How Complex are Real Algebraic Objects?

Introduction A standard technique to process non-linear curves and surfaces in geometric systems is to approximate them in terms of a piecewise linear object (a simplicial complex). A main goal is to preserve the topological properties of the input objects. Furthermore, geometric properties, such as the position of “critical” points of the object, are often of interest. For algebraic curves and surfaces as inputs, the former problem is usually called topology computation, the latter topological-geometric analysis of the object. Efficient techniques for curves (e.g, see [4][5], and references therein) and surfaces [2, 1] have been presented. In few words, we consider the following question: How many line segments/triangles are needed to approximate a real algebraic curve/surface of degree n? For curves, we are able to give sharp bounds: for a topological correct representation, Ω(n2) line segments are needed in the worst case, and we give an algorithm producing O(n2) line segments for all cases. Although the idea is simple, it seemingly does not appear in the literature yet. For geometrictopological representations, we construct a class of curves such that Ω(n3) line segments are necessary. This proves that the cylindrical algebraic decomposition [3] (“Find the critical x-coordinates of the curve; compute the fiber at these coordinates and at separating points in between; connect the fiber points by straight-line segments.” – compare the pictures on the next page) is asymptotically optimal. This is surprising, because the vertical decomposition strategy seems to introduce much more line segments than actually necessary. For surfaces, we still have gaps between lower and upper bounds. For the topological approximation, we get a lower bound of Ω(n3), and an upper bound of O(n5) triangles. For the geometric-topological approximation, the bounds are Ω(n4) and O(n7). A detailed version of this extended abstract will appear in the phd thesis of the first author [6].