## Supplemental Videos: Geodesic Distance and Beyond

In completing your assignment for finals next week, we thought you might find a couple videos helpful (though totally optional). The first was already linked to in the assignment writeup, which gives a reasonably short (18 minutes) motivation for and description of the algorithm you’ll be implementing. The second is a longer (50 minute) talk that covers the algorithm in more detail, and describes some fun extensions and applications of the same basic technique:

Enjoy!

## Lecture 20 — Geodesics

Our final lecture for the term focuses on geodesics, which generalize the notion of “straight line” to curved spaces; this material also connects to your final assignment, on computing geodesic distance. Once again we’ll play “The Game” of discrete differential geometry, and see how two natural characterizations from the smooth setting (straightest and locally shortest) lead to two distinct, and fascinating definitions in the discrete setting. Along the way we’ll also encounter many rich topics from (discrete) differential geometry including the cut locus, the medial axis, the exponential/log map, the covariant derivative, and the Lie bracket! We’ll also see how all this stuff connects with practical algorithms for things like surface reconstruction from points, and give two different algorithms for tracing out geodesics on curved surfaces.

Note that the lecture video comprises about two lectures (about two hours total). I’d recommend watching it in two logical chunks:

• Part I: Shortest Geodesics – 0:00—1:04
• Part II: Straightest Geodesics – 1:04–1:55

## Lecture 18 (revised): 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 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 14: 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 13: Smooth Surfaces II

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 $\mathbb{R}^n$. (Click below for the video!)