Below you will find a few selected final projects from our course. Students were asked to pick a topic from differential geometry and explore the different ways in which it can be explained and understood form a discrete, computational perspective. Some students devised new algorithms, while others proved new results in DDG. Overall, it’s been a very fun and stimulating semester! Thanks to everyone who participated.

*[Students: if you do not see your project below, but still would like it posted here, please send us an email!]*

**Mean and Principal Curvature Estimation from Noisy Point Cloud Data of Manifolds in \(\mathbb{R}^n\)**

*Yousuf Soliman*

**[Writeup] [Presentation] [Final Project] [Code]**

**Description:** *Data is often provided as a finite set of points embedded in Euclidean space. In many applications, we have reason to believe that these data points are generated from a distribution with support over a manifold, rather than from a distribution with support over all of \(\mathbb{R}^n\). In this paper, I will consider the problem of recovering the local curvature of an underlying manifold based on noisy point samples. I will then present an extension of integral invariants to submanifolds with arbitrary codimension as a methodology for inferring the curvature of point cloud data at different scales. Curvature is a widely used invariant feature in pattern classification and computer vision algorithms. Furthermore, understanding the curvature of the data gives one a better understanding of the local manifold geometry, which can then be used to better construct a sufficiently fine triangulation of the underlying manifold.*

**On the Conformal Maps of Triangle Linkages**

*(Anonymous)*

**[Writeup/Final Project] [Presentation]**

**Description:** *This project studies the nature of conformal maps, particularly in connection with discrete differential geometry. The discrete model we focus on is the triangular linkage geometry introduced by Konakovic et al. Abstractly, these linkages are equilateral triangles such that pairs of triangles meet at vertices and the triangles are connected in cycles of length six. In practice, such surfaces can be manufactured from flat “auxetic” (opening) materials with slits cut in them, providing many more degrees a freedom than ordinary developable (no-cut) surfaces. We present an overview of discretization of conformal geometry, both in the traditional Lagrangian element model as well as in the Crouzeix-Raviart element model. We describe how this theory connects to the geometry of triangular linkages, laying a foundation of discrete differential geometry for these structures. Furthermore, we propose a working definition of discrete conformal maps on triangular linkages, and prove some implications.*

**Heat Kernel Signature**

*Ye Han*

**[Writeup] [Presentation] [Final Project] [Code]**

**Description:** *I implement the computation of HKS by following the pipeline described by Sun et al, and apply it on several geometric models. The details involve computing the HKS of different feature vertices at different time step, comparing HKS of the same geometric models under isometric transformation, and comparing the HKS of different geometric models at the same scale. Examples showcase the multi scale property of HKS, where the chosen points are similar under local scale while the difference only appears at large time step.*

**Introduction to Spin Transformation and its Application on Shape Descriptor**

*Derek Liu*

**[Writeup] [Presentation] [Final Project]**

**Description:** *The topic of mapping shapes to different feature spaces is called shape descriptor design. The most common approach utilizes surface properties, such as curvature and geodesic distance, to construct shape descriptors. Another popular approach uses shape operators, such as Laplace-Beltrami operator to describe shape as a linear combination of basis functions. However, most descriptors were built heuristically and their performance is strongly task dependent. Which properties best represent a 3D shape? Bonnet proposed that mean curvature and metric should suffice to determine the surface generically. These two geometric objects are building blocks of conformal transformation and spin transformation in differential geometry. This article therefore aims to introduce spin transformations and their application as shape descriptors.*