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

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

Fig.6.5 Generating

Fig.6.5 Generating family F 2. Fig.6.6 Generating family F 3. 246 Now we formulate the analogous results in the case of an obstacle in 3-dimensional Euclidean space (cf. Arnold [1983], Scherbak [1988]). Here, on the basis of Scherbak [1988] p. 140, we have the following result (which is a reformulation of the analogous result for wavefront evolution also due to Scherbak [1988]). Proposition 6.4 (Scherbak [1988]) Let S be a generic surface in IR 3. Let the pair (A,L) be defined in a sufficiently small neighbourhood of a point on S. Thus, generically, the symplectic images (I~,A)(L) have the following generating families (normal forms): E l : F = ql 1 2 2 E 2 : F = ~- ~.5 + ~ qlk3 + qty. E 3 : F = I (t3+qlt+q2)2dt o A2: F = Z,3+ql~ 1 5+2 ^ .3 2 2 A 3 : F = ~ ~1 ~- ~2~'l+7~l~2+q1~2+q2~'2 A 4 : F = - ~ (~2 + t3 + ql t + q2) 2dt + q3Z'2 ft 3 : F = Z,4+ qlk2+q2~. A4: F= ~5+qt~3+q2~2+q3~.

247 1 5 2 3 q2)+kl(ql~.2+q2)2 H 4 : F = ~- ~.1 + ~- ~1(q1~2 + + X32+ q3~. 2. The corresponding wave fronts are illustrated in Fig. 6.7 a, b, c, d, e. a) b) ~'3 =-2 c) 2" .............. -, ,, A 3

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