Convexity-Increasing Morphs of Planar Graphs

We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201

An energetic stellar outburst accompanied by circumstellar light echoes

Some classes of stars, including supernovae and novae, undergo explosive outbursts that eject stellar material into space. In 2002, the previously unknown variable star V838 Monocerotis brightened suddenly by a factor of about 10^4. Unlike a supernova or nova, V838 Mon did not explosively eject its outer layers; rather, it simply expanded to become a cool supergiant with a moderate-velocity stellar wind. Superluminal light echoes were discovered as light from the outburst propagated into surrounding, pre-existing circumstellar dust. Here we report high-resolution imaging and polarimetry of the light echoes, which allow us to set direct geometric distance limits to the object. At a distance of >6 kpc, V838 Mon at its maximum brightness was temporarily the brightest star in the Milky Way. The presence of the circumstellar dust implies that previous eruptions have occurred, and spectra show it to be a binary system. When combined with the high luminosity and unusual outburst behavior, these characteristics indicate that V838 Mon represents a hitherto unknown type of stellar outburst, for which we have no completely satisfactory physical explanation.Comment: To appear in Nature, March 27, 2003. 9 pages, 6 figure

Exploring the Universe with Metal-Poor Stars

The early chemical evolution of the Galaxy and the Universe is vital to our understanding of a host of astrophysical phenomena. Since the most metal-poor Galactic stars (with metallicities down to [Fe/H]\sim-5.5) are relics from the high-redshift Universe, they probe the chemical and dynamical conditions of the Milky Way and the origin and evolution of the elements through nucleosynthesis. They also provide constraints on the nature of the first stars, their associated supernovae and initial mass function, and early star and galaxy formation. The Milky Way's dwarf satellites contain a large fraction (~30%) of the known most metal-poor stars that have chemical abundances that closely resemble those of equivalent halo stars. This suggests that chemical evolution may be universal, at least at early times, and that it is driven by massive, energetic SNe. Some of these surviving, ultra-faint systems may show the signature of just one such PopIII star; they may even be surviving first galaxies. Early analogs of the surviving dwarfs may thus have played an important role in the assembly of the old Galactic halo whose formation can now be studied with stellar chemistry. Following the cosmic evolution of small halos in simulations of structure formation enables tracing the cosmological origin of the most metal-poor stars in the halo and dwarf galaxies. Together with future observations and additional modeling, many of these issues, including the reionization history of the Milky Way, may be constrained this way. The chapter concludes with an outlook about upcoming observational challenges and ways forward is to use metal-poor stars to constrain theoretical studies.Comment: 34 pages, 11 figures. Book chapter to appear in "The First Galaxies - Theoretical Predictions and Observational Clues", 2012 by Springer, eds. V. Bromm, B. Mobasher, T. Wiklin

Evolutionary and pulsational properties of white dwarf stars

Abridged. White dwarf stars are the final evolutionary stage of the vast majority of stars, including our Sun. The study of white dwarfs has potential applications to different fields of astrophysics. In particular, they can be used as independent reliable cosmic clocks, and can also provide valuable information about the fundamental parameters of a wide variety of stellar populations, like our Galaxy and open and globular clusters. In addition, the high densities and temperatures characterizing white dwarfs allow to use these stars as cosmic laboratories for studying physical processes under extreme conditions that cannot be achieved in terrestrial laboratories. They can be used to constrain fundamental properties of elementary particles such as axions and neutrinos, and to study problems related to the variation of fundamental constants. In this work, we review the essentials of the physics of white dwarf stars. Special emphasis is placed on the physical processes that lead to the formation of white dwarfs as well as on the different energy sources and processes responsible for chemical abundance changes that occur along their evolution. Moreover, in the course of their lives, white dwarfs cross different pulsational instability strips. The existence of these instability strips provides astronomers with an unique opportunity to peer into their internal structure that would otherwise remain hidden from observers. We will show that this allows to measure with unprecedented precision the stellar masses and to infer their envelope thicknesses, to probe the core chemical stratification, and to detect rotation rates and magnetic fields. Consequently, in this work, we also review the pulsational properties of white dwarfs and the most recent applications of white dwarf asteroseismology.Comment: 85 pages, 28 figures. To be published in The Astronomy and Astrophysics Revie

Relativistic Binaries in Globular Clusters

Galactic globular clusters are old, dense star systems typically containing 10\super{4}--10\super{7} stars. As an old population of stars, globular clusters contain many collapsed and degenerate objects. As a dense population of stars, globular clusters are the scene of many interesting close dynamical interactions between stars. These dynamical interactions can alter the evolution of individual stars and can produce tight binary systems containing one or two compact objects. In this review, we discuss theoretical models of globular cluster evolution and binary evolution, techniques for simulating this evolution that leads to relativistic binaries, and current and possible future observational evidence for this population. Our discussion of globular cluster evolution will focus on the processes that boost the production of hard binary systems and the subsequent interaction of these binaries that can alter the properties of both bodies and can lead to exotic objects. Direct {\it N}-body integrations and Fokker--Planck simulations of the evolution of globular clusters that incorporate tidal interactions and lead to predictions of relativistic binary populations are also discussed. We discuss the current observational evidence for cataclysmic variables, millisecond pulsars, and low-mass X-ray binaries as well as possible future detection of relativistic binaries with gravitational radiation.Comment: 88 pages, 13 figures. Submitted update of Living Reviews articl

Generating surface crack patterns

We present a method for generating surface crack patterns that appear in materials such as mud, ceramic glaze, and glass. To model these phenomena, we build upon existing physically based methods. Our algorithm generates cracks from a stress field defined heuristically over a triangle discretization of the surface. The simulation produces cracks by evolving this field over time. The user can control the characteristics and appearance of the cracks using a set of simple parameters. By changing these parameters, we have generated examples similar to a variety of crack patterns found in the real world. We assess the realism of our results by comparison with photographs of real-world examples. Using a physically based approach also enables us to generate animations similar to time-lapse photography. © 2009 Elsevier Inc. All rights reserved

Generating surface crack patterns

We present a method for generating surface crack patterns that appear in materials such as mud, ceramic glaze, and glass. To model these phenomena, we build upon existing physically based methods. Our algorithm generates cracks from a stress field defined heuristically over a triangle discretization of the surface. The simulation produces cracks by evolving this field over time. The user can control the characteristics and appearance of the cracks using a set of simple parameters. By changing these parameters, we have generated examples similar to a variety of crack patterns found in the real world. We assess the realism of our results by comparison with photographs of real-world examples. Using a physically based approach also enables us to generate animations similar to time-lapse photography

Refolding planar polygons

This paper describes an algorithm for generating a guaranteed intersection-free interpolation sequence between any pair of compatible polygons. Our algorithm builds on prior results from linkage unfolding, and if desired it can ensure that every edge length changes monotonically over the course of the interpolation sequence. The computational machinery that ensures against self-intersection is independent from a distance metric that determines the overall character of the interpolation sequence. This decoupled approach provides a powerful control mechanism for determining how the interpolation should appear, while still assuring against intersection and guaranteeing termination of the algorithm. Our algorithm also allows additional control by accommodating a set of algebraic constraints that can be weakly enforced throughout the interpolation sequence

An energy-driven approach to linkage unfolding

We present a new algorithm for unfolding planar polygonal link-ages without self-intersection based on following the gradient flow of a "repulsive" energy function. This algorithm has several advantages over previous methods. (1) The output motion is represented explicitly and exactly as a piecewise-linear curve in angle space. As a consequence, an exact snapshot of the linkage at any time can be extracted from the output in strongly polynomial time (on a real RAM supporting arithmetic, radicals, and trigonometric functions). (2) Each linear step of the motion can be computed exactly in O(n2) time on a real RAM where n is the number of vertices. (3) We explicitly bound the number of linear steps (and hence the running time) as a polynomial in n and the ratio between the maximum edge length and the initial minimum distance between a vertex and an edge. (4) Our method is practical and easy to implement. We provide a publicly accessible Java applet that implements the algorithm