  This is a prejudiced survey on the Ahlfors (extremal) function and the weaker
{\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e.
those (branched) maps effecting the conformal representation upon the disc of a
{\it compact bordered Riemann surface}. The theory in question has some
well-known intersection with real algebraic geometry, especially Klein's
ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a
gallery of pictures quite pleasant to visit of which we have attempted to trace
the simplest representatives. This drifted us toward some electrodynamic
motions along real circuits of dividing curves perhaps reminiscent of Kepler's
planetary motions along ellipses. The ultimate origin of circle maps is of
course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass.
Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found
in Klein (what we failed to assess on printed evidence), the pivotal
contribution belongs to Ahlfors 1950 supplying an existence-proof of circle
maps, as well as an analysis of an allied function-theoretic extremal problem.
Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree
controls than available in Ahlfors' era. Accordingly, our partisan belief is
that much remains to be clarified regarding the foundation and optimal control
of Ahlfors circle maps. The game of sharp estimation may look narrow-minded
"Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to
contemplate how conformal and algebraic geometry are fighting together for the
soul of Riemann surfaces. A second part explores the connection with Hilbert's
16th as envisioned by Rohlin 1978.
