## About

22

Publications

3,389

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

289

Citations

Citations since 2017

Introduction

Additional affiliations

August 2015 - July 2016

## Publications

Publications (22)

We present geometric proofs of Menger’s results on isometrically embedding metric spaces in Euclidean space.

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper non-unitary c-polyhedra with M\"obius-congruent faces that are consistently oriented are M\"obius-congruent. Our r...

We present constructions inspired by the Ma-Schlenker example of~\cite{Ma:2012hl} that show the non-rigidity of spherical inversive distance circle packings. In contrast to the use in~\cite{Ma:2012hl} of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary constructions use only the inv...

The first phase of TreeMaker, a well-known method for origami design, decomposes a planar polygon (the “paper”) into regions. If some region is not convex, TreeMaker indicates it with an error message and stops. Otherwise, a second phases is invoked which computes a crease pattern called a “universal molecule”. In this paper we introduce and study...

We present a Java implementation of Lang's Universal Molecule algorithm, alongside with a visualization of its interconnected structures: the input metric tree and compatible convex polygon, whose 2D crease pattern and 3D uniaxial base are computed by the algorithm. The Java applet, the video, as well as further references and accompanying material...

Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies acro...

We verify the infinitesimal inversive rigidity of almost all triangulated circle polyhedra in the Euclidean plane \({\mathbb {E}}^{2}\), as well as the infinitesimal inversive rigidity of tangency circle packings on the 2-sphere \({\mathbb {S}}^{2}\). From this the rigidity of almost all triangulated circle polyhedra follows. The proof adapts Gluck...

In additively manufactured (AM) components, infill structure significantly affects the mechanical performance of the final printed part. However, mechanical stress induced by operation loads has been so far neglected for infill patterning. Most slicers currently available in the market provide infill patterns that are uniform in shape and size rega...

We verify the infinitesimal inversive rigidity of almost all triangulated circle polyhedra in the Euclidean plane $\mathbb{E}^{2}$, as well as the infinitesimal inversive rigidity of tangency circle packings on the $2$-sphere $\mathbb{S}^{2}$. From this the rigidity of almost all triangulated circle polyhedra follows. The proof adapts Gluck's proof...

First, we propose a meshing technique using a heuristic circle packing algorithm which generates meshes suitable for 3D printing. Second, we propose a discrete optimization approach to identify the optimal tool path in the mesh. Our approach is motivated by the Miller-Tucker-Zemlin formulation to the traveling salesman problem, and allows us to spe...

We present a new approach to incorporate an internal stress distribution into the design of infill via fused deposition modeling of additive manufacturing (AM). This design approach differs from topology optimization, since the topology optimization of AM focuses on changing the overall shape of the product, whereas the approach we propose in this...

KINARI-2 is the second release of the web server KINARI-Web for rigidity and flexibility of biomolecules. Besides incorporating new web technologies and making substantially improved tools available to the user, KINARI-2 is designed to automatically ensure the reproducibility of its computational experiments. It is also designed to facilitate incor...

We give the first deterministic $O(n\log n)$ time algorithm for computing the
straight-skeleton of a simple polygon given its induced motorcycle graph as
input. The previous best takes expected $O(n\log^2 n)$ time. Our algorithm is
reminiscent of Shamos and Hoey's divide and conquer algorithm for computing the
Voronoi diagram of a set of planar poi...

Lang's "universal molecule" algorithm solves a variant of the origami design problem. It takes as input a metric tree and a convex polygonal region (the "paper") having a certain metric relationship with the tree. It computes a crease- pattern which allows for the paper to "fold" to a uniaxial base, which is a 3-dimensional shape projecting onto th...

In a seminal paper from 1996 that marks the beginning of computational origami, R. Lang introduced TreeMaker, a method for designing origami crease patterns with an underlying metric tree structure. In this paper we address the foldability of paneled origamis produced by Lang’s Universal Molecule algorithm, a key component of TreeMaker.
We identify...

Diffusion curves [OBW*08] provide a flexible tool to create smooth-shaded images from curves defined with colors. The resulting image is typically computed by solving a Poisson equation that diffuses the curve colors to the interior of the image. In this paper we present a new method for solving diffusion curves by using ray tracing. Our approach i...

The ability to place surface samples with Poisson disk distribution can benefit a variety of graphics applications. Such a distribution satisfies the blue noise property, i.e. lack of low frequency noise and structural bias in the Fourier power spectrum. While many techniques are available for sampling the plane, challenges remain for sampling arbi...

The ability to place surface samples with Poisson disk distribution can benefit a variety of graphics applications. Such a distribution satisfies the blue noise property, i.e. lack of low frequency noise and structural bias in the Fourier power spectrum. While many techniques are available for sampling the plane, challenges remain for sampling arbi...

We construct a 1-parameter family of geodesic shape metrics on a space of closed parametric curves in Euclidean space of any
dimension. The curves are modeled on homogeneous elastic strings whose elasticity properties are described in terms of their
tension and rigidity coefficients. As we change the elasticity properties, we obtain the various ela...

We employ 3D arrangements of curves to represent and analyze biological shapes, in particular, the anatomy of the human brain. The arrangements of curves may vary from fairly sparse – such as a collection of sulcal lines that coarsely approximates the global shape of the brain – to very dense decompositions of the cortical surface into space curves...

In this supplemental document for [Bowers et al. 2010], we derive the analytic radial means and anisotropy of 1D white noise samples. We show that when using the real form of Fourier basis, using the radial mean is the same as using the complex form of Fourier basis; however, the anisotropy when using the former is twice as much as when using the l...

## Projects

Project (1)