Visualizing Cubic Algebraic Surfaces

Abstract

A cubic surface is defined as the set of zeroes of a homogeneous polynomial f of degree three in three-dimensional real projective space given by S = {(x : y : z : w) ∈ P³(R)| f (x, y, z, w) = 0}. The geometric classification of these objects remains an unsolved problem. Ideally, such a classification would incorporate information about the notable geometric properties of each surface, yet be general enough to encompass all cubic surfaces succinctly. Using new visualization tools, we review and develop methods to identify several of these properties; namely, the symmetry exhibited by a surface, the real valued lines on a surface, and the presence and number of singular points on a surface. We also experiment with the effect that deformation of the surface has on these properties, with the goal of studying their stability under such deformation.