Your first reading assignment will be to read an overview article on Discrete Differential Geometry. Since we know we have a diverse mix of participants in the class, you have several options (pick one):

**(pages 1–3)**Crane & Wardetzky, “A Glimpse into Discrete Differential Geometry”.

This article discusses the “no free lunch” story about curvature we looked at in class, plus a broader overview of the field.**(pages 1–5)**Pottman et al, “Architectural Geometry”.

This article discusses the beautiful tale of how discrete differential geometry is linked to modern approaches to computational design for architecture, as well as fabrication and “rationalization” of free-form designs.**(pages 5–9)**Bobenko & Suris, “Discrete Differential Geometry: Consistency As Integrability”.

This article provides another overview of discrete differential geometry, with an emphasis on*nets*and their connection to the notion of*integrability*in geometry and physics.

Though written for a broad audience, be warned that all of these articles are somewhat advanced—the goal here is not to understand every little detail, but rather just get a high-level sense of what DDG is all about.

**Assignment: **Pick one of the readings above, and write 2–3 sentences summarizing what you read, plus at least one question about something you didn’t understand, or some thought/idea that occurred to you while reading the article. For this first assignment, we are also **very** interested to know a little bit about **YOU!** E.g., why are you taking this course? What’s your background? What do you hope to get out of this course? What are your biggest fears about the course? Etc.

**Handin instructions can be found in the “Readings” section of the Assignments page. **Note that you must send your summary in no later than **10am Eastern** on the day of the next lecture (**September 7, 2017**).

Enjoy!

Crane & Wardetzky, “A Glimpse into Discrete Differential Geometry”, introduces the central idea of the “game” played by discrete differential geometers using examples across various applications. The goal of the game is to find one of the inequivalent discrete analogues of the equivalent characterization properties that suits best for the problems to be solved. The discretization scheme hence is problem-driven and cannot merely evaluated by (rates of) convergence.

Questions:

1. What distinguishes discrete differential geometry from a combination of (classical) differential geometry, algebraic topology (simplicial homology and cohomology), and geometric measure theory, other than explicit implementation considerations? In other words, what else can an expert in all three fields above gain by studying discrete differential geometry?

I would simply say: there is new mathematical knowledge being generated in the field of DDG that is not previously known from these more traditional areas (differential geometry, algebraic topology, and geometric measure theory). Some examples are mentioned in the “Glimpse” article—for instance, the connection of holomorphic functions to circle packings does not resemble anything in these more traditional fields (as one example).

With “A Glimpse into Discrete Differential Geometry”, Crane and Wardetzky present the goals and characteristics of this emerging field bridging analytical description of geometries and computation. Discrete Differential Geometry seeks discrete counterparts of smooth geometrical objects that preserve their geometrical invariances. Different discrete definitions lead to different discrete counterparts, so usually none of these embodies all the original properties, rather they preserve a subset of properties that turn out useful for specific purposes. Because convergence is not a good criterion to test which definition would be best, the answer to the question “which discretization is best?” depends on which goal we are pursuing in applying the discrete definition.

Question out of curiosity: given a specific problem, like the problem of finding curvature, are the possible discrete definitions finite or infinite? Can we come up with infinite definitions each of which preserves some useful analytic property?

About me: I’m a graduate student in Computational Design and a geometry nerd. My background is in Architecture and Statistics. With this class I hope to gain a deeper knowledge of discrete techniques to process 3D objects, and hope to apply it to my field of study.

I am reading Bobenko et al, “Discrete Differential Geometry: Consistency As Integrability”. I cannot find a definition of “integrability”, or “integrable systems” in the given pages. I searched the term online and got some definitions that require some differential topology. I can dig into whose definitions, but I wonder if there is an simpler way to describe it. Or maybe there is not a general satisfying definition, as the paper is kind of suggesting.

Architectural Geometry

Pottman et al present Architectural Geometry as an area of research that emerged from practical demands of contemporary architectural design – the recent integration of freeform modeling and parametric tools into the CAD workflow to design buildings with complex geometry. This area of research relies heavily on the combination of discrete differential geometry (DDG) and numerical optimization to rationalize freeform surfaces in relation to constraints of varied nature – such as design intention, statics, fabrication and energy performance. In the initial section, the authors review the properties, generation and rationalization of polyhedral surfaces, describing the advancements in the literature for triangle meshes, nearly rectangular meshes and hex-dominant meshes.

As an architect, I am very interested in the recent advancements in mesh rationalization and in the strategies that researchers adopt to solve it. Besides, I am always curious about how people approach problems, so I classified the strategies cited in the paper in two categories:

(1) A geometric algorithm that rationalizes a mesh based on its properties.

(2) The numerical optimization of a mesh following the existing constraints.

The approach 1 seems to be related to the definition of DDG presented in class. In contrast, the approach 2 corresponds to the consolidated field of numerical optimization. In the paper, the success of the approach 2 is associated with a good initialization that considers the knowledge of the geometrical properties. It reinforces the idea that the intersection between optimization and DDG is essential for Architectural geometry (and that my categorization might be fragile). However, how is this categorization and its intersection is understood from the field of DDG itself?

Crane & Wardetzky, “A Glimpse into Discrete Differential Geometry” discusses a variety of methods to define curvature in discrete geometry. One of the key issues in discrete geometry is the lack of a normal on vertices of discrete curves. After introducing methods of defining curvature, each method seems to be lacking properties that best describe curvature. This explains the “no free lunch idea”, that not one definition contains all properties and instead we have to pick and choose which properties are best suited to the task at hand.

I am an undergraduate mathematics major with a minor in computer science. I am taking this class to get an introductory idea of the whole field of processing 3D objects. The title of the class sounded neat as well.

Crane & Wardetzky, “A Glimpse into Discrete Differential Geometry” covers more mathematic details than in the introduction class. It first point out that the lacking of normal on vertices of discrete curve is the key issue in DDG. As discussed in class, there exists many definition but no one is perfect for all properties. So we shouldn’t use only convergence as the criteria.

My question would be, how is the discrete differential geometry applied in computer vision area, since I didn’t see many examples in the lecture.