A Geometrical Study of the Elementary Catastrophes by A.E.R. Woodcock

By A.E.R. Woodcock

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Figs. 22 and 23 show B n o n - z e r o for A p o s i t i v e and zero. 0 = -I. 0 Fig. 0 B : 0 . 0 E = -I. 0 E=+<0 Fig. 0 Fig. 0 E:+l. 0 E:+I. 0 E=-I,0 Fig. 0 B : 0 . 0 41 Ruled Surface Projections of the Star Catastrophe 8 V = ~ Ax 6 Bx 5 Cx 4 Dx 3 Ex 2 + --~-- + -~-- + --~- + --~- + --~- + Fx. onto the ~lane (E,F) When A is negative, C large and positive and B and D zero, the singularity is simply cuspoid (Fig. 24, 25 and 26). However, as C is reduced, two Swallowtails begin to develop one on either edge of the cusp (see, for example, Fig.

0 Fig. 0 F--5,0 Fig.

The general picture is of a Butterfly surface with a second small Butterfly replacing the central cusp in the area in which E and F are small also Figs. ) (Fig. 24 C = 20, for example, For C negative the Star Catastrophe collapses to become simply the Butterfly Catastrophe. For A zero or positive the pic- ture is also typical of the Butterfly with the two wings appearing when C is negative (Figs. 27 and 28). Figs 29 (A and B), 30, 31 and 32 show the complex changes that occur in the projected surface when B is varied (A negative, C positive D zero).

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