A picture of me
Mark Gillespie

I'm a postdoc in Computer Science at École Polytechnique, working with Mathieu Desbrun. I work on algorithms for reliably and efficiently processing geometric data, drawing inspiration from classical topology and differential geometry. I just received my PhD in computer science from Carnegie Mellon University, where I was advised by Keenan Crane. Previously I was an undergraduate at Caltech, where I worked on plasma simulation with Peter Schröder, chromosomal shape embeddings with Mathieu Desbrun, and interval analysis root finding techniques with Alan Barr.

In my spare time, I like to design origami models and knit.


Research
Mark Gillespie, Denise Yang, Mario Botsch, Keenan Crane
ACM Transactions on Graphics (SIGGRAPH 2024)
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.
ACM Transactions on Graphics (SIGGRAPH 2024)
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.
Mark Gillespie
PhD Thesis (Carnegie Mellon University)
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.
Nicole Feng, Mark Gillespie, Keenan Crane
ACM Transactions on Graphics (SIGGRAPH 2023)
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.
ACM Transactions on Graphics (SIGGRAPH 2023)
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.
Mark Gillespie, Nicholas Sharp, Keenan Crane
ACM Transactions on Graphics (SIGGRAPH Asia 2021)
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.
Geometry Processing with Intrinsic Triangulations
Nicholas Sharp, Mark Gillespie, Keenan Crane
ACM SIGGRAPH Courses (2021), SIAM IMR 2021 Courses
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.
Mark Gillespie, Boris Springborn, Keenan Crane
ACM Transactions on Graphics (SIGGRAPH 2021)
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.

Awards
2024

Two SIGGRAPH Best Paper Award Honorable Mentions

For Ray Tracing Harmonic Functions and Solid Knitting.
2019

NSF Graduate Research Fellowship

2017

SIGGRAPH ACM Turing Award Celebration Grant

I was one of 10 students sponsored by SIGGRAPH to attend the ACM Turing Award Celebration.
2017, 2016

Arthur R. Adams SURF Fellowship

Fellowship to fund my summer research.

Selected Talks
Nov. 2024

Solid Knitting and Harmonic Hitting

Institute of Science and Technology Austria (ISTA)
45 minute talk about my work on solid knitting and ray tracing harmnic functions. My slides are available as a pdf [38 mb].
Apr. 2024

Evolving Intrinsic Triangulations

Carnegie Mellon University (CMU)
Thesis defense. My slides are available here [76 mb] and my thesis document is available here [26 mb].
Dec. 2023

Dynamic Intrinsic Geometry Processing

Carnegie Mellon University (CMU)
Thesis proposal talk. My slides are available here [79 mb] and my proposal document is available here [13 mb].
Sept. 2023

Intrinsic Triangulations in Geometry Processing

Institute of Science and Technology Austria (ISTA)
90 minute talk about my work on intrinsic triangulations.
Aug. 2023

Intrinsic Triangulations in Geometry Processing

Geometry Workshop in Obergurgl
30 minute talk about my work on intrinsic triangulations.
Jul. 2023

Intrinsic Triangulations in Geometry Processing

TU Berlin SFB TRR 109 Colloquium
45 minute talk about some of my assorted work on intrinsic triangulations. The slides are available here [49 mb].
Apr. 2022

Discrete Conformal Equivalence of Polyhedral Surfaces

UCSD Pixel Cafe
45 minute talk about my paper of the same name. The slides are available here [32 mb].
Mar. 2022

Discrete Conformal Equivalence of Polyhedral Surfaces

Toronto Geometry Colloquium
10 minute talk about my paper of the same name. The slides are available here [19 mb], and the beautiful poster designed by Rachel Joan Wallis is available here.
Oct. 2019

Origami and Geometry

Carnegie Mellon University Graphics Lab
1 hour overview of algorithms for origami design, and how they relate to geometry. The slides are available here [37 mb].
Jun. 2019

Hyperbolic Geometry and Discrete Conformal Maps

Carnegie Mellon University Graphics Lab
1 hour talk about discrete conformal maps, length cross ratios, and hyperbolic geometry. The slides are available here [26 mb].
Jan. 2019

Magnetohydrodynamics and the Geometry of Conservation Laws

Carnegie Mellon University Graphics Lab
1 hour talk introducing geometric mechanics and magnetohydrodynamics. The slides are available here [22 mb].
Oct. 2017

2D Plasma Simulation via Discrete Exterior Calculus

Caltech Summer Research Seminar Day
15 minute presentation on the results of my summer research. The slides are available here [1 mb].
Sept. 2017

Combinatorics and the Probabilistic Method

Westfield High School Seminar in College Mathematics
30 minute presentation to a high school math class. Gave an introduction to elementary combinatorics and presented some simple applications of the probabilistic method. My notes are available here.

Miscellanea
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 turns 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.