Curves with rational chord-length parametrization

It has been recently proved that rational quadratic circles in standard Bezier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limacon of Pascal. The extension to 3D rational curves is also tackled

A Comparison of Three Curve Intersection Algorithms

An empirical comparison is made between three algorithms for computing the points of intersection of two planar Bezier curves. The algorithms compared are: the well known Bezier subdivision algorithm, which is discussed in Lane 80; a subdivision algorithm based on interval analysis due to Koparkar and Mudur; and an algorithm due to Sederberg, Anderson and Goldman which reduces the problem to one of finding the roots of a univariate polynomial. The details of these three algorithms are presented in their respective references

Binary Linear Classification and Feature Selection via Generalized Approximate Message Passing

For the problem of binary linear classification and feature selection, we propose algorithmic approaches to classifier design based on the generalized approximate message passing (GAMP) algorithm, recently proposed in the context of compressive sensing. We are particularly motivated by problems where the number of features greatly exceeds the number of training examples, but where only a few features suffice for accurate classification. We show that sum-product GAMP can be used to (approximately) minimize the classification error rate and max-sum GAMP can be used to minimize a wide variety of regularized loss functions. Furthermore, we describe an expectation-maximization (EM)-based scheme to learn the associated model parameters online, as an alternative to cross-validation, and we show that GAMP's state-evolution framework can be used to accurately predict the misclassification rate. Finally, we present a detailed numerical study to confirm the accuracy, speed, and flexibility afforded by our GAMP-based approaches to binary linear classification and feature selection

Germany as shipwreck: The “Robinson” trope in German diaries, 1943–1946

Drawing on unpublished diaries from the period 1943–46, this article shows how the “Robinson” trope of shipwreck and survival provided German civilians with a language to describe the transition from National Socialism to a radically open and anxiety-producing future. The identification with Robinson and his story makes the writers into agentic protagonists, at the same time that the metaphor of the “island” reflects an ambiguous position of isolation. Excerpts from diaries show how the “Robinson” trope combines dichotomies crucial to this period of turmoil: civilization and the primitive, individual and society, victimhood and agency, and how diarists rework this language to explore their place in a changing world.Drawing on unpublished diaries from the period 1943–46, this article shows how the “Robinson” trope of shipwreck and survival provided German civilians with a language to describe the transition from National Socialism to a radically open and anxiety-producing future. The identification with Robinson and his story makes the writers into agentic protagonists, at the same time that the metaphor of the “island” reflects an ambiguous position of isolation. Excerpts from diaries show how the “Robinson” trope combines dichotomies crucial to this period of turmoil: civilization and the primitive, individual and society, victimhood and agency, and how diarists rework this language to explore their place in a changing world

Build Strong Bodies and Minds Through Service Learning

In Physics and Astronomy at Purdue, service learning is an essential component to our outreach programs. While providing authentic deliverables to the “customer,” service learning engages individuals in ways through which they are able to make meaningful contributions, while at the same time developing their own knowledge and expertise in an area of personal interest or commitment. That learning may involve leadership, instructional design, communication of science fundamentals and applications to K-12 and general audiences, research, reflective practice, and the acquisition of skills that last a lifetime. This presentation illustrates ways in which we leverage resources between service learning and successful outreach programs in Physics and Astronomy. Keywords: outreach, service learning, science, physics, astronomy, SMAP, QuarkNe

Matrix representations for toric parametrizations

In this paper we show that a surface in P^3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over P^2 established in [BJ03] and [BC05]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case T = P^1 x P^1 in detail and give numerous examples.Comment: 20 page