Most special plane curves are of historical interest. Books on history of mathematics often include information on various curves sporadically. The following are some excellent math history books I've used. Significant part of classical math deal with curves. These books give a short account of many curve's history.
Contents are as accurate as historians can claim.
Easy reading but not trivial. This is an acclaimed book. Saunders College Publishing. Each chapter covers a period, followed by a fairly complete bibliography. First published 1968.