Course Description

Instructor: Keenan Crane (
Office hours: TBD, Smith 217

Daniel Li (
Office hours: Tuesday: 5-7, Friday:1-3, Smith Hall Graphics Lounge RIP
Friday OH Zoom Info:
Meeting ID: 858 742 537
Tuesday OH Zoom Info:
Meeting ID: 531 328 996

Alex Havrilla (
Office hours: Sunday: 3-5, Thursday: 5-7. Online through zoom. Zoom ID: 414-331-1363

Prerequisites: linear algebra, vector calculus, some programming experience. At CMU, this means: (15-110 or 15-112 or 15-122 or 02-201), and ((21-240 or 21-241 or 21-242 or 21-341), and (21-256 or 21-259 or 21-268 or 21-269)) or 21-254.

This course focuses on three-dimensional geometry processing, while simultaneously providing a first course in traditional differential geometry. Our main goal is to show how fundamental geometric concepts (like curvature) can be understood from complementary computational and mathematical points of view. This dual perspective enriches understanding on both sides, and leads to the development of practical algorithms for working with real-world geometric data. Along the way we will revisit important ideas from calculus and linear algebra, putting a strong emphasis on intuitive, visual understanding that complements the more traditional formal, algebraic treatment. The course provides essential mathematical background as well as a large array of real-world examples and applications. It also provides a short survey of recent developments in digital geometry processing and discrete differential geometry. Topics include: curves and surfaces, curvature, connections and parallel transport, exterior algebra, exterior calculus, Stokes’ theorem, simplicial homology, de Rham cohomology, Helmholtz-Hodge decomposition, conformal mapping, finite element methods, and numerical linear algebra. Applications include: approximation of curvature, curve and surface smoothing, surface parameterization, vector field design, and computation of geodesic distance.

Course material has been used for semester-long courses at CMU (2016, 2017, 2019), Caltech (2011, Harvey Mudd (2020), 2012, 2013, 2014, 2016, 2017), Columbia University (2013), and RWTH Aachen University (2014, 2015, 2016, 2017), as well as special sessions at SIGGRAPH (2013) and SGP (2012, 2013, 2014, 2017).