Thomas Hackl

Medial axis computation for planar free-form shapes

We present a simple, efficient, and stable method for computing-with any desired precision-the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the approximation of the boundary spline such that the medial axis is geometrically stable, and (2) the efficient decomposition of the domain into base cases where the medial axis can be computed directly and exactly. We solve these problems via spiral biarc approximation and a randomized divide & conquer algorithm.

On the number of plane graphs

We investigate the number of plane geometric, i.e., straight-line, graphs, a set <i>S</i> of <i>n</i> points in the plane admits. We show that the number of plane graphs and connected plane graphs as well as the number of cycle-free plane graphs is minimized when <i>S</i> is in convex position. Moreover, these results hold for all these graphs with an arbitrary but fixed number of edges. Consequently, we provide simple proofs that the number of spanning trees, cycle-free graphs (forests), perfect matchings, and spanning paths is also minimized for point sets in convex position.In addition we construct a new extremal configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ*(√72^<i>n</i>) = Θ*(8.4853^<i>n</i>) triangulations and Θ*(41.1889^<i>n</i>) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples.

Divide-and-conquer for Voronoi diagrams revisited

We show how to divide the edge graph of a Voronoi diagram into a tree that corresponds to the medial axis of an (augmented) planar domain. Division into base cases is then possible, which, in the bottom-up phase, can be merged by trivial concatenation. The resulting construction algorithm-similar to Delaunay triangulation methods-is not bisector-based and merely computes dual links between the sites, its atomic steps being inclusion tests for sites in circles. This guarantees computational simplicity and numerical stability. Moreover, no part of the Voronoi diagram, once constructed, has to be discarded again. The algorithm works for polygonal and curved objects as sites and, in particular, for circular arcs, which allows its extension to general free-form objects by Voronoi diagram preserving and data saving biarc approximations. The algorithm is randomized, with expected runtime O(nlogn) under certain assumptions on the input data. Experiments substantiate an efficient behavior even when these assumptions are not met. Applications to offset computations and motion planning for general objects are described.

On k-gons and k-holes in point sets

We consider a variation of the classical Erd?s-Szekeres problems on the existence and number of convex k-gons and k-holes (empty k-gons) in a set of n points in the plane. Allowing the k-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any k and sufficiently large n, we give a quadratic lower bound for the number of k-holes, and show that this number is maximized by sets in convex position.

Empty monochromatic triangles

We consider a variation of a problem stated by Erdos and Guy in 1973 about the number of convex k-gons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As a main result we show that any bi-colored set of n points in R^2 in general position determines a super-linear number of empty monochromatic triangles, namely @W(n^5^/^4).

Coloring hypergraphs induced by dynamic point sets and bottomless rectangles

We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color. We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k)=3k−2. This can be interpreted as coloring point sets in ℝ2 with k colors such that any bottomless rectangle containing at least 3k−2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set. For this problem, we also prove a lower bound p(k)>ck, where c>1.67. Hence, for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function p(k) exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).

Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).

Computational and structural advantages of circular boundary representation

Boundary approximation of planar shapes by circular arcs has quantitive and qualitative advantages compared to using straightline segments. We demonstrate this by way of three basic and frequent computations on shapes - convex hull, decomposition, and medial axis. In particular, we propose a novel medial axis algorithm that beats existing methods in simplicity and practicality, and at the same time guarantees convergence to the medial axis of the original shape.

4-Holes in point sets

We consider a variant of a question of Erd?s on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n ? 9 , the maximum number of general 4-holes is ( n 4 ) ; the minimum number of general 4-holes is at least 5 2 n 2 - ? ( n ) ; and the maximum number of non-convex 4-holes is at least 1 2 n 3 - ? ( n 2 log n ) and at most 1 2 n 3 - ? ( n 2 ) .

Packing Plane Spanning Trees and Paths in Complete Geometric Graphs

We consider the following question: How many edgedisjoint plane spanning trees are contained in a complete geometric graph GKn on any set S of n points in general position in the plane?