F/∂xi and using the matrix criterion, 6.1 Op or 6.10. If the criterion is satisfied then locally (near x0) the bifurcation set or discriminant set is diflFeomorphic to the standard model applicable to the values of r and k in question. These are listed in chapter 6, (6.16p–6.18p and 6.16–6.18).

Lest the reader think that we now have the answer to all questions of this kind, we give several examples where these methods give little or no information.