In the second half of the sixteenth century Latin editions of some classical works appeared, such as Federico Commandino’s editions of Books I–IV of Apollonius’s Conics and Pappus’s Collection, and Friederich Risner’s edition of Alhazem’s Optics. This period saw also the beginning of the tradition to restore some of lost Greek works, and several mathematicians participated in this work attesting to a renewed interest in the study of conic sections. The third chapter of the book is mainly focused on two figures: the geometer Francesco Maurolico, and the astronomer Johannes Kepler, who drastically changed the way of looking at conic sections. Both emphasized, in their own way, the optical generation of conics as “shadows” of a circle enlightened by a point source of light. Moreover, each of them conceived a plane system of such curves in which it was possible to continuously pass from one kind of conic to another; thus abandoning the static contemplation of such figures typical of Greek geometry. This point of view led to the consideration of points “infinitely far away”, and, in particular, led Kepler to the discovery of the second focus of the parabola. In this chapter Maurolico and Kepler’s achievements are discussed and commented in some depth, and it is also argued how their innovative ideas could have influenced the later work of Desargues and his followers.