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By A.E.R. Woodcock

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Ii II . -I- ii I ir 33 Ruled Surface Projections of B u t t e r f l y Sections of the Wi~am x7 Ax 5 Catastrophe v = -7 + -~-+ on to the ~ l a n e Bx 4 -~-+ Cx 3 Dx 2 c + - 7 + Ex (C,D) AS in Figure 8 (A and B), the surfaces c h a r a c t e r i s t i c of the B u t t e r f l y C a t a s t r o p h e exist w h e n A is negative. Fig. 17 shows a typical B u t t e r f l y s e c t i o n w h e n A is - 8 and B and E are zero. picture consists of two o v e r l a p p i n g cusps. with B zero and E negative the At E zero, the abutting edges fuse to give the B u t t e r f l y and at E positive the B u t t e r f l y degenerates into two o v e r l y i n g surfaces each containing a S w a l l o w t a i l configuration.

0 C=+12. D ' 0 . 0 Fig. 0 A = - 8 . 5 C ' + I 0 . 0 A ' - 8 . 0 A ' - 8 . 0 Fig. 30 50 A = - 8 . 0 B=+2. 0 A = - 8 . 0 A : - 8 . O A = -8. 0 A = - 8 . 0 B = - I . 0 C =+15,0 D : 0 . 0 D - 0 . O A : - 8 . 0 B : - 2 . 0 Fig. 0 A : - 8 . 0 D =0,0 Fig. 0 B A = -8. -I. 0 B = -I. 0 C =+20. 0 B : - I . 0 D : - 2 . 0 B = - I . O Fig. 0 B = - I . ed Surface Projections of Wigwam Sections of the Star 8 Catastrophe V = ~ on the p l ~ e Ax 6 Bx 5 + --~- + -~-- Cx 4 Dx 3 + -~- + -~-+ Ex 2 -~- + Fx. (D,E) A negative, B and F zero and C running from +20 to zero (Fig.

O Fig. O Fig. O C=-iO. 0 Fig. 0 C=-20. O F i g . 0 28 A=+5"O B'O'O C=+5"0 D=O'O ! i + c~ Q o ii o + ° 48 A : - 7 . 0 C=+12. D ' 0 . 0 Fig. 0 A = - 8 . 5 C ' + I 0 . 0 A ' - 8 . 0 A ' - 8 . 0 Fig. 30 50 A = - 8 . 0 B=+2. 0 A = - 8 . 0 A : - 8 . O A = -8. 0 A = - 8 . 0 B = - I . 0 C =+15,0 D : 0 . 0 D - 0 . O A : - 8 . 0 B : - 2 . 0 Fig. 0 A : - 8 . 0 D =0,0 Fig. 0 B A = -8. -I. 0 B = -I. 0 C =+20. 0 B : - I . 0 D : - 2 . 0 B = - I . O Fig. 0 B = - I . ed Surface Projections of Wigwam Sections of the Star 8 Catastrophe V = ~ on the p l ~ e Ax 6 Bx 5 + --~- + -~-- Cx 4 Dx 3 + -~- + -~-+ Ex 2 -~- + Fx.

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