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Geometry Processing

Fall Term 2020
CSC419H1 Geometry Processing
CSC2520H Geometry Processing
Prof. Alec Jacobson
TAs, Sarah Kushner, Jiayi Eris Zhang
W 3-5 Zoom
Office Hours: W 5-6 Zoom

The class is aimed at preparing students for working with geometric data via understanding fundamental theoretical concepts. Students should have a background in Linear Algebra and Computer Programming. Previous experience with Numerical Methods, Differential Equations, and Differential Geometry is appreciated but not required.

Extending traditional signal processing, geometry processing interprets three-dimensional curves and surfaces as signals. Just as audio and image signal data can be filtered, denoised and decomposed spectrally, so can the geometry of a three-dimensional curve or surface.

In this course, we study the algorithms and mathematics behind fundamental operations for interpreting and manipulating geometric data. These essential tools enable: geometric modeling for computer aided design, life-like animations for computer graphics, reliable physical simulations, and robust scene representations for computer vision.

Topics include: discrete curves and surfaces, curvature computation, surface reconstruction from point clouds, surface smoothing and denoising, mesh simplification, parameterization, symmetry detection, shape deformation and animation.


In lecture we will cover the mathematical background and visual intuition of the week's topic. Students will read academic papers and complete a small weekly programming assignment to implement a corresponding algorithm. Through the weekly seminars at the Toronto Geometry Colloquium, students will be exposed to a wide variety of cutting-edge research in geometry processing. Finally, students will implement a recent paper in the style of a libigl tutorial.


  1. Students should understand, derive, and implement solutions to the prominent challenges that arise in geometry processing applications.
  2. Students should create a final creative project showcasing their implementation of the different processing algorithms.
  3. Students should develop an understanding of the mathematical underpinnings of geometry processing including useful discretized operators and energies.
  4. Students should develop a working knowledge of libigl to develop these algorithms without worrying about the grunt-work of OpenGL viewers, quadrature, etc.


Students should have already taken Linear Algebra and Calculus.

Students should have already taken Introduction to Computer Science and should be proficient in computer programming (in any language) and should feel comfortable programming in C++.

While knowledge of Partial Differential Equations is not required, it will certainly be very handy for derivations. A quick survey will be posted to help students evaluate their readiness on these topics.

On the programming side, we will be coding mainly in C++ and using a libigl, an open-source geometry processing library. We will be using Eigen for computational linear algebra, and Cmake for building the coding assignments.

A Mathematical Foundation

Much of the framing for our techniques will be looking at the continuous analogue of our problem and discretizing it in an intrinsic way, preserving continuous theorems as much as possible. We will discretize continuous operators like the Laplacian and the Gradient, and we will find adequate representations of concepts like normal vectors and curvature. Perhaps surprisingly we will see that there are many choices of discretization, each with their own benefits and downsides, prompting us to choose appropriately for the particular application.


| Lecture Date | Tentative Topic | |:----------------------|:--| | Wednesday, 16/09/2020 | Geometry Processing Pipeline, "life of a shape", shapes, surface representations,
Polygon Mesh Processing [Botsch et al. 2008]
Intro Survey, due 12:00 noon ET 22/09/2020
HW 01 assigned, due 12:00 noon ET 22/09/2020 | Wednesday, 23/09/2020 | Representations, Topology vs Geometry tangents and normals, geometry vs. topology, orientability, discrete topology, graphs
HW 02 assigned, due 12:00 noon ET 02/10/2020 | Wednesday, 30/09/2020 | Acquisition & reconstruction, characteristic function, spatial gradient
HW 03 assigned, due 12:00 noon ET 10/10/2020 | Wednesday, 07/10/2020 | Alignment & registration Hausdorff distance, integrated closest point distance
"Object modelling by registration of multiple range images" [Chen & Medioni 1991]
"A method for registration of 3-D shapes" [Besl & McKay 1992]
"Efficient Variants of the ICP Algorithm" [Rusinkiewicz & Levoy 2001]
"Sparse Iterative Closest Point" [Bouaziz et al. 2013]
point-to-plane distance, iterative closest point, orthogonal Procrustes problem
HW 04 assigned, due 12:00 noon ET 19/10/2020 | Wednesday, 14/10/2020 | Surface fairing & denoising, 1D smoothing flow, 1D energy-based smoothing, PDE, Implicit Time integration
HW 05 assigned, tentatively due 12:00 noon ET 27/10/2020 | Wednesday, 21/10/2020 | Shape deformation, continuous deformation, pointwise mappings, energy-based model, handle-based deformation, local distortion mesure, gradient-based distortion, Laplacian-based distortion, as-rigid-as-possible
HW 06 assigned, due 12:00 noon ET 03/11/2020 | Wednesday, 28/10/2020 | Surface parameterization, texture mapping, mass-spring methods, graph drawing, harmonic maps, least squares conformal mapping, local parameterization, discrete exponential map, stroke parameterization
Final implementation paper signup assigned, due 12:00 noon ET 05/11/2020 | Wednesday, 04/11/2020 | Curvature & surface analysis Planar curves, tangents, arc-length parameterization, osculating circle, curvature, turning number theorem, discrete curvature, normal curvature, minimum/maximum/mean curvature Principal curvature, Gauss map, Euler's theorem, shape operator, Gaussian curvature
HW 07 assigned, due 12:00 noon ET 18/11/2020 | Wednesday, 11/11/2020 | No Lecture: FAS Reading week. | Wednesday, 18/11/2020 | Signed distances, Offset surfaces, inside-outside segmentation, medial axis, isosurface/level sets, signed distances to polyhedra, parity counting, winding number
"Signed Distance Computation using the Angle Weighted Pseudo-normal" [Baerentzen & Aanaes 2005]
"3D distance fields: a survey of techniques and applications" [Jones et al. 2006]
"Robust Inside-Outside Segmentation using Generalized Winding Numbers" [Jacobson et al. 2013] | Wednesday, 25/11/2020 | Guest lecture by Hsueh-Ti Derek Liu | Wednesday, 02/12/2020 | Signed distances continued. Outlook into geometry processing research. | Wednesday, 09/12/2020 | Video party
libigl-style paper implementation due 12:00 noon ET 09/12/2020

Cutting room floor: Quad meshing, Subdivision surfaces, Constructive solid geometry, Voxelization, Mesh decimation, simplification, remeshing


Lateness Policy

0.007% off for every minute late.

Academic Integrity

Any code must belong to the student submitting it. Submitted assignments may be automatically analyzed to identify suspicious levels of code similarity. Consequences of committing an academic offence can be severe.

By staying enrolled in this course, students acknowledge that they have read and understand the University of Toronto's definitions and policy on Academic Integrity.

Supplemental Textbook

Polygon Mesh Processing. Mario Botsch, Leif Kobbelt, Mark Pauly, Pierre Alliez, and Bruno Levy, 2008.


Final Paper Implementation

Sign up by 12 noon 05/11/2020

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