� Jacobi Curves: Computing the Exact Topology of Arrangements of Non-Singular Algebraic Curves Nicola Wolpert Max-Planck-Institut f¨ ur Informatik Stuhlsatzenhausweg 85 66123 Saarbr¨ ucken, Germany nicola@mpi-sb.mpg.de March 26, 2003 Abstract lines defined by rational numbers all computations can be done over the field of rational numbers avoid- ing numerical errors and leading to exact mathemat- We present an approach that extends the Bentley- ical results. Ottmann sweep-line algorithm [3] to the exact com- As soon as higher degree algebraic curves are con- putation of the topology of arrangements induced by sidered, instead of linear ones, things become more non-singular algebraic curves of arbitrary degrees. difficult. In general, the intersection points of two Algebraic curves of degree greater than 1 are diffi- planar curves defined by rational polynomials have cult to handle in case one is interested in exact and irrational coordinates. That means instead of ra- efficient solutions. In general, the coordinates of in- tional numbers one now has to deal with algebraic tersection points of two curves are not rational but numbers. One way to overcome this difficulty is algebraic numbers and this fact has a great negative to develop algorithms that use floating point arith- impact on the efficiency of algorithms coping with metic. These algorithms are quite fast but in degen- them. The most serious problem when computing erate situations they can lead to completely wrong arrangements of non-singular algebraic curves turns out be the detection and location of tangential inter- results because of approximation errors, rather than just slightly inaccurate outputs. Assume that for two section points of two curves. The main contribution planar curves one is interested in the number of in- of this paper is a solution to this problem, using only tersection points. If the curves have tangential inter- rational arithmetic. We do this by extending the con- section points, the slightest inaccuracy can lead to a cept of Jacobi curves introduced in [12]. Our algo- wrong output. rithm is output-sensitive in the sense that the alge- A second approach besides using floating point arith- braic effort we need for sweeping a tangential inter- metic is to use exact algebraic computation methods section point depends on its multiplicity. like the use of the gap theorem [5] or multivariate Sturm sequences [17]. Then of course the results are 1 Introduction correct, but the algorithms in general are very slow. We consider arrangements of non-singular curves in Computing arrangements of curves is one of the fun- the real plane defined by rational polynomials. Al- damental problems in computational geometry as though the non-singularity assumption is a strong well as in algebraic geometry. For arrangements of restriction on the curves we consider, this class of ✁ Partially supported by the IST Programme of the EU as curves is worthwile to be studied because of the gen- a Shared-cost RTD (FET Open) Project under Contract No eral nature of the main problem that has to be solved. IST-2000-26473 (ECG – Effective Computational Geometry for Two algebraic curves can have tangential intersec- Curves and Surfaces) 1
✆ Arrangements of quadric surfaces in IR 3 are consid- tions and it is inevitable to determine them precisely in the case we are interested in exact computation. ered by Wolpert [24] and Dupont et al. [9]. By pro- As a main tool for solving this problem we will in- jection the first author reduces the spatial problem to troduce generalized Jacobi curves, for more details the one of computing planar arrangements of alge- consider [24]. Our resulting algorithm computes the braic curves of degree at most 4. The second authors exact topology using only rational arithmetic. It is directly work in space determining a parameteriza- output-sensitive in the sense that the algebraic degree tion for the intersection curve of two arbitrary im- of the Jacobi curve that is constructed to locate a tan- plicit quadrics. gential intersection point depends on its multiplicity. For computing planar arrangements of arbitrary pla- nar curves very little is known. An exact approach 2 Previous work using rational arithmetic to compute the topological configuration of a single curve is done by Sakkalis As mentioned, methods for the calculation of ar- [21]. For computing arrangements of curves we are rangements of algebraic curves are an important also interested in intersection points of two or more area of research in computational geometry. For curves. Of course we could interpret these points an overview consider the articles of Halperin [14] as singular points of the curve that is the union of and Agarwal and Sharir [1]. A great focus is on both. But the approach of Sakkalis for determining arrangements of linear objects. Algorithms coping singular points with the help of negative polynomial with linear primitives can be implemented using ra- remainder sequences is not very efficient, at least if tional arithmetic, leading to exact mathematical re- singular points occur frequently. sults in any case. For fast filtered implementations MAPC [15] is a library for exact computation and see for example the ones in LEDA [16] and CGAL manipulation of algebraic points. It includes a pack- [11]. There are also some geometric methods dealing age for determining arrangements of planar curves. with arbitrary curves, see for example [18], [8], [22], For degenerate situations like tangential intersections [2], [19]. But all of them neglect the problem of ex- the use of the gap theorem [5] or multivariate Sturm act computation in the way that they are based on an sequences [17] is proposed. Both methods, like the idealized real arithmetic provided by the real RAM one of Sakkalis, are not efficient. model of computation [20]. The assumption is that all, even irrational, numbers are representable and 3 Notation that one can deal with them in constant time. This postulate is not in accordance with real computers. The objects we consider and manipulate in our work Recently the exact computation of arrangements of are non-singular algebraic curves represented by ra- non-linear objects has come into the focus of re- tional polynomials. We define an algebraic curve in search. Wein [23] extended the CGAL implemen- � x the following way: Let f be a polynomial in Q ✁ y ✂ . tation of planar maps to conic arcs. Berberich et ✆✞✝✟✄ α ✁ β ☛ f ✄ α ✁ β IR 2 We set Z ERO ✄ f ☎ : 0 ✌ and al. [4] made a similar approach for conic arcs ☎✡✠ ☎☞✆ call Z ERO ✄ f ☎ the algebraic curve defined by f . If based on the improved LEDA [16] implementation the context is unambiguous, we will often identify of the Bentley-Ottmann sweep-line algorithm [3]. the defining polynomial of an algebraic curve with For conic arcs the problem of tangential intersection its zero set. points is not serious because the coordinates of every such point are one-root expressions of rational num- For an algebraic curve f we define its gradient vector � x ∂ f to be ∇ f : ✁ y ✂✑☎ 2 with f x : bers. Eigenwillig et al. [10] extended the sweep-line ✆✍✄ f x ✁ f y ☎✎✠✏✄ Q ∂ x . We as- approach to cubic arcs. All tangential intersection sume the set of input curves to be non-singular , that ✄ α ✁ β IR 2 with f ✄ α ✁ β points in the arrangements of cubic arcs either have means for every point 0 ☎✒✠ ☎✡✆ ✄ ∇ f ☎✓✄ α ✁ β ✄ α ✁ β ✄ α ✁ β coordinates that are one-root expressions or they are we have ☎✔✆✍✄ f x ☎✕✁ f y ✆✙✄ 0 ✁ 0 ☎ . A ☎✖☎✘✗ ✄ α ✁ β ✄ ∇ f ☎✓✄ α ✁ β of multiplicity 2 and therefore can be solved using point ☎ with ☎✘✆✚✄ 0 ✁ 0 ☎ we would call the Jacobi curve introduced in [12]. singular . The geometric interpretation is that for ev- 2
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