#)} 6uHuHV$@@ @`uFuJ@RJubuK\uK4=@   g;PThis takes place in the upper half-plane. The dark red line is the x-axis, and the magenta ray is the positive y-axis. The sketch examines the transformation f(z) = -1/z (denoted z# above), which can be decomposed as inversion in the unit circle (black) followed by reflection in the y-axis, i.e., a composition of two hyperbolic reflections. Since the lines of reflection intersect orthogonally, one might suspect that the composition is a rotation by 180 degrees. By dragging point z, one sees some evidence that this is true (the blue circle is the hyperbolic line through z and z#).CgCC9KCS>y >y ?n @??)H~@p3K-"J 3%* ACC BCC zCgCq)j  "H 1CCC ?5%F?5%F' jCCCC?'L mCCCgCq)j?$  rCgCq)jCC?$'*H kCCCC?R' nCgCq)jD iC|i?BTGY84 FCCZ+0a J D )hC DI D XC¨C23  2CCZB?5%F?5%F *'0 uD )hCD )hC? %*?t P CC ' pC¨C23CC?   > GCCX$*]0  x CCCC?Z'[H qC©C23C=Co? ;' tCCXBĶC0?#<c ECZCS;W@\* HC|CP#cT z#C9KCS ;@9 z#CtCP 4{  a#CgCC9KCS?@  y#CgCq)jCtCP?-2V T#CC;@ R#C[Ck'< b#CCC9BY4?'W z#C[CCpiC? UU U#C|C~ S#CC{%3 7#C|CC4?5%F?5%F <&8 6#CCC$y?5%F?5%F!