CS 15-458/858: Discrete Differential Geometry
CARNEGIE MELLON UNIVERSITY | SPRING 2022 | TUE/THU 11:50-1:10 | GHC 4215
In this lecture we continue our discussion of conformal maps, and see how the different characterizations we saw in the smooth setting lead to different algorithms in the computational setting.
The video covering both this lecture and the previous one (on smooth conformal maps) can be found here.
A basic task in geometric algorithms is finding mappings between surfaces, which can be used to transfer data from one place to another. A particularly nice class of mappings (both from a mathematical and computational perspective) are conformal maps, which preserve angles between vectors, and are generally very well-behaved. In this first lecture we’ll take a look at smooth characterizations of conformal maps, which will ultimately inspire the way we talk about conformal maps in the discrete/computational setting.
The video covering both today and Thursday’s lecture (on discrete aspects of conformal maps) can be found here.
A basic task in geometric algorithms is finding mappings between surfaces, which can be used to transfer data from one place to another. A particularly nice class of algorithms (both from a mathematical and computational perspective) are conformal mappings, which preserve angles between vectors, and are generally very well-behaved. In this lecture we’ll take a look at smooth characterizations of conformal maps, which will ultimately inspire the way we talk about conformal maps in the discrete/computational setting.
The video covering both this and the next lecture (on discrete aspects of conformal maps) can be found here.
In the last lecture we introduced the Laplace operator from the smooth point of of view; in this lecture we talk about how to discretize it, and show that from computational point of view it really is the —Swiss army knife— of geometry processing algorithms—essentially playing the role of the discrete Fourier transform from classical signal processing.
You can find a video lecture for these slides, from a talk given by Etienne Vouga, here. (We’ll have our own lecture in-class! This one is just for reference/for anyone who is sick, etc.)
In this lecture we take a close look at the Laplacian, and its generalization to curved spaces via the Laplace-Beltrami operator. The Laplacian is one of the most fundamental objects in geometry and physics, and which plays a major role in algorithms. Here we consider several perspectives to build up some basic intuition about what the Laplacian is, and what it means.
In this lecture we wrap up our discussion of discrete curvature, and see how it all fits together into a single unified picture that connects the integral viewpoint, the variational viewpoint, and the Steiner formula. Along the way we’ll touch upon several of the major players in discrete differential geometry, including a discrete version of Gauss-Bonnet, Schläfli’s polyhedral formula, and the cotan Laplace operator—which will be the focus of our next set of lectures.
Just as curvature provides powerful ways to describe and analyze smooth surfaces, discrete curvatures provide a powerful way to encode and manipulate digital geometry—and is a fundamental component of many modern algorithms for surface processing. This first of two lectures on discrete curvature from the integral viewpoint, i.e., integrating smooth expressions for discrete curvatures in order to obtain curvature formulae suitable for discrete surfaces. In the next lecture, we will see a complementary variational viewpoint, where discrete curvatures arise by instead taking derivatives of discrete geometry. Amazingly enough, these two perspectives will fit together naturally into a unified picture that connects essentially all of the standard discrete curvatures for triangle meshes.
Much of the geometry we encounter in everyday life (such as curves and surfaces sitting in space) is well-described by it curvatures. For instance, the fundamental theorem for plane curves says that an arc-length parameterized plane curve is determined by its curvature function, up to rigid motions. Similar statements can be made about surfaces and their curvatures, which we explore in this lecture.
We’ll follow up our lecture on smooth surfaces with a view of surfaces from the discrete point of view. Our goal will be to translate basic concepts (such as the differential, immersions, etc.) into a purely discrete language. Here we’ll also start to see the benefit of developing discrete differential forms: many of the statements we made about surfaces in the smooth setting can be translated into the discrete setting with minimal effort. As we move forward with discrete differential geometry, this “easy translation” will enable us to take advantage of deep insights from differential geometry to develop practical computational algorithms.
In this lecture we continue our discussion of smooth surfaces, introducing some key concepts like the Gauss map and the area vector. We’ll also sketch out how to finally talk about differential forms on curved surfaces, rather than in flat \(\mathbb{R}^n\).