5 years ago

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

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

234 and v is a smooth

234 and v is a smooth function in the neighbourhood of V~ . Note that ~ is not necessarily a continuous function in our approach. The configuration space {q : ql < q'l ) is called the object space, and the space {q :q 1 > q" 1 } the image space, see Luneburg [1964]. The corresponding spaces of fight rays (oriented lines) we denote by (P,c0) and (P,cB) respectively. They form the open subsets in (M,o~) for which Pl > 0 (i.e. transverse to the obstacle boundary). The optical instrument between {ql = q'l } and {ql --- q"l } determines a transformation from the straight fines in object space to the straight lines in image space. To find this transformation, we must look at the corresponding Riemannian geometry of W. The appropriate Riemannian metric on W is 2 2 2 do 2 = "9(q) (dql + dq2 + dq3 ). We see that light rays in object space (P,o~) that meet the plane {q : ql = q' 1 } form a 4-dimensional open subset of M: the initial conditions for the equation of geodesics in W. These initial conditions propagate symplectomorphically along the geodesic flow to the plane {q : ql = q"l }, where they are considered as elements of the image space (IS,05) c_ (M,o~). The pairs of light rays connected to each other in this way form the corresponding canonical varietyof rays associated to the optical instrument. In this case we have, for a sufficiently small neighbourhood (in the C Oo Whitney topology) of the constant function "v:W ~ 1, that for each element of this neighbourhood, the corresponding canonical variety of rays forms a Lagrangian submanifold of $~. It is the graph of a symplectomorphism q~ : P ~ t S defined, at least, on a sufficiently small open subset of M (say the neighbourhood of a principal ray, Luneburg [1964]). The focusing structure of the canonical variety is determined by the common position of the cut-locus of W and the boundaries Reflection in the Plane {q:ql = q'l} u {q : ql = q"l )" Consider a general mirror (curve) in ~2, given parametrically by t ~ (cp(t),t), tp ~ Coo, tp(0) = 0. Consider the general incident ray £ ~ P, parametrized by a: £ = (cp(t),t) + u(1,a), u E IR with (r,s)-coordinates on (P,0~), Then £ has a reflected ray corresponding to a r= __ ,s=t-cp(t)a . l+a 2 "£ =(tp(t), t)+ u (1 + 2(a~'(O-1) , a 2(acp'(t)-l) tp'(t)) I + tp'(02 1 + tp'(t) 2

where m ~t-- 1 +,~2 = a9 '2 - 2 9' - a 9 ,2 + 2a 9' - l 235 (a- cp'(t)- (p'(t)3)p(tXl+tp'(t) 2) ) E 9'(t) 2 + 2aq)'(t) - 1 See Fig. 4.3. The pairs conjugate under reflection (£,~) ~ P x P define the canonical variety which is the graph of a symplectomorphism. In the case of a plane mirror, {q : ql = 0}, this symplectomorphism is the identity, i.e. the canonical variety {((r,s), (~,~)) : r = "f = ~ a ,s=~=t} ,,/7S describes completely the reflecting properties of the mirror. Transformation of Systems of Rays The general systems of rays in (M,o~) that can be produced by general sources of light are represented by Lagrangian submanifolds of (M,co) (e.g. see Fig. 4.4). L 1,L 2 are examples of Lagrangian submanifolds in (M,¢0). FigA.3 Reflection in a mirror. Fig. 4.4 Systems of rays. (P,o~) q2 ' q!

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