The space of circles in S3 completed to include points is identified with a complex quadric hypersurface [Hscr ] of CP4. A conformal foliation by arcs of circles of a domain in R3 now corresponds to a complex holomorphic curve in [Hscr ]. The space [Hscr ] minus a hyperplane section fibres in a natural way over the space of lines in R3, each fibre consisting of the well known family of circles of Villarceau transformed by a homothety. As a consequence we show that there is a correspondence between harmonic morphisms on the 3-dimensional space forms and demonstrate the existence of a distinguished surface associated to a conformal foliation by arcs of circles.

We generalize Cartan's isoparametric functions to complex valued functions and classify all those horizontally conformal complex isoparametric functions defined on domains of R3 with arcs of circles or lines as fibres.