The algebra and geometry of Steiner and other quadratically
parametrizable surfaces

Adam Coffman, Art J. Schwartz, Charles M. Stanton

article abstract

Quadratically parametrizable surfaces
(x1,x2,x3,x4)=(f1(u),f2(u),f3(u),f4(u)) where fk are homogeneous
functions of u=(u0,u1,u2) are studied in P^3(R).  These correspond to
rationally parametrizable surfaces in R^3.  All such surfaces of order
greater than two are completely catalogued and described.  The
geometry of the parametrizations as well as the geometry of the
surfaces are revealed by the use of basic matrix algebra.  The
relationship of these two geometries is briefly discussed.  The
presentation is intended to be accessible to applied mathematicians
and does not presume a knowledge of algebraic geometry.