- 0}. 243 Taking A i in general position with respect to A and finding" />
 
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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

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

a) ~ -~ c) aperture

a) ~ -~ c) aperture 5/,3, " -NN~. 4 ..,} 242 C 3 /i\ ~ t i o n r ........ .j," 2 -~ ~) "1' T v d) B3 ~z+X 3 B2+A 3 B 3 T g Fig.5.3 Generic one-parameter families of caustics by diffraction in a half-line aperture. because K(r,q 1,q2) + Fi(r) are stable unfoldings of Fi(v). Let the aperture be defined in normal form by the equations q 1 -- 0, q2 < 0. Thus we have the boundary singularities (as images, Janeczko [1986]) A(L) defined in (1VI,~3) by the following generating functions: A T

A1 :FlCO = -7 i2" A3 : F3f/) = - ¼ #, >- 0}. 243 Taking A i in general position with respect to A and finding the corresponding normal forms we obtain part 1 of Proposition 5.2. Part 2 follows by checking all the possible one-parameter evolutions (where the rays passing through an edge are not parallel to the aperture) of the stable caustic on the plane, in the presence of a half-line aperture. Two possible directions of intersection of the A2-caustic by an edge of the aperture give the cases (a) and (b) in Figure 5.3. The evolution when an edge of the aperture passes through the ray tangent to the cusp caustic A 3 is illustrated in Fig. 5.3c. Finally evolution through the intersection point A 2 + A2-caustic gives Figure 5.3d. 6. Diffraction at Smooth Obstacles Consider an open subset S of an obstacle surface in N3. Denote by ~ 1 the initial tangent line to a geodesic segment T on S. Let 12 be a tangent line to S. We say that £2 is subordinate to l 1 with respect to an obstacle S if 12 (or a piece in (IR 3, S)) belongs to the geodesic segment with the same initial point and the same tangent vector as T, see Alexander, Berg, and Bishop [1987]. By direct checking we have the following (see also Janeczko [1987]): Proposition 6.1 Let T be a geodesic flow on S. Then the set A --- {(~,~ ) E P:3 is subordinate to I with respect to S and geodesic flow ~/} is a Lagrangian subvariety of P defining the diffraction process at the obstacle S. First we consider the planar case. Here we have Proposition 6.2 For a generic obstacle curve on the plane the only possible canonical varieties A c_ p have the following normal forms of generating families 1. G(r~) = - 1~ (r3 +~3), obstacle curve q2 = - ql 2 . 9 5 5 43 _3 1 24 +~2~.2, 2. G0tl,X2,r~) = q-ff (X1 + ~.2) -rA, 1 -~.2 + ~-r t~ t obstacle curve q2 = ql 3 • . G(r~) = (d r I +~t~ I ), obstacle curve with a double tangent (see Fig. 6.1).

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