International Journal of Computational Geometry & Applications

Path planning can be subject to different types of optimization. Some years ago a German researcher, U. Leuthäusser, proposed a new variational method for reducing most types of optimization criteria to one and the same: minimization of path length. This can be done by altering the Riemannian metric of the domain, so that optimal paths (with respect to whatever criterion) are simply seen as shortest. This method offers an extra feature, which has not been exploited so far: it admits direction–dependent criteria.

In this paper we make this feature explicit, and apply it to two different anisotropic settings. One is that of different costs for different directions: E.g. the situation of a countryside scene with ploughed fields. The second is dependence on oriented directions, which is called here "strong" anisotropy: the typical scene is that of a hill side. A covering projection solves the additional difficulty. We also provide some experimental results on synthetic data.

Let R and B be two disjoint sets of red points and blue points in the plane, respectively, such that no three points of R ∪ B are collinear, and let a,b and g be positive integers. We show that if ag ≤ |R| < (a + 1)g and bg ≤ |B| < (b + 1)g, then we can subdivide the plane into g convex polygons so that every open convex polygon contains exactly a red points and b blue points and that the remaining points lie on the boundary of the subdivision. This is a generalization of equitable subdivision of ag red points and bg blue points in the plane.

In recent years, there has been a resurgence in the use of raw point cloud data as the geometric primitive of choice for several modeling tasks such as rendering, editing and compression. Algorithms using this representation often require reliable additional information such as the curve tangent or surface normal at each point. Estimation of these quantities requires the selection of an appropriate scale of analysis to accommodate sensor noise, density variation and sparsity in the data. To this goal, we present a new class of locally semi-parametric estimators that allows analysis of accuracy with finite samples, as well as explicitly addresses the problem of selecting optimal support volume for local fitting. Experiments on synthetic and real data validate the behavior predicted by the model, and show competitive performance and improved stability over leading alternatives that require a preset scale.

This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that:

• If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each.

• If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8.

• If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2^k).

We propose heuristics for visibility coverage of a polygon with the fewest point guards. This optimal coverage problem, often called the "art gallery problem", is known to be NP-hard, so most recent research has focused on heuristics and approximation methods. We evaluate our heuristics through experimentation, comparing the upper bounds on the optimal guard number given by our methods with computed lower bounds based on heuristics for placing a large number of visibility-independent "witness points". We give experimental evidence that our heuristics perform well in practice, on a large suite of input data; often the heuristics give a provably optimal result, while in other cases there is only a small gap between the computed upper and lower bounds on the optimal guard number.