Approximating the Maximum Overlap of Polygons under Translation

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

Two-dimensional visibility charts for continuous curves

This paper considers computation of visibility for two-dimensional shapes whose boundaries are C1 continuous curves. We assume we are given a one-parameter family of candidate viewpoints, which may be interior or exterior to the object, and at finite or infinite locations. We consider how to compute whether the whole boundary of the shape is visible from some finite set of viewpoints taken from this family, and if so, how to compute a minimal set of such viewpoints. The viewpoint families we handle include (i) the set of viewing directions from infinity, (ii) viewpoints on a circle located outside the object (for inspection from a turntable), and (iii) viewpoints located on the walls of the shape itself. We compute a structure called a visibility chart, which simultaneously encodes the visible part of the shape's boundary from every view in the family. Using such a visibility chart, finding a minimal set of viewpoints reduces to the set-covering problem over the reals. Practical algorithms are obtained by a discrete sampling of the visibility chart. For exterior visibility problems, a reasonable approach is to compute an almost-optimal solution (in terms of number of viewpoints), which can be done in almost-linear time. For interior visibility problems, or when a more correct solution is required, we solve the general set-covering problem, guaranteeing an optimal solution but taking exponential time

Halving Balls in Deterministic Linear Time

Let \D be a set of $n$ pairwise disjoint unit balls in $\R^d$ and $P$ the set of their center points. A hyperplane \Hy is an \emph{$m$-separator} for \D if each closed halfspace bounded by \Hy contains at least $m$ points from $P$. This generalizes the notion of halving hyperplanes, which correspond to $n/2$-separators. The analogous notion for point sets has been well studied. Separators have various applications, for instance, in divide-and-conquer schemes. In such a scheme any ball that is intersected by the separating hyperplane may still interact with both sides of the partition. Therefore it is desirable that the separating hyperplane intersects a small number of balls only. We present three deterministic algorithms to bisect or approximately bisect a given set of disjoint unit balls by a hyperplane: Firstly, we present a simple linear-time algorithm to construct an $\alpha n$-separator for balls in $\R^d$, for any $0<\alpha<1/2$, that intersects at most $cn^{(d-1)/d}$ balls, for some constant $c$ that depends on $d$ and $\alpha$. The number of intersected balls is best possible up to the constant $c$. Secondly, we present a near-linear time algorithm to construct an $(n/2-o(n))$-separator in $\R^d$ that intersects $o(n)$ balls. Finally, we give a linear-time algorithm to construct a halving line in $\R^2$ that intersects $O(n^{(5/6)+\epsilon})$ disks. Our results improve the runtime of a disk sliding algorithm by Bereg, Dumitrescu and Pach. In addition, our results improve and derandomize an algorithm to construct a space decomposition used by L{\"o}ffler and Mulzer to construct an onion (convex layer) decomposition for imprecise points (any point resides at an unknown location within a given disk)

(Non)Existence of Pleated Folds: How Paper Folds Between Creases

We prove that the pleated hyperbolic paraboloid, a familiar origami model known since 1927, in fact cannot be folded with the standard crease pattern in the standard mathematical model of zero-thickness paper. In contrast, we show that the model can be folded with additional creases, suggesting that real paper “folds” into this model via small such creases. We conjecture that the circular version of this model, consisting simply of concentric circular creases, also folds without extra creases. At the heart of our results is a new structural theorem characterizing uncreased intrinsically flat surfaces—the portions of paper between the creases. Differential geometry has much to say about the local behavior of such surfaces when they are sufficiently smooth, e.g., that they are torsal ruled. But this classic result is simply false in the context of the whole surface. Our structural characterization tells the whole story, and even applies to surfaces with discontinuities in the second derivative. We use our theorem to prove fundamental properties about how paper folds, for example, that straight creases on the piece of paper must remain piecewise-straight (polygonal) by folding.National Science Foundation (U.S.) (CAREER Award CCF-0347776

Three-Dimensional Object Registration Using Wavelet Features

Recent developments in shape-based modeling and data acquisition have brought three-dimensional models to the forefront of computer graphics and visualization research. New data acquisition methods are producing large numbers of models in a variety of fields. Three-dimensional registration (alignment) is key to the useful application of such models in areas from automated surface inspection to cancer detection and surgery. The algorithms developed in this research accomplish automatic registration of three-dimensional voxelized models. We employ features in a wavelet transform domain to accomplish registration. The features are extracted in a multi-resolutional format, thus delineating features at various scales for robust and rapid matching. Registration is achieved by using a voting scheme to select peaks in sets of rotation quaternions, then separately identifying translation. The method is robust to occlusion, clutter, and noise. The efficacy of the algorithm is demonstrated through examples from solid modeling and medical imaging applications

Computational Structural Analysis: Multiple Proteins Bound to DNA

BACKGROUND: With increasing numbers of crystal structures of proteinratioDNA and proteinratioproteinratioDNA complexes publically available, it is now possible to extract sufficient structural, physical-chemical and thermodynamic parameters to make general observations and predictions about their interactions. In particular, the properties of macromolecular assemblies of multiple proteins bound to DNA have not previously been investigated in detail. METHODOLOGY/PRINCIPAL FINDINGS: We have performed computational structural analyses on macromolecular assemblies of multiple proteins bound to DNA using a variety of different computational tools: PISA; PROMOTIF; X3DNA; ReadOut; DDNA and DCOMPLEX. Additionally, we have developed and employed an algorithm for approximate collision detection and overlapping volume estimation of two macromolecules. An implementation of this algorithm is available at http://promoterplot.fmi.ch/Collision1/. The results obtained are compared with structural, physical-chemical and thermodynamic parameters from proteinratioprotein and single proteinratioDNA complexes. Many of interface properties of multiple proteinratioDNA complexes were found to be very similar to those observed in binary proteinratioDNA and proteinratioprotein complexes. However, the conformational change of the DNA upon protein binding is significantly higher when multiple proteins bind to it than is observed when single proteins bind. The water mediated contacts are less important (found in less quantity) between the interfaces of components in ternary (proteinratioproteinratioDNA) complexes than in those of binary complexes (proteinratioprotein and proteinratioDNA).The thermodynamic stability of ternary complexes is also higher than in the binary interactions. Greater specificity and affinity of multiple proteins binding to DNA in comparison with binary protein-DNA interactions were observed. However, protein-protein binding affinities are stronger in complexes without the presence of DNA. CONCLUSIONS/SIGNIFICANCE: Our results indicate that the interface properties: interface area; number of interface residues/atoms and hydrogen bonds; and the distribution of interface residues, hydrogen bonds, van der Walls contacts and secondary structure motifs are independent of whether or not a protein is in a binary or ternary complex with DNA. However, changes in the shape of the DNA reduce the off-rate of the proteins which greatly enhances the stability and specificity of ternary complexes compared to binary ones

Some Causes of the Variable Shape of Flocks of Birds

Flocks of birds are highly variable in shape in all contexts (while travelling, avoiding predation, wheeling above the roost). Particularly amazing in this respect are the aerial displays of huge flocks of starlings (Sturnus vulgaris) above the sleeping site at dawn. The causes of this variability are hardly known, however. Here we hypothesise that variability of shape increases when there are larger local differences in movement behaviour in the flock. We investigate this hypothesis with the help of a model of the self-organisation of travelling groups, called StarDisplay, since such a model has also increased our understanding of what causes the oblong shape of schools of fish. The flocking patterns in the model prove to resemble those of real birds, in particular of starlings and rock doves. As to shape, we measure the relative proportions of the flock in several ways, which either depend on the direction of movement or do not. We confirm that flock shape is usually more variable when local differences in movement in the flock are larger. This happens when a) flock size is larger, b) interacting partners are fewer, c) the flock turnings are stronger, and d) individuals roll into the turn. In contrast to our expectations, when variability of speed in the flock is higher, flock shape and the positions of members in the flock are more static. We explain this and indicate the adaptive value of low variability of speed and spatial restriction of interaction and develop testable hypotheses