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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

Proof / obstacle curve

Proof / obstacle curve 244 Fig.6.1 Obstacle curve with a double tangent. Consider a non-inflection point of the generic curve. Parametrically the curve is given by (q 1,q2) = (v, - v 2) , v ¢ IR, and the family of tangent lines corresponding to the given incident ray has the form £1 : (ql,q2) --" (0, v 2) + u(1, -2v), u ¢ ~q. So by identification in Darboux coordinates we have and P2, q2 qlP2 ) t = (r,s), ql = 0 r ll: (ql'q2) = (O,s)+t (1, ) = + t(1,-=v, -2v s=v 2, r= X/~+4v 2. 1 Thus locally r 2 = 4s and F(r) = - -~- r 3, so we obtain case 1, which corresponds to the cartesian product of two ordinary folds (Fig. 6.2). Fig.6.2 Cartesian product of two folds.

245 Now taking an inflection point on an obstacle curve we obtain, in a similar way, the following parametrization of A G $~, namely (Fig. 6.3) and its generating family. 3v 2 3~ 2 s=-2v3,r = ~ ,s= 2~3,r = 1X~+~4 Fig.6.3 Lagrangian variety for an inflection point on an obstacle curve. Now we have the following: Corollary 6.3 For generic pairs (A,L) we have the following stable images A(L) and their corresponding generating families (functions): I ~3 A 2 : FI(~) = _ ~- H 3 : F2(X~) = 9X5 -~X3 + ~2X A2,2 : ~3(~) = 1 The generating families for their corresponding configurational images are the following: 1 1 2 a) Fl(3.,ql,q2) = _ .~.X3 + q2 x _ ~_ql~. (Fig. 6.4) I ~.~1 + I 2 b) F2(Xl,~.2,q1,q2 ) = 9X15 - ~.2~31 + ~- q2~.2 - ~q1~2 (Fig. 6.5) 1 1 c) F3(~',ql,q2) = ~ ~-[ ~.i+ q2 ~. - ~- q1~L 2 (Fig. 6.6) Fig. 6.4 Generating family F I.

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