Certified simultaneous isotopic approximation of curves via subdivision

TL;DR

This work extends the Plantinga–Vegter subdivision framework to certify topologically correct approximations of multiple plane curves by introducing a new $C_1^\times$ predicate that controls pairwise gradient relations and avoids missing or spurious intersections. A dedicated subdivision step for a pair of curves ensures transversal crossings of the approximations correspond bijectively to the true crossings, while handling excursions and shared edges via the snake construct. The authors provide adaptive and nonadaptive complexity bounds and a bit-complexity analysis, showing the method scales with degree and coefficient size, and extend the approach to arbitrary numbers of curves under clear transverse-intersection assumptions. The results yield practical, certified topology-preserving curve approximations with explicit complexity guarantees and demonstrate the approach on illustrative examples, including near-parallel and multi-curve configurations. Overall, the paper delivers a rigorous, scalable framework for globally correct isotopic approximations of multiple plane curves using interval arithmetic and subdivision-based predicates.

Abstract

We present a certified algorithm based on subdivision for computing an isotopic approximation to any number of curves in the plane. Our algorithm is based on the certified curve approximation algorithm of Plantinga and Vegter. The main challenge in this algorithm is to correctly and efficiently identify and isolate all intersections between the curves. To overcome this challenge, we introduce a new and simple test that guarantees the global correctness of our output. A main step in our algorithm for approximating any number of curves is to correctly approximate a pair of curves. In addition to developing the details of this special case, we provide complexity analyses for both the number of steps and the bit-complexity of this algorithm using both worst-case bounds as well as those based on continuous amortization.

Figures (15)

• Figure 1: Two approximations (thick lines) of a pair of curves (thin lines). The approximations and curves are paired by color and line style. A naive approach (a) misses a pair of intersections due to excursions while our approach (b) correctly approximates the curves and their intersections.
• Figure 2: Illustrations of (a) Lemma \ref{['lem:PVcorrectness']} and (b) Lemma \ref{['lem:multicrossings']}. In (a), the existence of two perpendicular segments meeting the curve twice implies that $C_1$ fails in this box. In (b), the existence of two parallel segments each meeting a curve twice implies that $C_1^\times$ fails in this box.
• Figure 3: An illustration of the two types of extensions of a curve $\gamma$, denoted by the thickened curve, with a region $S$. The first extension is $\overline{\gamma}$, which is denoted by the thin curve. $\overline{\gamma}$ extends $\gamma$ to the first time that the curve containing $\gamma$ leaves the region of interest. The second extension is $\widetilde{\gamma}$, which is denoted by the dashes. $\widetilde{\gamma}$ includes all excursions until the curve containing $\gamma$ leaves the region of interest and does not immediately return. The part of the curve denoted by dots is beyond the extensions $\overline{\gamma}$ and $\widetilde{\gamma}$.
• Figure 4: Two approximations (thick lines) of a pair of curves (thin lines). The approximations and curves are paired by color and line style. The naive approach to approximating the pairs of curves include (a) an extra intersection or (b) several shared edges of the approximation.
• Figure 5: The neighborhood of the box $B$ with two crossing components. The curves intersect in the neighborhood of $B$ while the approximations (not shown) intersect in $B$.

Theorems & Definitions (58)

• Definition 1
• Lemma 2: PV:2004PV:2007
• Corollary 3: PV:2004PV:2007
• Corollary 4: PV:2004PV:2007
• Corollary 5: PV:2004PV:2007
• Definition 6
• Lemma 7: PV:2004PV:2007
• Lemma 8: PV:2004PV:2007
• Lemma 9
• proof