In this paper we present a new algorithm for smoothing arbitrary triangle meshes while satisfying G¹ boundary conditions. The algorithm is based on solving a non-linear fourth order partial differential equation (PDE) that only depends on intrinsic surface properties instead of being derived from a particular surface parameterization. This continuous PDE has a (representation-independent) welldefined solution which we approximate by our triangle mesh. Hence, changing the mesh complexity (refinement) or the mesh connectivity (remeshing) leads to just another discretization of the same smooth surface and doesn't affect the resulting geometric shape beyond this. This is typically not true for filter-based mesh smoothing algorithms. To simplify the computation we factorize the fourth order PDE into a set of two nested second order problems thus avoiding the estimation of higher order derivatives. Further acceleration is achieved by applying multigrid techniques on a fine-to-coarse hierarchical mesh representation.

Abstract. We minimize a discrete version of the squared mean curvature integral for polyhedral surfaces in three-space using Brakke’s Surface Evolver. Our experimental results support the conjecture that the smooth minimizers exist for each genus and are stereographic projections of certain minimal surfaces in the three-sphere.

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this paper extends the technology in several directions. In Section 3.1 we study the collapse of a surface under motion by mean curvature. In [Sethian 1989] it was shown experimentally that the handle of a dumbbell pinches off, splitting the dumbbell into two surfaces, each of which collapses to a point. Here we show that an extension of this problem produces an interesting result: a multi-armed dumbbell leaves a separate, residual closed object at the center after the singularity forms. We verify this by studying a series of similar numerical problems, each showing this detached surface. In Section 3.2 we briefly consider flow under Gaussian curvature. In Section 4 we use the level set approach to generate minimal surfaces attached to a given onedimensional closed curve (wire frame) in R

Shape optimization and surface fairing for polygon meshes have been active research areas for the last few years. Existing approaches either require the border of the surface to be fixed, or are only applicable to closed surfaces. In this paper, we propose a new approach, that computes natural boundaries. This makes it possible not only to smooth an existing geometry, but also to extrapolate its shape beyond the existing border. Our approach is based on a global parameterization of the surface and on a minimization of the squared curvatures, discretized on the edges of the surface. The soconstructed surface is an approximation of a minimal energy surface (MES). Using a global parameterization makes it possible to completely decouple the outer fairness (surface smoothness) from the inner fairness (mesh quality). In addition, the parameter space provides the user with a new means of controlling the shape of the surface. When used as a geometry filter, our approach computes a smoothed mesh that is discrete conformal to the original one. This allows smoothing textured meshes without introducing distortions.

The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in three-dimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in three-dimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or self-intersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a

We accept this thesis as conforming to the required standard

. We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The proof is based on a theorem of L uk o on average chord lengths of closed curves. 1.

In this paper we present a technique for simulating and rendering liquid foams. We are aiming at a functional realism that allows our simulation to be consistent with the physical effects in real liquid foam while avoiding the prohibitive computational cost of a physically accurate simulation. To this end, we have to recreate two important attributes of foam. The dynamic behaviour of the simulated foam must be based on the physics of real foam, and the characteristic interior structures of foam and their optical properties must be reproduced. We tackle these requirements by introducing a two part hybrid rendering approach. The first stage is geometric and determines the dynamic behaviour of the foam by simulating structural forces on a set of spheres, which represent the foam bubbles. In the second stage we render these spheres using a special surface shader that implicitly reconstructs the foam surfaces and performs the shading calculations. This two step approach allows us to easily integrate our technique into existing ray-tracing systems. We include images of an example animation to demonstrate the visual quality.

We propose a general paradigm for computing optimal coordinate frame fields that may be exploited to visualize curves and surfaces. Parallel-transport framings, which work well for open curves, generally fail to have desirable properties for cyclic curves and for surfaces. We suggest that minimal quaternion measure provides an appropriate heuristic generalization of parallel transport. Our approach differs from minimal-tangential-acceleration approaches due to the addition of "sliding ring" constraints that fix one frame axis, but allow an axial rotational freedom whose value is varied in the optimization process. Our fundamental tool is the quaternion Gauss map, a generalization to quaternion space of the tangent map for curves and of the Gauss map for surfaces. The quaternion Gauss map takes 3D coordinate frame fields for curves and surfaces into corresponding curves and surfaces constrained to the space of possible orientations in quaternion space. Standard optimization tools provid...