Month: April 2021

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.

• Video
• Slides

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.

• Video
• Slides

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 \(\mathbb{R}^n\).

• Video
• Slides