6 research outputs found
The Graphs of Planar Soap Bubbles
We characterize the graphs formed by two-dimensional soap bubbles as being
exactly the 3-regular bridgeless planar multigraphs. Our characterization
combines a local characterization of soap bubble graphs in terms of the
curvatures of arcs meeting at common vertices, a proof that this
characterization remains invariant under Moebius transformations, an
application of Moebius invariance to prove bridgelessness, and a
Moebius-invariant power diagram of circles previously developed by the author
for its applications in graph drawing.Comment: 16 pages, 9 figure
Drawing Trees with Perfect Angular Resolution and Polynomial Area
We study methods for drawing trees with perfect angular resolution, i.e.,
with angles at each node v equal to 2{\pi}/d(v). We show:
1. Any unordered tree has a crossing-free straight-line drawing with perfect
angular resolution and polynomial area.
2. There are ordered trees that require exponential area for any
crossing-free straight-line drawing having perfect angular resolution.
3. Any ordered tree has a crossing-free Lombardi-style drawing (where each
edge is represented by a circular arc) with perfect angular resolution and
polynomial area. Thus, our results explore what is achievable with
straight-line drawings and what more is achievable with Lombardi-style
drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
A model for soap film dynamics with evolving thickness
Previous research on animations of soap bubbles, films, and foams largely focuses on the motion and geometric shape of the bubble surface. These works neglect the evolution of the bubble’s thickness, which is normally responsible for visual phenomena like surface vortices, Newton’s interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. In this paper, we model these natural phenomena by introducing the film thickness as a reduced degree of freedom in the Navier-Stokes equations and deriving their equations of motion. We discretize the equations on a nonmanifold triangle mesh surface and couple it to an existing bubble solver. In doing so, we also introduce an incompressible fluid solver for 2.5D films and a novel advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance state-of-the-art bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film
Arquitectura paramétrica y diseño paramétrico. Aplicación al diseño de mobiliario urbano
A lo largo de la presente memoria de Trabajo fin de grado se profundiza en diferentes técnicas digitales existentes en el campo del diseño arquitectónico asà como su aplicación. En primer lugar se hace un breve recorrido por cinco ejemplos paradigmáticos del uso de técnicas digitales que permiten tanto al autor como al lector aproximarse a las diferentes posibilidades que existen. Posteriormente se elige el caso concreto del edificio Water Cube de PekÃn y se estudia el modo en que fue diseñada su fachada y por ultimo se realiza un caso de aplicación de los conceptos aprendidos al diseño de una pieza de arte urbano a colocar en un espacio público urbano. Finalmente se dan unas indicaciones para que una persona parta en el futuro del trabajo realizado y continúe con la investigación