
I'm a sixth year Computer Science PhD student at Carnegie Mellon University, advised by Keenan Crane. I work on algorithms for reliably and efficiently processing geometric data, drawing inspiration from classical topology and differential geometry. 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 fold origami and knit. You can find more here.
Research

Winding Numbers on Discrete Surfaces
ACM TOG (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.
project
pdf (11 mb)
supplement (2.7 mb)
video (20 s)
doi
bibtex
@article{Feng:2023:WND,
author = {Feng, Nicole and Gillespie, Mark and Crane, Keenan},
title = {Winding Numbers on Discrete Surfaces},
journal = {ACM Trans. Graph.},
volume = {42},
number = {4},
year = {2023},
publisher = {ACM},
address = {New York, NY, USA},
issn = {0730-0301},
url = {https://doi.org/10.1145/3592401},
doi = {10.1145/3592401},
month = {jul},
articleno = {36},
}

Surface Simplification using Intrinsic Error Metrics
ACM TOG (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.
project
pdf (15.6 mb)
poster (9 mb)
arxiv
code
doi
bibtex
@article{Liu:2023:SSI,
author = {Liu, Hsueh-Ti Derek and Gillespie, Mark and Chislett, Benjamin and Sharp, Nicholas and Jacobson, Alec and Crane, Keenan},
title = {Surface Simplification Using Intrinsic Error Metrics},
journal = {ACM Trans. Graph.},
volume = {42},
number = {4},
year = {2023},
publisher = {ACM},
address = {New York, NY, USA},
issn = {0730-0301},
url = {https://doi.org/10.1145/3592403},
doi = {10.1145/3592403},
month = {jul},
articleno = {118},
}
Integer Coordinates for Intrinsic Geometry Processing
ACM TOG (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.
project
pdf (7.6 mb)
arxiv
video (20 min)
video (5 min)
video (45 s)
code (C++ demo app)
code (C++ library)
doi
bibtex
@article{Gillespie:2021:ICI,
author = {Gillespie, Mark and Sharp, Nicholas and Crane, Keenan},
title = {Integer Coordinates for Intrinsic Geometry Processing},
journal = {ACM Trans. Graph.},
volume = {40},
number = {6},
year = {2021},
publisher = {ACM},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/3478513.3480522},
doi = {10.1145/3478513.3480522},
}
Geometry Processing with Intrinsic Triangulations
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.
pdf (25 mb)
video (3 hour)
coding tutorial
doi
bibtex
@article{Sharp:2021:GPI,
author = {Sharp, Nicholas and Gillespie, Mark and Crane, Keenan},
title = {Geometry Processing with Intrinsic Triangulations},
booktitle = {ACM SIGGRAPH 2021 courses},
series = {SIGGRAPH '21},
year = {2021},
publisher = {ACM},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/3450508.3464592},
doi = {10.1145/3450508.3464592},
}

Discrete Conformal Equivalence of Polyhedral Surfaces
ACM TOG (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.
project
pdf (16 mb)
video (5 min)
video (20 min)
code (C++ demo app)
doi
bibtex
@article{Gillespie:2021:DCE,
author = {Gillespie, Mark and Springborn, Boris and Crane, Keenan},
title = {Discrete Conformal Equivalence of Polyhedral Surfaces},
journal = {ACM Trans. Graph.},
volume = {40},
number = {4},
year = {2021},
publisher = {ACM},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/3450626.3459763},
doi = {10.1145/3450626.3459763},
}


Magnetohydrodynamics Simulation via Discrete Exterior Calculus
Senior Thesis (Caltech)
Magnetohydrodyamics (MHD) models the behavior of current-carrying fluids such as plasma on the surface of the sun, or the earth's molten core. In this thesis, I gave an integrator for ideal MHD in two-dimensional domains with boundary and showed that the integrator preserves total energy and cross helicity.
pdf (6.6 mb)
bibtex
@mastersthesis{Gillespie:2018:MHD,
author = {Gillespie, Mark},
title = {Magnetohydrodynamics Simulation via Discrete Exterior Calculus},
type = {Bachelor's Thesis},
school = {California Institute of Technology},
year = {2018}
}
Awards
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. | |
2016 | William Lowell Putnam Mathematics Competition |
31 points (rank: 365/3214) |
Education
Schools
- Carnegie Mellon University
- 2018-present
- PhD student in the Computer Science Department
- California Institute of Technology
- 2014-2018
- Majors: Computer Science, Mathematics
- GPA: 4.1
Talks
Sept. 2023 | Intrinsic Triangulations in Geometry Processing |
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. | |
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. | |
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, and the beautiful poster designed by Rachel Joan Wallis is available here. | |
Dec. 2021 | Integer Coordinates for Intrinsic Geometry Processing |
Siggraph Asia Technical Papers | |
20 minute talk about my paper of the same name. The full talk is available on YouTube here, and a 5-minute version is available here. | |
Aug. 2021 | Discrete Conformal Equivalence of Polyhedral Surfaces |
Siggraph Technical Papers | |
20 minute talk about my paper of the same name. The full talk is available on YouTube here, and a 5-minute version is available here. | |
Oct. 2019 | Origami and Geometry |
CMU Graphics Lab | |
1 hour talk overview of assorted algorithms for origami design, and how they relate to geometetry. The slides are available here. | |
Jun. 2019 | Hyperbolic Geometry and Discrete Conformal Maps |
CMU Graphics Lab | |
1 hour talk about discrete conformal maps, length cross ratios, and hyperbolic geometry. The slides are available here. | |
Jan. 2019 | Magnetohydrodynamics and the Geometry of Conservation Laws |
CMU Graphics Lab | |
1 hour talk on my previous summer research. Focused on general background about geometric mechanics and MHD. The slides are available here. | |
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. | |
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. | |
Mar. 2017 | Continuous and Discrete Mechanics for Variational Integrators |
Caltech CS 177b | |
1.5 hour final presentation for a computer graphics class. Gave an overview of Hamiltonian and Lagrangian mechanics, and discussed how to discretize them to produce variational time integrators. Extended notes are available here. |
Miscellanea

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

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