A BSP-based algorithm for dimensionally nonhomogeneous planar implicit curves with topological guarantees

Mathematical systems (e.g., Mathematica, Maple, Matlab, and DPGraph) easily plot planar algebraic curves implicitly defined by polynomial functions. However, these systems, and most algorithms found in the literature, cannot draw many implicit curves correctly; in particular, those with singularities (self-intersections, cusps, and isolated points). They do not detect sign-invariant components either, because they use numerical methods based on the Bolzano corollary, that is, they assume that the curve-describing function f flips sign somewhere in a line segment &ABhorbar; that crosses the curve, or f(A)·f(B) Generalized False Position (GFP) method. It allows us to sample an implicit curve against the Binary Space Partitioning (BSP), say bisection lines, of a rectangular region of R2. Interestingly, the GFP method can also be used to determine isolated points of the curve. The result is a general algorithm for sampling and rendering planar implicit curves with topological guarantees.