Sphere tracing is a fast and effective algorithm for visualizing surfaces encoded by signed distance functions (SDFs), which have become a centerpiece in a wide range of visual computing algorithms. We introduce an analogous algorithm for a completely different class of functions, harmonic functions, opening up a whole new set of possibilities. We show how our new algorithm can be used to directly visualize smooth surfaces reconstructed from point clouds (via Poisson surface reconstruction) or polygon soup (via generalized winding numbers) without performing linear solves or mesh extraction. We also show how it can be used to render nonplanar polygons (including those with holes), and to visualize key objects from mathematics, including knots, links, spherical harmonics, and Riemann surfaces.

This paper introduces a new fabrication technique called solid knitting. Unlike standard knitting, which makes hollow surfaces, solid knitting creates dense volumes by layering knit sheets—much as 3D printers layer plastic sheets. We envision a future where everyday objects like furniture or shoes—including soles—can be knit as one piece. We define the basic building blocks of solid knitting and demonstrate a working prototype of a solid knitting machine controlled by a low-level instruction language, along with a volumetric design tool for creating machine-knittable patterns.

This thesis presents algorithms and data structures for computing on surfaces whose intrinsic geometry evolves over time. We take as examples the problems of mesh simplification and surface parameterization—in both cases, we find that the intrinsic perspective leads to simple algorithms which are robust and efficient on a variety of challenging examples.

Winding numbers are a basic component of geometric algorithms such as point-in-polygon tests, and their generalization to data with noise or topological errors has proven invaluable in robust geometry processing. However, standard definitions do not immediately apply on surfaces, where not all curves bound regions. We develop a meaningful generalization, starting with the well-known relationship between winding numbers and harmonic functions. Ultimately, our algorithm yields (i) a closed completion the input curves, (ii) integer labels for regions that are meaningfully bounded by these curves, and (iii) the complementary curves that do not bound any region.

This paper describes a method for fast simplification of surface meshes. Rather than approximate the extrinsic geometry, we construct a coarse intrinsic triangulation of the input domain. In the spirit of the quadric error metric (QEM), we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature “drifts” during simplification. The overall payoff is a “black box” approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations.

This paper describes a numerically robust data structure for encoding intrinsic triangulations of polyhedral surfaces. Our starting point is the framework of normal coordinates from geometric topology, which we extend to a broader set of operations needed for mesh processing. As a stress test, we successfully compute an intrinsic Delaunay refinement and associated subdivision for all manifold meshes in the Thingi10k dataset.

Intrinsic triangulations de-couple the mesh used to encode geometry from the one used for 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. This course provides a first introduction to intrinsic triangulations and their use in mesh processing algorithms, covering the theory, practical details, and some cutting-edge research in the area.

We present a numerical method for surface parameterization, leveraging hyperbolic geometry to yield maps that are locally injective and discretely conformal in an exact sense. Stress tests involving difficult cone configurations and near-degenerate triangulations indicate that the method is extremely robust in practice, and provides high-quality interpolation even on meshes with poor elements.

Web demo which traces out geodesics on a mesh. The code is available here.

Notes on hyperbolic geometry and discrete conformal maps.

A shader to make meshes tartan.

GLSL implementation of BPM: Blended Piecewise Möbius Maps by Rorberg, Vaxman & Ben-Chen, incorporated into polyscope.

Takes any triangle mesh and turn it into an orientable manifold mesh by constructing the orientable double cover.

This is a rough implementation of the Delaunay edge split algorithm presented in Efficient construction and simplification of Delaunay meshes by Yong-Jin Liu, Chunxu Xu, Dian Fan, and Ying He. It takes in a triangle mesh and then performs edge splits to make the mesh Delaunay.

Web demo computing the Karcher mean of points on a sphere.

Visualization of the phase space of a pendulum as a cylinder.

Comparison of explicit, implicit, and symplectic Euler integrators for a pendulum.

Demo of an implementation of the combinatorial map data structure for n-dimensional simplicial complexes that I experimented with in geometry-central. It has not yet made its way into the library itself, but the implementation can be found here.

Online tool to check if a word is in the dictionary.