Surface rendering or point location on a surface can easier be accomplished in an implicit rather than parametric representation. This observation has been the key motivation for the introduction of algebraic surfaces in geometric modelling. Cyclids, spheres and quadrics are algebraic surfaces of low degree with nice geometric properties:

On each cyclide the lines of curvature form two families of circles. Further we can obtain each cyclide of Dupin as image of the torus by inversion.

Quadrics have only two points of intersection with an arbitrary line, which can be used for a simple parametrization of the surface. Joining triangular quadric segments smoothly we obtain so called quadric splines, which can be used to interpolate points and corresponding tangent planes arranged in a triangular net.