complex mapping with cosine function from a conic

Locus of the point u=cos(z) is illustrated with z on a conic centred origin. Move point F , point D , point E to alter the conic. point I represents complex number u point H represents complex number z other variables are unnecessary, a trial in layer(1) with straight line instead of conic some other loci are included
illustrate the property of the locus corresponding to different kinds of conic.