Convex drawings of the complete graph: topology meets geometry

In this work, we introduce and develop a theory of convex drawings of the complete graph $K_n$ in the sphere. A drawing $D$ of $K_n$ is convex if, for every 3-cycle $T$ of $K_n$, there is a closed disc $\Delta_T$ bounded by $D[T]$ such that, for any two vertices $u,v$ with $D[u]$ and $D[v]$ both in $\Delta_T$, the entire edge $D[uv]$ is also contained in $\Delta_T$. As one application of this perspective, we consider drawings containing a non-convex $K_5$ that has restrictions on its extensions to drawings of $K_7$. For each such drawing, we use convexity to produce a new drawing with fewer crossings. This is the first example of local considerations providing sufficient conditions for suboptimality. In particular, we do not compare the number of crossings {with the number of crossings in} any known drawings. This result sheds light on Aichholzer's computer proof (personal communication) showing that, for $n\le 12$, every optimal drawing of $K_n$ is convex. Convex drawings are characterized by excluding two of the five drawings of $K_5$. Two refinements of convex drawings are h-convex and f-convex drawings. The latter have been shown by Aichholzer et al (Deciding monotonicity of good drawings of the complete graph, Proc.~XVI Spanish Meeting on Computational Geometry (EGC 2015), 2015) and, independently, the authors of the current article (Levi's Lemma, pseudolinear drawings of $K_n$, and empty triangles, \rbr{J. Graph Theory DOI: 10.1002/jgt.22167)}, to be equivalent to pseudolinear drawings. Also, h-convex drawings are equivalent to pseudospherical drawings as demonstrated recently by Arroyo et al (Extending drawings of complete graphs into arrangements of pseudocircles, submitted)

Algorithms for Colourful Simplicial Depth and Medians in the Plane

The colourful simplicial depth of a point x in the plane relative to a configuration of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point, called a median, that maximizes colourful simplicial depth. For computing the colourful simplicial depth of x, our algorithm runs in time O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n^4). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O(n log(n)) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure

Gravitational Microlensing Near Caustics I: Folds

We study the local behavior of gravitational lensing near fold catastrophes. Using a generic form for the lensing map near a fold, we determine the observable properties of the lensed images, focusing on the case when the individual images are unresolved, i.e., microlensing. Allowing for images not associated with the fold, we derive analytic expressions for the photometric and astrometric behavior near a generic fold caustic. We show how this form reduces to the more familiar linear caustic, which lenses a nearby source into two images which have equal magnification, opposite parity, and are equidistant from the critical curve. In this case, the simplicity and high degree of symmetry allows for the derivation of semi-analytic expressions for the photometric and astrometric deviations in the presence of finite sources with arbitrary surface brightness profiles. We use our results to derive some basic properties of astrometric microlensing near folds, in particular we predict for finite sources with uniform and limb darkening profiles, the detailed shape of the astrometric curve as the source crosses a fold. We find that the astrometric effects of limb darkening will be difficult to detect with the currently planned accuracy of the Space Interferometry Mission. We verify our results by numerically calculating the expected astrometric shift for the photometrically well-covered Galactic binary lensing event OGLE-1999-BUL-23, finding excellent agreement with our analytic expressions. Our results can be applied to any lensing system with fold caustics, including Galactic binary lenses and quasar microlensing.Comment: 37 pages, 7 figures. Revised version includes an expanded discussion of applications. Accepted to ApJ, to appear in the August 1, 2002 issue (v574