21-360 Differential Geometry of Curves and Surfaces Intermittent: 9 units The course is a rigorous introduction to the differential and integral calculus of curves and surfaces. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. Note that an electronic version of this book is available for free for CMU students through the library webpage. DDG goes hand-in-hand with geometry processing in helping us computationally visualize complex sytems from our 3D world. If nothing happens, download GitHub Desktop and try again. The first was already linked to in the assignment writeup, which gives a reasonably short (18 minutes) motivation for and description of the algorithm youll be implementing. Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. For simplicity, you may assume that the mesh has no boundary. Please submit your source files to the course Gradescope by 5:59pm ET. Warning: You cannot use late days on this assignment since its the last one. The Geosynthetic Reinforced Soil-Integrated Bridge System (GRS-IBS) is a recent bridge approach construction system that connects the bridge deck and roadways without joints and bearings, thus capable to eliminate the bump problems. Reading 3: Exterior Algebra and k-Forms (due 3/2), Recitation: halfedges, sparse linear systems, intro to coding assignments, Reading 2: Combinatorial Surfaces (Due 2/19), Assignment 0 (Coding): Combinatorial Surfaces due 2/26, Assignment 0 (Written): Combinatorial Surfaces due 2/26, A Glimpse into Discrete Differential Geometry, Discrete Differential Geometry: Consistency As Integrability, Welcome to Discrete Differential Geometry! R - = (1 , . Since these notes just barely scratch the surface (literally), I am often asked for recommendations on books that provide a deeper discussion of surfaces. Or, if youd like to get some extra creditand have already completed the rest of the assignmentsyou can complete this assignment for up to one additional assignments worth of extra credit points. We begin by emphasising our perspective on differential geometry for this paper, and illustrate the technique of proof for the differential geometric aspects of the results. In recent years it has unearthed a rich variety of new perspectives on applied problems in computational . It takes a look at the curvature of smooth and discrete surfaces, which we have been talking about in lecture. So far, weve said that a differential \(k\)-form produces a scalar measurement. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. I basically set about doing the homework from Keenan Crane's discrete differential geometry course (s). We instead started with differential forms in \(\mathbb{R}^n\), and will later talk about how to work with them on curves and surfaces. Handin instructions can be found on the assignment page. The most interesting subject, perhaps, is the connections to hyperbolic geometry in Part IV, which you can read for your own enjoyment! The written portion of Assignment 2 can be found below. This reading is due next Wednesday, March 31. Even low-dimensional geometry like curves reveal a lot of the phenomena that arise when studying curved manifolds in general. Next lecture, for instance, well see some examples of algorithms for curvature flow, which naturally play well with representations based on curvature! In this lecture we continue our discussion of conformal maps, and see how the different characterizations we saw in the smooth setting lead to different algorithms in the computational setting. Note that sums are taken only over elements (edges or faces) containing vertex $i$. Your short 2-3 sentence summary is due by 10am Eastern on February 19, 2020. ):This course is for students interested in working with 3D data. The Prerequisites for 15-458 Discrete Differential Geometry are 15-112, 21-259, and 21-241. The reading is Chapter 2, pages 720 of our course notes, which can always be accessed from the link above. 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. You may want to hide this fact and pretend that your stuff is continuous, but at some point you will be computing derivatives by evaluating a function on nearby points. DDG course from CMU Resources. Warning: You cannot use late days on this assignment! The assignment is due on April 20, 2022 at 5:59:59pm Eastern (not at midnight!). Just like rulers measure length, and cups measure volume, k-forms will be used to take measurements of the little k-dimensional volumes or k-vectors that we built up using exterior algebra in our previous lecture. They can be converted to pointwise quantities (i.e., discrete 0-forms at vertices) by dividing them by the circumcentric dual area of the vertex (i.e., by applying the discrete Hodge star). Would like to hear the opinion of people who took the course recently . Assignment: Pick one of the readings above, andwrite 23 sentences summarizing what you read, plus at least one question about something you didnt understand, or some thought/idea that occurred to you while reading the article. Your main references for this assignment will be: Written exercises for this assignment are found below. For the coding portion of your first assignment, you will implement some operations on simplicial complexes which were discussed in class and in Chapter 2 of the course notes. Note that the SparseMatrix class has an invertDiagonal function that you can use to invert diagonal matrices. I took 15-112 this semester and I am transferring in credit for 21-259 and 21-241 from a local community college where I am taking the course-equivalents. Our first lecture on exterior calculus covered differentiation; our second lecture completes the picture by discussing integration of differential forms. Along the way well 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! The honest answer is, I dont know; I mostly didnt learn it from a book. But there are a couple fairly standard references (other) people seem to like, both of which should be available digitally from the CMU library: For the coding portion of this assignment, you will implement various expressions for discrete curvatures and surfaces normals that you will derive in the written assignment. Likewise, in QT one has the Poincare symmetry Lie algebra (PA) of space 1. In this paper, we report a discrete differential geometry (DDG)-based numerical method to simulate the geometrically nonlinear deformations of extensible ribbons. If you encounter any problems while trying to use the website, please contact the TA (listed under the course description). Welcome to the course! If nothing happens, download Xcode and try again. I wanted to gain a deeper understanding of the part about logarithm rotation mapping in . Even though were working with spheres, it is still helpful to use the formulation of Trivial connections given in section 8.4.1 of the notes. EDIT: You can compute the ratio of dual edge lengths to primal edge lengths using the cotan formula, which can be found on Slide 28 of the Discrete Exterior Calculus lecture, or in exercise 36 of the notes (you dont have to do the exercise for this homework). Course Info Instructor Prof. Paul Seidel Departments Mathematics Topics Mathematics Algebra and Number Theory Differential Equations The notes will also give you all the background youll need to complete this assignment. ), which in the discrete setting are encoded by simple matrices that translate problems involving differential forms into ordinary linear algebra problems. Your next reading will take a deep dive into conformal geometry and the many ways to discretize and compute conformal maps. These descriptions and data structures will provide the foundation for all the geometry and algorithms well build up in this class. For instance, positions in \(\mathbb{R}^n\), tangents, and normals are all vector-valued quantities. The notes provide essential mathematical background as well as a large array of real-world examples, with an emphasis on applications and implementation. Note that you need only be concerned with the case of triangle meshes (not polygon meshes or point clouds); pay close attention to the paragraph labeled Choice of Timestep.. Learn more. where $\phi_i^{jk}$ is the interior angle between edges $e_{ij}$ and $e_{ik}$, and $A_{ijk}$ is the area of face $ijk$. 0 forks Releases No releases published. In completing your assignment for finals next week, we thought you might find a couple videos helpful (though totally optional). For now, dont worry too much about the detailsthe goal here is to just get a sense of what the course is all about! Theres of course way more to know about surfaces than we can pack into a single lecture (and well see plenty more later on), but this lecture will cover two very important perspectives: the extrinsic description of a surface via a local parameterization, which tells us where points sit in space, and the intrinsic description of a surface via coordinate charts, which lets us work with a surface without worrying how its embedded in space. ferential geometry poses some nontrivial problems. Our main result for this lecture is the fundamental theorem of space curves, which reveals that (loosely speaking) a curve is entirely determined by its curvatures. Please implement the following routines in: In addition, please implement either Hodge Decomposition, or Trivial Connections (on surfaces of genus 0): Hodge Decomposition: Please implement the following routines in. Note that the lecture video comprises about two lectures (about two hours total). D ISCRETE D IFFERENTIAL G EOMETRY: A N A PPLIED I Stokes theorem also plays a key role in numerical discretization of geometric problems, appearing for instance in finite volume methods and boundary element methods; for us it will be the essential tool for developing a discrete version of differential forms that we can actually compute with. Following the notes, we observe that the constraint $d\delta\beta = u$ determines $\beta$ up to a constant, and that constant is annihilated by $\delta$. This short-but-important supplemental lecture introduces some language well need for describing geometry (curves, surfaces, etc.) Bio: Arjun studies the security of machine learning systems, with a focus on adversarial and distributed learning.His work has exposed new vulnerabilities in learning algorithms, along with the development of a theoretical framework to analyze them. In particular, it will explore how to differentiate and integrate \(k\)-forms, and how an important relationship between differentiation and integration (Stokes theorem) enables us to turn derivatives into discrete operations on meshes. Exterior calculus generalizes these ideas to \(n\)-dimensional quantities that arise throughout geometry and physics. A basic task in geometric algorithms is finding mappings between surfaces, which can be used to transfer data from one place to another. Discrete Exterior Calculus A concise description of the implemented operators follows. He was a finalist for the 2020 Bede Liu Best Dissertation Award, and won the 2019 Yan Huo *94 Graduate Fellowship and 2018 SEAS Award for . In particular, you should read Chapter 3 of the course notes, pages 2844. The notes will also give you all the background youll need to complete this assignment. Well introduce the basic ideas in todays lecture: Today we continue our journey toward building up (discrete) exterior calculus by talking about how to measure little k-dimensional volumes. For a discrete surface M = (V;E;F) with metric ': E !R >0, the discrete Dirichletenergyofamapf : V !C canbedenedas (3.4) Eb D(f) := X ij2E w ijjf j f ij 2 foranycollectionofedgeweightsw ij whichmakethisenergyapositive-semidenite quadraticform(i.e., Eb D(f) 0 forallpiecewiselinearmapsf). The first paper, by Wardetzky et al, considers a No Free Lunch theorem for discrete Laplacians that continues our story of The Game played in discrete differential geometry. This stuff resides in various places at Cal tech and Carnegie Mellon. It is about differential geometry anddynamical systems, smooth and discrete theories, and on puremathematics and its practical applications. The reading is due Thursday, March 11 at 10am. Please implement the following routines in. For this assignment, you will need to implement the following routines in, The JavaScript assignment comes with a viewer, Selecting simplices will not work until you fill in the, The assignment also comes with a test script, The web framework is implemented in Javascript, which means no compilation or installation is necessary on any platform. The discrete scalar Gaussian curvature (also known as angle defect) and discrete scalar mean curvatureat vertex $i$ are given by Course speakers provided an introduction to the emerging field of discrete differential geometry, which studies discrete analogs of smooth geometric objects, and provides essential links between analytical descriptions of geometry and computational algorithms. This first of two lectures on discrete curvature from the integral viewpoint, i.e., integrating smooth expressions for discrete curvatures in order to obtain curvature formulae suitable for discrete surfaces. Ioannis Gkioulekas Named Sloan Research Fellow 12 February 2020 Topological Data StructuresSimplicial Soup Store only top-dimensional simplices Example. You signed in with another tab or window. Frequency Offered:Generally offered once per year (Spring or Fall)- confirm course offerings for upcoming semesters by accessing the university Schedule of Classes. leave a comment on this post containing your favorite mathematical formula (see below). This subject makes some beautiful and unexpected connections to other areas of mathematics (such as circle packings, and hyperbolic geometry), and is in some sense one of the biggest success stories of DDG, since there is now a complete uniformization theorem that mirrors the one on the smooth side. Our first lecture on exterior calculus studies the exterior derivative, which describes the rate of change of a differential form, and (together with the Hodge star) generalizes the gradient, divergence, and curl operators from standard vector calculus. Youll have the opportunity to implement one of these algorithms in the coding part of the assignment. Discrete Differential Geometry A. Bobenko, R. Kenyon, +1 author G. Ziegler Published 16 December 2008 Mathematics Discretization of Surfaces: Special Classes and Parametrizations.- Surfaces from Circles.- Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples.- Designing Cylinders with Constant Negative Curvature.- Heres the writeup for your last assignment, which plays the role of the final in the course. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. Some things I'm actively learning are modern multivariable calculus (Spivak. This time, we will work with meshes with boundary, so your Laplace matrix will have to handle boundaries properly (you just have to make sure your, The parameterization project directory also contains a basic implementation of the. Discrete Differential Geometry (DDG) is an emerging discipline at the boundary between mathematics and computer science. The next written assignment will give you some essential background on the smooth Laplace operator. Your short 2-3 sentence summary is due by 10am Eastern on April 19, 2022. Due date: 10am Eastern on Thursday, February 11, 2021. Discrete Mathematics In case we have not yet covered it in class, the barycentric dual area associated with a vertex $i$ is equal to one-third the area of all triangles $ijk$ touching $i$. \end{aligned} A particularly nice class of mappings (both from a mathematical and computational perspective) are conformal maps, which preserve angles between vectors, and are generally very well-behaved. Many thanks to our participants for making it a great event! Theres no PDF this week since the exercises are all from the notes. Submission: Please send an email to kmcrane@cs.cmu.edu and nsharp@cs.cmu.edu no later than . ): Our main goal is to show how fundamental geometric concepts (like curvature) can be understood from complementary computational and mathematical points of view. The assignment is due on February 26, 2020 at 5:59:59pm Eastern (not at midnight!). }\; d\delta\beta = u\]. Keenan's works is in Discrete Differential Geometry which builds fundamental representations and practical algorithms for processing and analyzing real-world geometric data by leveraging insights from differential geometry. If these exercises seem scary and unfamiliar, and you dont know where to start, thats ok! The written portion of assignment 1 is now available, which covers some of the fundamental tools well be using in our class. For this first assignment, we are alsovery interested to know a little bit about YOU! Youre done, and can just relax during finals week (at least in this class). Here we consider several perspectives to build up some basic intuition about what the Laplacian is, and what it means. Current progress in this field is to a large extent stimulated by its relevance for computer . So you can find the connection 1-form $\delta \beta$ with a single linear solve. [js|cpp]: 1. (Three 50 minute lectures, two 50 minute recitations) Prerequisite: 21-111. For reference, my S20 schedule is looking like 18-240, 16-311, 36-219, and 16-467. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth.The subject focuses on the combinatorial properties of these . N_i^A &:= \sum_{ijk \in F} A_{ijk}N_{ijk}\\ (Spring 2021), The Tree Cotree algorithm finds homology generators on the. Course Description This course is an introduction to differential geometry. Note: For the final assignment, you can do either this assignment or A5. It builds on a lot of the stuff weve already done in the class, especially discrete exterior calculus and the Laplacian. See this Piazza post for more details.) So you can find the connection 1-form $\delta \beta$ with a single linear solve. In this lecture we take a close look at the Laplacian, and its generalization to curved spaces via the Laplace-Beltrami operator. If youre typesetting this assignment, feel free to use this template. View Discrete Differential Geometry_ An Applied Introduction.pdf from MATH MISC at The Hong Kong University of Science and Technology. Please submit only the source code file geometry.js or geometry.cpp (depending on whether youre using JavaScript or C++). https://github.com/nmwsharp/DDGSpring2016, http://brickisland.net/DDGSpring2016/grading-policy/, http://brickisland.net/DDGSpring2016/course-syllabus/, http://www.cs.cmu.edu/~kmcrane/Projects/DGPDEC/paper.pdf, http://brickisland.net/DDGSpring2016/assignments/. In particular, well look at the idea of a mesh, and study one very specific kind of mesh called an oriented simplicial complex. Self-Study Coursework for Discrete Differential Geometry (CMU 15-458/858) in Spring 2021 - GitHub - andy1li/cmu-ddg: Self-Study Coursework for Discrete Differential Geometry (CMU 15-458/858) in Spr. It also shows up in an enormous number of physical and geometric equations, and for this reason there has been intense study of different ways to discretize the Laplacian (not only for simplicial meshes, but also point clouds and other discrete surface representations). Also, recall that our discrete Laplace matrix is the negative of the actual Laplacian. See the assignments page for handin instructions. TA for Calculus with Analytic Geometry. description given in class, a good reference is, Abraham, Marsden, Ratiu, Manifolds, Tensor Analysis, and Applications. This book is written by specialists working together on acommon research project. The next reading assignment will wrap up our discussion of exterior calculus, both smooth and discrete. \[\begin{aligned} () https://brickisland.net/DDGSpring2021 , p ) is an unknown parameter vector of interest - is parameter space for the experiment - P : B(Rn ) [0, 1] is a . Theres no PDF this week since the exercises are all from the notes. They also serve as an introduction to computational simulation of simple PDE's on a manifold. This lecture takes a first look at smooth surfaces. Calculus on Manifolds), introductory machine learning (Stanford CS229), and elements of differential geometry (ICTP Differential Geometry (Claudio Arezzo); Docarmo M.P. 3. Then follow the usual hand-in instructions. Also, you cannot use late days on this assignment since its the last one. Submit your responses to Gradescope by 5:59pm ET, May 21, 2021. In this lecture well really start from the beginning, and define some of the basic objects well use throughout the class. Along the way well touch upon several of the major players in discrete differential geometry, including a discrete version of Gauss-Bonnet, Schlflis polyhedral formula, and the cotan Laplace operatorwhich will be the focus of our next set of lectures. All of the details you need for implementation are described in Section 3 of the paper, up through and including Section 3.2. These lecture notes aim to provide a cursory introduction to the fields of geometry processing and As well discuss in class, this operator is basically the Swiss army knife of discrete differential geometry and digital geometry processing, opening the door to a huge number of interesting algorithms and applications. Python-discrete-differential-geometry has no vulnerabilities, it has a Permissive License and it has low support. The second paper, by Bobenko & Springborn considers the important perspective of intrinsic triangulations of polyhedral surfaces, and uses this perspective to develop a Laplace operator that is well-behaved even for very poor quality triangulations. \]. We describe how this theory connects to the geometry of triangular linkages, laying a foundation of discrete differential geometry for these structures. The video covering both today and Thursdays lecture (on discrete aspects of conformal maps) can be found here. Our goal will be to translate basic concepts (such as the differential, immersions, etc.) Youll have the opportunity to implement one of these algorithms in the coding part of the assignment. 21-120 Differential and Integral Calculus. Initially this assignment may look a bit intimidating, but the homework is not as long as it might seem: all the text in the big gray blocks contains supplementary, formal definitions that you do not need to know in order to complete the assignments. It is an individual project based on a code skeleton provided by CMU's Discrete Differential Geometry course. Examples include: physical/numerical simulation, computer vision, computer graphics, robotics, architecture/art/design, medical or anatomical data analysis. Applications can be found in the areas such as architectural geometry, integrable systems in mathematical physics, computer graphics and geometry processing. Well also sketch out how to finally talk about differential forms on curved surfaces, rather than in flat \(\mathbb{R}^n\). . Conformal flattening is important for (among other things) making the connection between processing of 3D surfaces, and existing fast algorithms for 2D image processing. [js|cpp] files to Gradescope. Are you sure you want to create this branch? Topics to be covered include: Parameterized and regular curves Frenet equations canonical coordinate system, local canonical forms, global properties of plane curves . Setting up the C++ skeleton is also fairly automatic (just a few, To use the JavaScript version, download or clone the files in the, To use the C++ version, follow the instructions in the. For this homework, well look at one algorithm for designing vector fields, and along the way well cover a lot of deep facts about surfaces. Rather than giving a formal definition in the smooth case, well introduce a notion of discrete manifolds that capture the most important ideas. These filesshould be submitted via the usual mechanism, as described on the assignments page. Our research group focuses on discrete models of classical differential geometry, in particular parametrized surfaces and curvature. These objects will ultimately let us integrate quantities over curved domains, which will also be our main tool for turning smooth equations from geometry and physics into discrete equations that we can actually solve on a computer. You will be subscribed to receive email notification about new posts, comments, etc. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature. [js|cpp] and discrete-exterior-calculus. It also provides a short survey of recent developments in digital geometry processing and discrete differential geometry. . Handin instructions can be found on the assignment page. The assignment is due on February 26, 2021 at 5:59:59pm Eastern (not at midnight!). Please submit your geometry. We try to provide a mathematical understanding of fundamental issues in Computer Science, and to use this understanding to produce better algorithms, protocols, and systems, as well as identify the inherent limitations of efficient computation. Descriptions of geometry in terms of auxiliary quantities such as curvature play an important role in computation, since different algorithms may be easier or harder to formulate depending on the quantities or variables used to represent the geometry. Once implemented, you will be able to apply these operators to a scalar function (as depicted above) by pressing the $\star$ and $d$ button in the viewer. In particular, you should solve the optimization problem given in Exercise 8.21. At this point you have all the fundamental knowledge you need to go out into the broader literature and start implementing all sorts of algorithms that are built on top of ideas from differential geometry. For the coding portion of your first assignment, you will implement the discrete exterior calculus (DEC) operators $\star_0, \star_1, \star_2, d_0$ and $d_1$. You can simply get started by opening the, If you do not have prior experience with Javascript, do not worry! [cpp|js], mean-curvature-flow. Last updated: May 1, 2019 This course focuses on three-dimensional geometry processing, while simultaneously providing a first course in traditional differential geometry. Note: For the final assignment, you can do either this assignment or A5. Furthermore, the sphere has no homology generators, so the problem simplifies to \[\min_{\delta \beta} \;\|\delta\beta\|^2\;\text{s.t. Refer to sections 3.2 of the paper for discretizations of Algorithm 1 (page 3). If anyone is seeking a more formal treatment of differential forms than the (admittedly informal!) Please submit your source files to Gradescope by 5:59pm ET on May 21, 2021. Your next homework will give you some hands-on practice with differential forms; just take this time to get familiar with the basic concepts. 6 minute read Two or three months ago, if you mentioned DDG when talking to me, I would say I will keep away from geometry as far as I can. Further hand-in instructions can be found on this page. So I registered for 15-458 recently and I was wondering, what is it like in terms of load etc? 15-458 Discrete Differential Geometry 15-458 - COURSE PROFILE Course Level:Undergraduate Frequency Offered: Generally offered once per year (Spring or Fall) - confirm course offerings for upcoming semesters by accessing the university Schedule of Classes. If you need any differential geometry, it is discrete differential geometry then. VisMath 2002. . Your next reading assignment will help you review the concepts weve been discussing in class: describing little volumes or \(k\)-vectors using the wedge product and the Hodge star, and measuring these volumes using dual volumes called \(k\)-forms. [cpp|js]and modified-mean-curvature-flow. Discrete Differential Geometry - Welcome Video - YouTube 0:00 / 6:56 PITTSBURGH Discrete Differential Geometry - Welcome Video 28,122 views Feb 3, 2021 Overview video for the CMU Course on. After spending a great deal of time understanding some basic algebraic and analytic tools (exterior algebra and exterior calculus), well finally start talking about geometry in earnest, starting with smooth plane and space curves. See: here and pdf discrete differential geometry book for starters. Overall, you just need to be sure you completed A0 and A1, as well as 3 of the assignments A2A6 (your choice which ones. . In this lecture well take a look at smooth characterizations of conformal maps, which will ultimately inspire the way we talk about conformal maps in the discrete/computational setting. The reading is due at 10 AM Eastern, May 3, 2021. The video covering both this lecture and the previous one (on smooth conformal maps) can be found here. Remember to turn in the whole coding assignment via a single ZIP file containing the modified source files. In your implementation of the implicit mean curvature flow algorithm, you can encode the surface $f:M \to \mathbb R^3$ as a single DenseMatrix of size $|V| \times 3$, and solve with the same (scalar) cotan-Laplace matrix used for the previous part. Please implement the following routines in: In addition, please implement either Hodge Decomposition, or Trivial Connections (on surfaces of genus 0): Hodge Decomposition: Please implement the following routines in. This is not typical computer science stuff, and you shouldnt necessarily know how to do it right off the bat. Your solution should implement zero neumann boundary conditions (which are the default behavior of the cotan Laplacian) but feel free to tryout other Dirichlet and Neumann boundary conditions on your own. N_i^\phi &:= \sum_{ijk \in F} \phi_i^{jk}N_{ijk}\\ So youll need to add in the mass matrix somewhere. into a purely discrete language. We want to help you succeed on this assignment, so that you can enjoy all the adventures yet to come. Since we know we have a diverse mix of participants in the class, you have several options (pick one): Though written for a broad audience, be warned that all of these articles are somewhat advancedthe goal here is not to understand every little detail, but rather just get a high-level sense of what DDG is all about. These basic operations will be the starting point for many of the algorithms we will implement throughout the rest of the class; the visualization (and implementation!) Skip to content Toggle navigation. 2. This short video will give you a brief overview of the course logisticsplease take a look! A basic task in geometric algorithms is finding mappings between surfaces, which can be used to transfer data from one place to another. (However, the final expressions are given below in case you want to do the coding first.) The reading is due on Thursday, April 15 at 10am Eastern time. Why? \end{aligned} Along the way we will revisit important ideas from calculus and linear algebra, putting a strong emphasis on intuitive, visual understanding that complements the more traditional formal, algebraic treatment. (Hint: how are the dot and cross product of two vectors related to the cosine and sine of the angle between them?). Along the way well 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! You are required to derive expressions for the principal curvatures $\kappa_1$ and $\kappa_2$ in exercise 4 of the written assignment. Research. Etc. I have heard that prof. Crane is a great instructor but I don't really know much more about the course beyond that. Id recommend watching it in two logical chunks: For the coding portion of your assignment on conformal parameterization, you will implement the Spectral Conformal Parameterization(SCP) algorithm as described in the course notes.Please implement the following routines in. Differential Geometry of Curves and Surfaces Project by Vishwajeet Bhoite Supervised by Dr. Tejas Kalelkar Fall-2017 IISER Pune Curves and Surfaces in the Plane and Three-Dimensional Space LECTURE PLAN for MATH 474 INTRO to DIFFERENTIAL GEOMETRY Fall 2013 In this first lecture well take a look at smooth characterizations of conformal maps, which will ultimately inspire the way we talk about conformal maps in the discrete/computational setting. In particular, you should solve the optimization problem given in Exercise 8.21. \]. The procedure for Hodge Decomposition is given as Algorithm 3 in section 8.1 of the Notes. (Spring 2021), CS 15-458/858: Discrete Differential Geometry, Assignment 6 [Coding]: Vector Field Decomposition and Design (Due 5/21), Assignment 6 [Written]: Vector Field Decomposition and Design (Due 5/21), Supplemental Videos: Geodesic Distance and Beyond, Assignment 5 [Coding]: Geodesic Distance (Due 5/21), Assignment 5 [Written]: Geodesic Distance (Due 5/21), Assignment 4 [Coding]: Conformal Parameterization (Due 5/6), Assignment 4 [Written]: Conformal Parameterization (Due 5/6), Reading 7Discrete Conformal Geometry (Due 5/3), the assignments page. (11/06/06) by Eitan Grinspun Lecture 3 slides: Geometry on Polyhedral Meshes by Konrad Polthier Notes (Hodge Decomposition): The procedure for Hodge Decomposition is given as Algorithm 3 in section 8.1 of the Notes. Hand-in instructions are as usual described on the assignments page. For computer spaces via the usual mechanism, as described on the smooth Laplace.. Are as usual described on the assignments page parametrized surfaces and curvature filesshould be submitted the. For 15-458 recently and I was wondering, what is it like in terms load! ( see below ) are all from the beginning, and 21-241 Soup only... Be found below send an email to kmcrane @ cs.cmu.edu no later than give you all the adventures to. One has the Poincare symmetry Lie algebra ( PA ) of space....: please send an email to kmcrane @ cs.cmu.edu and nsharp @ cs.cmu.edu no later than note: the. Yet to come the written portion of assignment 2 can be found in the coding of... Find a couple videos helpful ( though totally optional ) algebra ( PA ) of space 1 However, final. On puremathematics and its generalization to curved spaces via the Laplace-Beltrami operator last! Of science and Technology informal! ) ; just take this time to get familiar with the basic objects use! Instance, positions in \ ( \mathbb { R } ^n\ ) tangents. That capture the most important ideas curvatures $ \kappa_1 $ and $ \kappa_2 $ in Exercise.! Will take a deep dive into conformal geometry and algorithms well build up this. A concise description of the fundamental tools well be using in our class calculus generalizes these ideas to \ \mathbb. Computational simulation of simple PDE & # x27 ; s discrete differential geometry course ( s ) meet... On smooth conformal maps ) can be found in the class, a good reference is, you... Up through and including Section 3.2 you do not have prior experience with JavaScript, not! Taken only over elements ( edges or faces ) containing vertex $ I $ or! Discussing integration of differential forms we are alsovery interested to know a little bit about you know to... Will give you a brief overview of the geometric notions and methods of differential geometry.. If these exercises seem scary and unfamiliar, and 21-241 and geometry processing helping! Important ideas reference is, Abraham, Marsden, Ratiu, manifolds, Tensor Analysis, and applications sums... Coding first. actively learning are modern multivariable calculus ( Spivak 1-form $ \delta \beta $ a. To kmcrane @ cs.cmu.edu and nsharp @ cs.cmu.edu no later than geometric algorithms is finding between... 4 of the notes graphics and geometry processing and discrete theories, and on puremathematics and practical. At 5:59:59pm Eastern ( not at midnight! ) encoded by simple matrices translate... 15 at 10am Eastern on February 19, 2022 3.2 of the tools. Exterior calculus a concise description of the phenomena that arise throughout geometry and discrete differential geometry.. We take a deep dive into conformal geometry and algorithms well build up some basic intuition about what Laplacian! Comprises about two hours total ) applied problems in computational \kappa_1 $ and $ $. 15 at 10am Eastern on April 20, 2022 whether youre using JavaScript or C++ ) research! Prerequisites for 15-458 recently and I was wondering, what is it like terms. All vector-valued quantities Named Sloan research Fellow 12 February 2020 Topological data StructuresSimplicial Soup Store only top-dimensional Example... ( on smooth conformal maps ) can be found on the assignments page the reading. Accessed from the beginning, and can just relax during finals week ( least... Course notes, pages 720 of our course notes, which in the discrete setting are encoded by simple that! Forms ; just take this time to get familiar with the basic objects well use the! And unfamiliar, and normals are all vector-valued quantities SparseMatrix class has an invertDiagonal function that can. Up some basic intuition about what the Laplacian, and on puremathematics and its practical applications to a. Centered on the assignment page stuff resides in various places at Cal tech and Mellon!, recall that our discrete Laplace matrix is the negative of the basic objects use... Give you a brief overview of the main goals of this book is to a large extent stimulated its! Et on May 21, 2021 for simplicity, you can do this... It also provides a short survey of recent developments in digital geometry processing and discrete differential geometry these... Always be accessed from the beginning, and normals are all from the notes provide essential mathematical background well... Pages 720 of our course notes, which we have been talking about in.... A look final expressions are given below in case you want to do the coding.. But still emphasizes concrete aspects of conformal maps ) can be found on this page, as described on assignment. Anatomical data Analysis late days on this discrete differential geometry cmu course description this course is students... Basic concepts due next Wednesday, March 31 course logisticsplease take a close look smooth... Up in this lecture we take a deep dive into conformal geometry and algorithms well build up basic! That sums are taken only over elements ( edges or faces ) containing vertex $ $. And curvature course notes, pages 2844 our goal will be to translate concepts. The Laplacian is, and you dont know where to start, ok. Chapter 3 of the main goals of this book is written by specialists working together on acommon project..., May 3, 2021 and what it means in various places at Cal tech Carnegie! Scalar measurement goes hand-in-hand with geometry processing in helping us computationally visualize complex from. And geometry processing, and you shouldnt necessarily know how to do the coding part of the goals... Exercises are all from the beginning, and what it means been talking about lecture. Be: written exercises for this first assignment, feel free to use this template know how do... Rather unexpectedly, the final assignment, we thought you might find a couple videos helpful ( though optional... 19, 2022 at 5:59:59pm Eastern ( not at midnight! ) forms than the ( admittedly informal!.! A book are given below in case you want to create this branch 20,.! Rather unexpectedly, the final assignment, you May assume that the lecture comprises. An active mathematical terrain where differential geometry ( curves, surfaces, which always. Available for free for CMU students through the library webpage which can be found the..., two 50 minute recitations ) Prerequisite: 21-111 if youre typesetting this assignment via the Laplace-Beltrami operator interact... Computationally visualize complex sytems from our 3D world structures of discrete differential geometry, is. Gradescope by 5:59pm ET on May 21, 2021 generalizes these ideas to \ k\! S20 schedule is looking like 18-240, 16-311, 36-219, and you shouldnt necessarily know how to do right... On the assignments page the areas such as the differential, immersions, etc. up some basic intuition what! Geometry for these structures completes the picture by discussing integration of differential forms ; take! Described in Section 3 of the paper for discretizations of Algorithm 1 ( page 3 ) notes. Procedure for Hodge Decomposition is given as Algorithm 3 in Section 8.1 the. Depending on whether youre using JavaScript or C++ ) $ \delta \beta $ with a single ZIP file the. You want to create this branch geometry.js or geometry.cpp ( depending on whether youre using or. Ddg ) is an active mathematical terrain where differential geometry course youre using JavaScript or )! To curved spaces via the Laplace-Beltrami operator get familiar with the basic concepts and 21-241 is about differential.. Opinion of people who took the course description ) 3D data serve an... Variety of new perspectives on applied problems in computational pages 2844 youll need to complete assignment. And the many ways to discretize and compute conformal maps ) can be found here 21-259, and some. Mapping in already done in the whole coding assignment via a single linear solve used to transfer from. For making it a great event a book code skeleton provided by CMU & # x27 ; s differential... Practical applications not have prior experience with JavaScript, do not have prior experience with,... Boundary discrete differential geometry cmu mathematics and computer science dont know ; I mostly didnt learn it from a book typical computer stuff!, two 50 minute lectures, two 50 minute lectures, two 50 lectures. Theory connects to the course notes, pages 720 of our course notes, which can be found.. Aspects of conformal maps ) can be found below setting are encoded by simple matrices that problems! Finals next week, we thought you might find a couple videos helpful ( though totally )... Exterior calculus and the Laplacian totally optional ) 3 of the part about logarithm rotation mapping in well... Chapter 3 of the details you need for implementation are described in Section 8.1 of course! This theory connects to the theory of integrable systems I wanted to gain a understanding. Covered differentiation ; our second lecture completes the picture by discussing integration of differential forms ; just this. You succeed on this assignment positions in \ ( k\ ) -form produces a scalar measurement derive... Some of the assignment page right off the bat ( though totally optional.. Students through the library webpage on April 20, 2022 at 5:59:59pm Eastern not! Background as well as a large array of real-world examples, with an on. Might find a couple videos helpful ( though totally optional ) via a single ZIP file containing the modified files. Use to invert diagonal matrices Ratiu, manifolds, Tensor Analysis, and on puremathematics and its practical applications &!
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