Slides — Conformal Geometry

Conformal geometry is, in a sense, the study of geometry when you can measure angles, but not lengths. Though this viewpoint may seem a bit abstract, it plays an surprisingly interesting and important role in both smooth and discrete differential geometry. For one thing, it provides a setting for working with surfaces that is both very simple and very regular—recall for instance that we typically like to work with regular curves, because we can be confident that subsequent quantities will be well-defined (tangents, normals, curvatures, etc.); if we assume curves have an arc length parameterization, then life becomes particularly simple because we don’t have to worry about accounting for “stretching” as we go from the domain to the image of the curve. Likewise, with surfaces, we tend to work with immersions, which provide a useful notion of regularity. What’s the analogue of an arc-length parameterization for surfaces? In general it’s impossible to find a parameterization that has no stretching whatsoever, but we can always find one that at least preserves angles—a so-called conformal parameterization. Akin to arc-length parameterized curves, the amount of extra information about “stretching” that we need to carry around is now minimal. Moreover, the condition of angle preservation automatically gives us even more regularity than even a plain immersion: it automatically guarantees that our map is smooth. Note that this whole story applies equally well in both the smooth and discrete case: just as we had notions of discrete regularity for polygonal curves and discrete immersions for simplicial surfaces, we will also have a notion of discrete conformal equivalence for triangle meshes. Beyond these analytical properties, discrete conformal geometry leads to a huge number of fast, useful, and beautiful algorithms, which we will study and implement in the next few weeks.

Slides — Discrete Surfaces

Below are our slides about discrete surfaces. Here our hard work starts to pay off: since we’ve already discretized exterior calculus, and have described smooth surfaces in terms of (smooth) exterior calculus, the transition to the discrete setting takes very little additional work.

Slides — Curves

After our long journey to understand exterior calculus (and its discrete counterpart), we will start putting these tools to work to manipulate real curves and surfaces. This lecture studies smooth and discrete curves, which illustrate many of the important features of geometry embedded in $\mathbb{R}^n$.

Slides—Exterior Calculus in $R^n$

Later this week we’ll start talking about exterior calculus, which is a modern language used across differential geometry, mathematical physics, geometric computation… and the rest of our class! :-). Initially this language can look a bit daunting, but by making some connections with familiar ideas from vector calculus (like grad, div, and curl), we’ll see that it’s actually not so bad once you get down to concrete calculations. Slides here:

Slides—Differential Forms in $R^n$

Following our lecture on exterior algebra, we will start building up differential forms, which is the next step on our journey toward doing computation on meshes with discrete exterior calculus. This material may be helpful for those of you working through the second part of the written homework: