A remark on parameterizing nonsingular cubic surfaces

Article history: Received 13 June 2008 Received in revised form 29 May 2009 Accepted 2 June 2009 Available online 6 June 2009 Extending a geometric construction due to Sederberg and to Bajaj, Holt, and Netravali, an algorithm is presented for parameterizing a nonsingular cubic surface by polynomials of degree three. The fact that such a parametrization exists is classical. The present algorithm is, by its purely geometric nature, a very natural one. Moreover, it contains a practical way of finding all lines in an implicitly given cubic surface. Two explicit examples are presented, namely the classical Clebsch diagonal surface and the cubic Fermat surface.  2009 Elsevier B.V. All rights reserved.