Topology and arrangement computation of semi-algebraic planar curves

TLDR: It is shown on examples that this algorithm is able to handle curves defined by high degree polynomials with large coefficients, to identify regions of interest and use the resulting structure for either efficient rendering of implicit curves, point localization or boolean operation computation.

About: This article is published in Computer Aided Geometric Design.The article was published on 2008-11-01 and is currently open access. It has received 67 citations till now. The article focuses on the topics: Extension topology & General topology.

Figures

Figure 10: Topology of a curve with hidden cusp points.

Figure 5: From left to right: A simply singular domain, a domain that fails to be simply singular because there are too many points on its boundary, another one that fails to be simply singular because there are several extremal points of f inside it, an x-regular domain, and a domain that fails to be x-regular because there are vertical tangents to C inside it.

Figure 3: Building the region segmentation.

Figure 6: x-indices of an x-regular domain

Figure 9: Topology of a high degree curve in Axel.

Figure 8: Computing the topological degree

Frequently asked questions

Q1. What are the contributions in "Topology and arrangement computation of semi-algebraic planar curves" ?

The authors describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. The authors use enveloping techniques of the polynomial represented in the Bernstein basis to achieve both efficiency and certification. Although the authors focus on the implicit case, the algorithm can also treat parametric or piecewise linear curves without much additional work and no theoretical difficulties. 

Q2. What are the objects that the authors consider in this framework?

The objects that the authors consider in this framework are semi-algebraic curves whose representation can either be discrete, parametric or implicit. 

Q3. How can the authors compute a RUR of the roots of E?

A RUR of the roots of E can be computed by finding a separating linear function and using resultant or Groebner basis techniques. 

Q4. What is the main drawback of projections methods?

Its complexity is intrinsic to the geometry of the curve (like the subdivision methods) and it avoids the main drawback of projections methods because it does not need to lift points. 

Q5. What is the originality of the subdivision algorithm?

Its originality is to prevent useless computation by stopping the subdivision as soon as the topology of the object is known in a cell of subdivision. 

Q6. What is the x-regular tangent of a curve?

As the curve is x-regular, it has no vertical tangent and thus no closed loop in D. Consequently, each of the interior connected components of C ∩D intersects ∂D in two distinct points p, q ∈ C ∩ ∂D. 

Q7. What is the main contribution of the algorithm?

Another contribution is that thanks to its generic approach, it can be used to compute an heterogeneous arrangement, i.e. an arrangement of objects having different representations. 

Q8. What is the general operation that merges regions stored in a quadtree?

The algorithm takes as input a quadtree obtained from the subdivision, in which, only leaves contain regions, and propagates these regions from the leaves to the root. 

Q9. What is the connection algorithm for (convex) simply singular domains?

Therefore the connection algorithm for (convex) simply singular domains is to first compute the points qi of C ∩∂D, then choose an arbitrary point p inside D and finally for every qi, connect qi and p by a half branch segment bi = [p, qi]. 

Q10. What is the way to solve multivariate problems?

Multivariate solving is a difficult problem in itself that can be solved efficiently in most cases by sleeve methods or subdivision methods for example [48, 15, 39], but no matter the approach it seems necessary, in some cases, to resort to purely algebraic techniques such as rational univariate representations of roots [6, 17]. 

Q11. how many polynomials are expressed in the Bernstein basis?

Each of these polynomials is expressed in the Bernstein basis on D0:h(x, y) =dx ∑i=0dy ∑j=0γi,j B i dx (x; a, b)Bjdy(y; c, d),where h ∈ {f, ∂xf, ∂yf} and dx is the degree of h in x, dy the degree of h in y.