Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure-preserving: it reproduces curl-free fields precisely, and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation

Vector field processing on triangle meshes

While scalar fields on surfaces have been staples of geometry processing, the use of tangent vector fields has steadily grown in geometry processing over the last two decades: they are crucial to encoding directions and sizing on surfaces as commonly required in tasks such as texture synthesis, non-photorealistic rendering, digital grooming, and meshing. There are, however, a variety of discrete representations of tangent vector fields on triangle meshes, and each approach offers different tradeoffs among simplicity, efficiency, and accuracy depending on the targeted application. This course reviews the three main families of discretizations used to design computational tools for vector field processing on triangle meshes: face-based, edge-based, and vertex-based representations. In the process of reviewing the computational tools offered by these representations, we go over a large body of recent developments in vector field processing in the area of discrete differential geometry. We also discuss the theoretical and practical limitations of each type of discretization, and cover increasingly-common extensions such as n-direction and n-vector fields. While the course will focus on explaining the key approaches to practical encoding (including data structures) and manipulation (including discrete operators) of finite-dimensional vector fields, important differential geometric notions will also be covered: as often in Discrete Differential Geometry, the discrete picture will be used to illustrate deep continuous concepts such as covariant derivatives, metric connections, or Bochner Laplacians

Methodological Fundamentalism: or why Batterman’s Different Notions of ‘Fundamentalism’ may not make a Difference

I argue that the distinctions Robert Batterman (2004) presents between ‘epistemically fundamental’ versus ‘ontologically fundamental’ theoretical approaches can be subsumed by methodologically fundamental procedures. I characterize precisely what is meant by a methodologically fundamental procedure, which involves, among other things, the use of multilinear graded algebras in a theory’s formalism. For example, one such class of algebras I discuss are the Clifford (or Geometric) algebras. Aside from their being touted by many as a “unified mathematical language for physics,” (Hestenes (1984, 1986) Lasenby, et. al. (2000)) Finkelstein (2001, 2004) and others have demonstrated that the techniques of multilinear algebraic ‘expansion and contraction’ exhibit a robust regularizablilty. That is to say, such regularization has been demonstrated to remove singularities, which would otherwise appear in standard field-theoretic, mathematical characterizations of a physical theory. I claim that the existence of such methodologically fundamental procedures calls into question one of Batterman’s central points, that “our explanatory physical practice demands that we appeal essentially to (infinite) idealizations” (2003, 7) exhibited, for example, by singularities in the case of modeling critical phenomena, like fluid droplet formation. By way of counterexample, in the field of computational fluid dynamics (CFD), I discuss the work of Mann & Rockwood (2003) and Gerik Scheuermann, (2002). In the concluding section, I sketch a methodologically fundamental procedure potentially applicable to more general classes of critical phenomena appearing in fluid dynamics

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics and locking issues. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling, but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling based representations of developable surfaces, while still obtaining developable, curvature aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its level sets are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight level sets is nonlinear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle based interactive freeform modeling of developable shapes

The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations

This survey article begins with a discussion of the original form of the Strominger-Yau-Zaslow conjecture, surveys the state of knowledge concering this conjecture, and explains how thinking about this conjecture naturally leads to the program initiated by the author and Bernd Siebert to study mirror symmetry via degenerations of Calabi-Yau manifolds and log structures.Comment: 44 pages, to appear in the Proceedings of the 2005 AMS Symposium on Algebraic Geometry, Seattl