The geometry of flex tangents to a cubic curve and its parameterizations

International audienceWe show how the study of the geometry of the nine flex tangents to a cubic produces many pseudo-parameterizations, including the ones given by Icart, Kammerer, Lercier, Renault and Farashahi

Maximally inflected real rational curves

We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in the real Schubert calculus. We establish restrictions on the number of real nodes of such curves and construct curves realizing the extreme numbers of real nodes. These constructions imply the existence of real solutions to some problems in the Schubert calculus. We conclude with a discussion of maximally inflected curves of low degree.Comment: Revised with minor corrections. 37 pages with 106 .eps figures. Over 250 additional pictures on accompanying web page (See http://www.math.umass.edu/~sottile/pages/inflected/index.html

The geometry of some parameterizations and encodings

We explore parameterizations by radicals of low genera algebraic curves. We prove that for $q$ a prime power that is large enough and prime to $6$, a fixed positive proportion of all genus 2 curves over the field with $q$ elements can be parameterized by $3$-radicals. This results in the existence of a deterministic encoding into these curves when $q$ is congruent to $2$ modulo $3$. We extend this construction to parameterizations by $\ell$-radicals for small odd integers $\ell$, and make it explicit for $\ell=5$

Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects

In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.Comment: Minor revisions. Now 49 pages, 4 figures. To appear in European Journal of Mathematics, special issue in memory of W.L. Edg

Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects

In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.Comment: Minor revisions. Now 49 pages, 4 figures. To appear in European Journal of Mathematics, special issue in memory of W.L. Edg

Programmed design of ship forms

This paper describes a new category of CAD applications devoted to the definition and parameterization of hull forms, called programmed design. Programmed design relies on two prerequisites. The first one is a product model with a variety of types large enough to face the modeling of any type of ship. The second one is a design language dedicated to create the product model. The main purpose of the language is to publish the modeling algorithms of the application in the designer knowledge domain to let the designer create parametric model scripts. The programmed design is an evolution of the parametric design but it is not just parametric design. It is a tool to create parametric design tools. It provides a methodology to extract the design knowledge by abstracting a design experience in order to store and reuse it. Programmed design is related with the organizational and architectural aspects of the CAD applications but not with the development of modeling algorithms. It is built on top and relies on existing algorithms provided by a comprehensive product model. Programmed design can be useful to develop new applications, to support the evolution of existing applications or even to integrate different types of application in a single one. A three-level software architecture is proposed to make the implementation of the programmed design easier. These levels are the conceptual level based on the design language, the mathematical level based on the geometric formulation of the product model and the visual level based on the polyhedral representation of the model as required by the graphic card. Finally, some scenarios of the use of programmed design are discussed. For instance, the development of specialized parametric hull form generators for a ship type or a family of ships or the creation of palettes of hull form components to be used as parametric design patterns. Also two new processes of reverse engineering which can considerably improve the application have been detected: the creation of the mathematical level from the visual level and the creation of the conceptual level from the mathematical level. © 2012 Elsevier Ltd. All rights reserved. 1. Introductio

Asymptotics, Geometry, and Soft Matter

This dissertation is concerned with two problems that lie at the interface of soft-matter physics, geometry, and asymptotic analysis, but otherwise have no bearing on one another. In the first problem, I consider the equilibrium thermal fluctuations of deformable mechanical frameworks. These frameworks have served as highly idealized representations of mechanical structures that underlie a plethora of soft, few-body systems at the submicron scale such as colloidal clusters and DNA origami. When the holonomic constraints in a framework cease to be linearly independent, singularities can appear in its configuration space, where it becomes energetically softer. Consequently, the framework\u27s free-energy landscape becomes dominated by the neighborhoods of points corresponding to these singularities. In the second problem, I study the localization of elastic waves in thin elastic structures with spatially varying curvature profiles, using a curved rod and a uniaxially-curved shell as concrete examples. Waves propagating on such structures have multiple components owing to the curvature-mediated coupling of the tangential and normal components of the displacement field. Here, using the semiclassical approximation, I show that these waves form localized, bound states around points where the absolute curvature of the structure has a minimum. Both these problems exemplify the subtle interplay between the mechanical properties of soft materials and their geometry, which further sets the stage for many interesting consequences

Computer-Aided Geometry Modeling

Techniques in computer-aided geometry modeling and their application are addressed. Mathematical modeling, solid geometry models, management of geometric data, development of geometry standards, and interactive and graphic procedures are discussed. The applications include aeronautical and aerospace structures design, fluid flow modeling, and gas turbine design

Isogeometric Approximation of Variational Problems for Shells

The interaction of applied geometry and numerical simulation is a growing field in the interplay of com- puter graphics, computational mechanics and applied mathematics known as isogeometric analysis. In this thesis we apply and analyze Loop subdivision surfaces as isogeometric tool because they provide great flexibility in handling surfaces of arbitrary topology combined with higher order smoothness. Compared with finite element methods, isogeometric methods are known to require far less degrees of freedom for the modeling of complex surfaces but at the same time the assembly of the isogeo- metric matrices is much more time-consuming. Therefore, we implement the isogeometric subdivision method and analyze the experimental convergence behavior for different quadrature schemes. The mid-edge quadrature combines robustness and efficiency, where efficiency is additionally increased via lookup tables. For the first time, the lookup tables allow the simulation with control meshes of arbitrary closed connectivity without an initial subdivision step, i.e. triangles can have more than one vertex with valence different from six. Geometric evolution problems have many applications in material sciences, surface processing and modeling, bio-mechanics, elasticity and physical simulations. These evolution problems are often based on the gradient flow of a geometric energy depending on first and second fundamental forms of the surface. The isogeometric approach allows a conforming higher order spatial discretization of these geometric evolutions. To overcome a time-error dominated scheme, we combine higher order space and time discretizations, where the time discretization based on implicit Runge-Kutta methods. We prove that the energy diminishes in every time-step in the fully discrete setting under mild time-step restrictions which is the crucial characteristic of a gradient flow. The overall setup allows for a general type of fourth-order energies. Among others, we perform experiments for Willmore flow with respect to different metrics. In the last chapter of this thesis we apply the time-discrete geodesic calculus in shape space to the space of subdivision shells. By approximating the squared Riemannian distance by a suitable energy, this approach defines a discrete path energy for a consistent computation of geodesics, logarithm and exponential maps and parallel transport. As approximation we pick up an elastic shell energy, which measures the deformation of a shell by membrane and bending contributions of its mid-surface. Bézier curves are a fundamental tool in computer-aided geometric design. We extend these to the subdivision shell space by generalizing the de Casteljau algorithm. The evaluation of Bézier curves depends on all input data. To solve this problem, we introduce B-splines and cardinal splines in shape space by gluing together piecewise Bézier curves in a smooth way. We show examples of quadratic and cubic Bézier curves, quadratic and cubic B-splines as well as cardinal splines in subdivision shell space