Tcd phd thesis

Tcd phd thesis

Arduino /* Blink Turns on an LED on for one second, then off for one second, repeatedly. This example code is in the public domain. */ // Pin 13 has an LED connected on most Arduino boards. // give it a name: int led = 13; // the setup routine runs once when you press reset: void setup() { // Aug 12,  · 3, Likes, 39 Comments - William & Mary (@william_and_mary) on Instagram: “Move-In looks a little different this year, and we know there are mixed emotions right now. We want ” Aug 01,  · The chemical structure of the three components was analyzed using FTIR through pelleting the sample with KBr powder, the method was described in detail in our previous blogger.com IR spectra of cellulose, hemicellulose and lignin are shown in Fig. blogger.com typical functional groups and the IR signal with the possible compounds are listed in Table 1 for a reference

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This paper describes a numerical method for surface parameterization, yielding maps that are locally injective and discretely conformal in an exact sense. Unlike previous methods for discrete conformal parameterization, the method is guaranteed to work for any manifold triangle mesh, with no restrictions on triangulation quality or cone singularities.

In particular we consider maps from surfaces of any genus with or without boundary to the plane, or globally bijective maps from genus zero surfaces to the sphere, tcd phd thesis. Recent theoretical developments show that each task can be formulated as a convex problem where the triangulation is allowed to change—we complete the picture by introducing the machinery needed to actually construct a discrete conformal map.

In particular, we introduce a new scheme for tracking correspondence between triangulations based on normal coordinatesand a new interpolation procedure based on layout in the light cone. Stress tests involving difficult cone configurations and near-degenerate triangulations indicate that the method is tcd phd thesis robust in practice, and provides high-quality interpolation even on meshes with poor elements. This 3-hour course provides a first introduction to intrinsic triangulations and their use in mesh processing algorithms.

As geometric data becomes more ubiquitous, e. Intrinsic triangulations provide a powerful framework for these problems, by de-coupling the tcd phd thesis used to encode geometry from the one used tcd phd thesis computation. The basic shift in perspective is to encode the geometry of a mesh not in terms of ordinary vertex positions, but instead only in terms of edge lengths. Intrinsic triangulations have a long history in mathematics, but only in recent years have been applied to practical geometric computing.

The course begins by giving motivation for intrinsic triangulations in terms of recent tcd phd thesis in computer graphics, followed by an interactive coding session where participants can make first contact with the idea of intrinsic meshes. We then give some mathematical background, and describe key data structures overlay, signpost, normal coordinates.

Using this machinery, we translate algorithms from computational geometry and scientific computing into cutting-edge algorithms for curved surfaces. For instance, tcd phd thesis, we look at mesh parameterization, vector field processing, finding geodesics, solving partial differential equations PDEstcd phd thesis, tcd phd thesis more.

We also discuss processing of nonmanifold meshes and point clouds; participants can explore these algorithms via interactive demos. We conclude with a discussion of open questions and opportunities for future work. Curves play a fundamental role across computer graphics, physical simulation, and mathematical visualization, yet most tools for curve tcd phd thesis do nothing to prevent crossings or self-intersections.

This paper develops efficient algorithms for self- repulsion of plane and space curves that are well-suited to problems in computational design. Our starting point is the so-called tangent-point energywhich provides an infinite barrier to self-intersection. In contrast to local collision detection strategies used in, e, tcd phd thesis. A reformulation of gradient descent, based on a Sobolev-Slobodeckij inner product enables us to make rapid progress toward local minima—independent of curve resolution.

We also develop a hierarchical multigrid scheme that significantly reduces the per-step cost of optimization. The energy is easily integrated with a variety of constraints and penalties e. Discrete Differential Geometry DDG is an emerging discipline at the boundary between mathematics and computer science. It aims to translate concepts from classical differential geometry into a language that is purely finite and discrete, and can hence be used by algorithms to reason about geometric data.

In contrast to standard numerical approximation, the central philosophy of DDG is to faithfully and exactly preserve key invariants of geometric objects at the discrete level, tcd phd thesis. This process of translation from smooth to discrete helps to both illuminate the fundamental meaning behind geometric ideas and provide useful algorithmic guarantees. This volume illustrates the principles of DDG via several recent topics: discrete nets, discrete differential operators, tcd phd thesis, discrete mappings, discrete conformal geometry, and discrete optimal transport.

This paper introduces a new approach to computing geodesics on polyhedral surfaces—the basic idea is to iteratively perform edge flipsin the same spirit as the classic Delaunay flip algorithm. This process also produces a triangulation conforming to the output geodesics, which is immediately useful for tasks in geometry processing and numerical simulation. More precisely, our FlipOut algorithm transforms a given sequence of edges into a locally shortest geodesic while avoiding self-crossings formally: it finds a geodesic in the same isotopy class.

The algorithm is guaranteed to terminate in a finite number of operations; practical runtimes are on the order of a few milliseconds, even for meshes with millions of triangles. The same approach is easily applied to curves beyond simple paths, including closed loops, curve networks, and multiply-covered curves.

We explore how the method facilitates tasks such as straightening cuts and segmentation boundaries, computing geodesic Bézier curves, extending the notion of constrained Delaunay triangulations CDT to curved surfaces, and providing accurate boundary conditions for partial differential equations PDEs. Evaluation on challenging datasets such as Thingi10k indicates that the method is both robust and efficient, tcd phd thesis, even for low-quality triangulations. We describe a discrete Laplacian suitable for any triangle mesh, tcd phd thesis those that are nonmanifold or nonorientable with or without boundary.

Our Laplacian is a robust drop-in replacement for the usual cotan matrix, and is guaranteed to have nonnegative edge weights on both interior and boundary edges, even for extremely poor-quality meshes. Since all edges are manifold, we can flip to an intrinsic Delaunay triangulation; our Laplacian is then the cotan Laplacian of this new triangulation.

This construction also provides a high-quality point cloud Laplacian, via a nonmanifold triangulation of the point set, tcd phd thesis. We validate our Laplacian on a variety of challenging examples including all models from Thingi10kand a variety of standard tasks including geodesic distance computation, surface deformation, parameterization, and computing minimal surfaces.

This paper explores how core problems in PDE-based geometry processing can be efficiently and reliably solved via grid-free Monte Carlo methods. Modern geometric algorithms often need to solve Poisson-like equations on geometrically intricate domains, tcd phd thesis.

Conventional methods most often mesh the domain, which is both challenging and expensive for geometry with fine details or imperfections holes, self-intersections, etc, tcd phd thesis. In contrast, tcd phd thesis, grid-free Monte Carlo methods avoid mesh generation entirely, and instead just evaluate closest point queries. They hence do not discretize space, tcd phd thesis, time, nor even function spaces, and provide the exact solution in expectation even on extremely challenging models.

More broadly, they share many benefits with Monte Carlo methods from photorealistic rendering: excellent scaling, trivial parallel implementation, view-dependent evaluation, and the ability to work with any kind of geometry including implicit or procedural descriptions.

In particular, we consider several fundamental linear elliptic PDEs with constant coefficients on solid regions of R n. Overall we find that Monte Carlo methods significantly broaden the horizons of geometry processing, since they easily handle problems of size and complexity that are essentially hopeless for conventional methods. We introduce a system called Penrose for creating mathematical diagrams.

Its basic functionality is to translate abstract statements written in familiar math-like notation into one or more possible visual representations. Rather than rely on a fixed library of visualization tools, the visual representation is user-defined in a constraint-based specification language; diagrams are then generated automatically via constrained numerical optimization. The system is user-extensible to many domains of mathematics, and is fast enough for iterative design exploration.

In contrast to tools that specify diagrams via direct manipulation or low-level graphics programming, Penrose enables rapid creation and exploration of diagrams that faithfully preserve the underlying mathematical meaning.

We demonstrate the effectiveness and generality of the system by showing how it can be used to illustrate a diverse set of concepts from mathematics and computer graphics. We present a data structure that makes it easy to run a large class of algorithms from computational geometry and scientific computing on extremely poor-quality surface meshes, tcd phd thesis.

Rather than changing the geometry, as in traditional remeshing, we consider intrinsic triangulations which connect vertices by straight paths along the exact geometry of the input mesh. Our key insight is that such a triangulation can be encoded implicitly by storing the direction and distance to neighboring vertices, tcd phd thesis. The resulting signpost data structure then allows geometric and topological queries to be made on-demand by tracing paths across the surface.

Existing algorithms can be easily translated into the intrinsic setting, since this data structure supports the same basic operations as an ordinary triangle mesh vertex insertions, tcd phd thesis, edge splits, tcd phd thesis, etc.

The output of intrinsic algorithms can then be stored on an ordinary mesh for subsequent use; unlike previous data structures, tcd phd thesis, we use a constant amount of memory and do not need to explicitly construct an overlay tcd phd thesis unless it is specifically requested.

Working in the intrinsic setting incurs little computational overhead, yet we can run algorithms on extremely degenerate inputs, including all manifold meshes from the Thingi10k data set.

To evaluate our data structure we implement several fundamental geometric algorithms including intrinsic versions of Delaunay refinement and tcd phd thesis Delaunay triangulation, approximation of Steiner trees, adaptive mesh refinement for PDEs, and computation of Poisson equations, geodesic distance, and flip-free tangent vector fields.

A basic challenge in field-guided hexahedral meshing is to find a spatially-varying frame that is adapted to the domain geometry and is continuous up to symmetries of the cube. We introduce a fundamentally new representation of such 3D cross fields based on Cartan's method of moving frames. Our key observation is that cross fields and ordinary frame fields are locally characterized by identical conditions on their Darboux derivative.

Hence, by using derivatives as the principal representation and only later recovering the field itselfone avoids the need to tcd phd thesis account for symmetry during optimization. At the discrete level, derivatives are encoded by skew-symmetric matrices associated with the edges of a tetrahedral mesh; these matrices encode arbitrarily large rotations along each edge, and can robustly capture singular behavior even on coarse meshes.

We apply this representation to compute 3D cross fields that are as smooth as possible everywhere but on a prescribed network of singular curves—since these fields are adapted to curve tangents, they can be directly used as input for field-guided mesh generation algorithms. Optimization amounts to an easy nonlinear least squares problem that behaves like a convex program in the sense that it always appears to produce the same result, independent of initialization.

We study the numerical behavior of this procedure, and perform some preliminary experiments with mesh generation. This paper describes a method for efficiently tcd phd thesis parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold, tcd phd thesis.

More precisely, it extends a vector field defined over any region to the tcd phd thesis of the domain via parallel transport along shortest geodesics. Rather than evaluate parallel transport by explicitly tracing geodesics, tcd phd thesis, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem.

We also study the numerical behavior tcd phd thesis our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations iDT so that they can be used in the context of tangent vector field processing. Conformal geometry studies geometric properties that are invariant with respect to angle-preserving transformations.

In the discrete setting of polyhedral surfaces, the idea of naively preserving angles leads to a notion of conformal geometry that is far too rigid, i. These notes explore several alternative notions of discrete conformal structure, culminating with a recent uniformization theorem for general simplicial surfaces.

Topics covered include circle packings, circle patterns, inversive distance, discrete Yamabe flow, and connections to variational principles for ideal hyperbolic polyhedra. This chapter is tcd phd thesis extended version of tcd phd thesis notes developed for the Tcd phd thesis Short Course on Discrete Differential Geometry. Developable surfaces are those that can be made by smoothly bending flat pieces without stretching or shearing.

We introduce a definition of developability for triangle meshes which exactly captures two key properties of smooth developable surfaces, namely flattenability and presence of straight ruling lines. This definition provides a starting point for algorithms in developable surface modeling—we consider a variational approach that drives a given mesh toward developable pieces separated by regular seam curves. Computation amounts to gradient descent on an energy with support in the vertex star, without the need to explicitly cluster patches or identify seams.

We briefly explore applications to developable design and manufacturing. This paper develops a global variational approach to cutting curved surfaces so that they can be flattened into the plane with low metric distortion, tcd phd thesis. Such cuts are a critical component in a variety of algorithms that seek to parameterize surfaces over flat domains, or fabricate structures from flat materials. Rather than evaluate the quality of a cut solely based on properties of the curve itself e.

Notably, we do not have to explicitly parameterize the surface in order to evaluate the cost of a cut, but can instead integrate a simple evolution equation defined on tcd phd thesis cut curve itself.

We arrive at this flow via a novel application of shape derivatives to the Yamabe equation from conformal geometry. We then develop an Eulerian numerical integrator on triangulated surfaces, which does not restrict cuts to mesh edges and can incorporate user-defined data such as importance or occlusion, tcd phd thesis.

The resulting cut curves can be used to drive distortion to arbitrarily low levels, and have a very different character from cuts obtained via purely discrete formulations.

We briefly explore potential applications to computational design, as well as connections to space filling curves and the problem of uniform heat distribution, tcd phd thesis. Angle-preserving or conformal surface parameterization has proven to be a powerful tool across applications ranging from geometry processing, to digital manufacturing, to machine learning, yet conformal maps can still suffer from severe area distortion.

Cone singularities provide a way to mitigate this distortion, but finding the best configuration of cones is notoriously difficult. This paper develops a strategy that is globally optimal in the sense that it minimizes total area distortion among all possible cone configurations number, placement, and size that have no more than a fixed total cone angle. A key insight is that, for the purpose of optimization, one should not work directly with curvature measures which naturally represent cone configurationsbut can instead apply Fenchel-Rockafellar duality to obtain a formulation involving only ordinary functions.

The result is a convex optimization problem, which can be solved via a sequence of sparse linear systems easily built from the usual cotangent Laplacian, tcd phd thesis.

MPhil in Children's Literature

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Arduino /* Blink Turns on an LED on for one second, then off for one second, repeatedly. This example code is in the public domain. */ // Pin 13 has an LED connected on most Arduino boards. // give it a name: int led = 13; // the setup routine runs once when you press reset: void setup() { // Abstract. This 3-hour course provides a first introduction to intrinsic triangulations and their use in mesh processing algorithms. As geometric data becomes more ubiquitous, e.g., in applications such as augmented reality or machine learning, there is a pressing need to develop algorithms that work reliably on low-quality data Aug 12,  · 3, Likes, 39 Comments - William & Mary (@william_and_mary) on Instagram: “Move-In looks a little different this year, and we know there are mixed emotions right now. We want ”