5 - Envelopes

Summary

Holmes laughed. ‘It is quite a pretty little problem,’ said he.

(A Scandal in Bohemia)

What have light caustics, grass fires, gunnery ranges and embroidery in common? The canny reader, glancing at the title of this chapter, will immediately answer ‘They are all connected with envelopes (whatever those are)’ – and indeed that is exactly right. We suggest that you try to relate each of the following pictures (fig. 5.1) with one of the above topics.

In each case there are a lot of curves (they might be straight lines), which represent light rays or trajectories or threads or whatever. These appear to cluster along another curve, which the eye immediately picks out; they also touch this other curve. The new curve may look very different from those which gave birth to it; we hope you agree that it can be very beautiful. The new curve is called the envelope of the others.

In chapter 1 we considered all the normals to a given parabola, where the envelope is a cuspidal cubic curve – see 1.5 and 1.7. There we spread out the normals to form a surface in ℝ³: the envelope then appeared as the contour of this surface when viewed from above. (See figs. 1.2, 1.5.) It is actually this idea which is formalized in the definition of envelope (5.3), but in an optimal section we also formulate and compare some other definitions – see 5.8 et seq.