Webs of rational curves on surfaces

What are the possible shapes of surfaces that are defined as the solution set of polynomial equations with real coefficients? It is classically known that a non-planar surface containing two lines through each point must be shaped like either a cooling tower or a horse saddle. Such doubly ruled quadrics are a union of lines in two different ways and are of interest to architects as the lines can be realized as straight beams that support a building. Instead of lines, we more generally consider mira curves, namely curves on surfaces that behave like lines in the plane but may look like tangled shoelaces ("mira" stands for the specialist terms "minimal degree" and "rational"). Now suppose that S is a surface that is a union of mira curves in at least two different ways. Unlike doubly ruled quadrics, this surface typically has self-intersections and therefore determining its possible shapes is very hard. Even if the mira curves are circles this is an unsolved problem and the starting point of this project. Our hypothesis is that there exist two mira curves A and B on the surface S together with an equivalence relation = and an algebraic operation * such that S=A*B. From such a decomposition of S into the curves A and B, we obtain additional insight in for example its self-intersections. Closely related is the problem of decomposing complicated motions of robots with two degrees of freedom into revolutions along axes. Another method to reveal the geometry of the surface S, we imagine a spider web lying on this surface except that the silk strings coincide with mira curves. Such webs admit a deep combinatorial structure and if the mira curves are lines this structure can be traced back to the Greek mathematicians of antiquity. We envision a revival of ideas with classical roots by using modern mathematical and computational methods. This research project is mostly of theoretical nature, but has potential applications in architecture, kinematics, computer vision and physics.