Category: Slides

Lecture 10: Smooth Curves

After spending a great deal of time understanding some basic algebraic and analytic tools (exterior algebra and exterior calculus), we’ll finally start talking about geometry in earnest, starting with smooth plane and space curves. Even low-dimensional geometry like curves reveal a lot of the phenomena that arise when studying curved manifolds in general. Our main result for this lecture is the fundamental theorem of space curves, which reveals that (loosely speaking) a curve is entirely determined by its curvatures. Descriptions of geometry in terms of “auxiliary” quantities such as curvature play an important role in computation, since different algorithms may be easier or harder to formulate depending on the quantities or variables used to represent the geometry. Next lecture, for instance, we’ll see some examples of algorithms for curvature flow, which naturally play well with representations based on curvature!

Lecture 11: Discrete Curves

This lecture presents the discrete counterpart of the previous lecture on smooth curves. Here we arrive at a discrete version of the fundamental theorem for plane curves: a discrete curve is completely determined by its discrete parameterization (a.k.a. edge lengths) and its discrete curvature (a.k.a. exterior angles). We’ll also cook up a discrete version of the fundamental theorem for space curves, and give a bunch of neat examples of discrete curves in geometry processing and physical simulation.

Lecture 12: Smooth Surfaces I

This lecture takes a first look at smooth surfaces. There’s of course way more to know about surfaces than we can pack into a single lecture (and we’ll see plenty more later on), but this lecture will cover two very important perspectives: the extrinsic description of a surface via a local parameterization, which tells us where points sit in space, and the intrinsic description of a surface via coordinate charts, which lets us work with a surface without worrying how it’s embedded in space.

Lecture 14: Discrete Surfaces

Discrete Surfaces

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.

Lecture 15: Curvature of Surfaces

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.

Lecture 16—Discrete Curvature I (Integral Viewpoint)

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.

Lecture 17—Discrete Curvature II (Variational Viewpoint)

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.

Lecture 18: The Laplace-Beltrami Operator

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.

Lecture 19—The Discrete Laplacian

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.)