## Steiner Formula for Surfaces in $\mathbb{R}^3$

In the slides we derived a Steiner formula for polyhedral surfaces in $\mathbb{R}^3$, by considering the Minkowski sum with a ball and working out expressions for the areas and curvatures associated with vertices, edges, and faces. But we can also get a Steiner formula for smooth surfaces, using the expressions already derived in class. In particular, recall that for a closed surface $f: M \to \mathbb{R}^3$ with Gauss map $N$, we can obtain the basic curvatures by just wedging together $df$ and $dN$ in all possible ways:
$\begin{array}{rcl} df \wedge df &=& 2NdA \\ df \wedge dN &=& 2HNdA \\ dN \wedge dN &=& 2KNdA \\ \end{array}$
Here $H$ and $K$ denote the mean and Gauss curvature (resp.), and dA is the area form induced by $f$. For sufficiently small $t$, taking a Minkowski sum with a ball is the same as pushing the surface in the normal direction a distance $t$. In other words, the surface
$f_t := f + tN$
will describe the “outer” boundary of the Minkowski sum; this surface has the same Gauss map $N$ as the original one. To get its area element, we can take the wedge product
$\begin{array}{rcl} df_t \wedge df_t &=& (df + t dN) \wedge (df + t dN) \\ &=& df \wedge df + 2t df \wedge dN + t^2 dN \wedge dN, \end{array}$
where we have used the fact that $\alpha \wedge \beta = \beta \wedge \alpha$ when $\alpha,\beta$ are both $\mathbb{R}^3$-valued 1-forms. The list of identities above then yields
$df_t \wedge df_t = (2 + 4tH + 2t^2 K)NdA,$
or equivalently,
$\fbox{$dA_t = (1+2tH+t^2K)dA.$}$
In other words, just as in the polyhedral case, the rate at which the area is growing is a polynomial in the ball radius $t$; the coefficients of this polynomial are given by the basic curvatures of the surface (also known as quermassintegrals!).

## Lecture 16—Discrete Curvature II (Variational Viewpoint)

In this lecture we wrap up our discussion of discrete curvature, and see how it all fits together into a single unified picture that connects the integral viewpoint, the variational viewpoint, and the Steiner formula. Along the way we’ll touch upon several of the major players in discrete differential geometry, including a discrete version of Gauss-Bonnet, Schläfli’s polyhedral formula, and the cotan Laplace operator—which will be the focus of our next set of lectures.

## Lecture 15—Discrete Curvature I (Integral Viewpoint)

Just as curvature provides powerful ways to describe and analyze smooth surfaces, discrete curvatures provide a powerful way to encode and manipulate digital geometry—and is a fundamental component of many modern algorithms for surface processing. 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. In the next lecture, we will see a complementary variational viewpoint, where discrete curvatures arise by instead taking derivatives of discrete geometry. Amazingly enough, these two perspectives will fit together naturally into a unified picture that connects essentially all of the standard discrete curvatures for triangle meshes.

## Lecture 14—Smooth Curvature

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.

## Lecture 13—Discrete Surfaces

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.

## Lecture 12—Smooth Surfaces

This lecture gives a crash course in the differential geometry of surfaces. There’s of course way more to know about surfaces than we can pack into a single lecture (and we’ll see plenty more later on), but this lecture will cover basic concepts like how to describe a surface and its normals. It also starts to connect surface theory to the other tools we’ve been building up, via vector-valued differential forms.

## Assignment 2 (Written): Investigating Curvature (Due 3/19)

The written portion of Assignment 2 can be found below. It takes a look at the curvature of smooth and discrete surfaces, which we have been talking about in lecture. Note that you are required to complete only two problems from each section!

Warning: We renumbered the Exercises in the course notes to make more sense, so you make sure you refer to the updated notes when doing these exercises.

## Assignment 2 (Coding): Investigating Curvature (Due 3/19)

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. (However, the final expressions are given below in case you want to do the coding first.) Once implemented, you will be able to visualize these geometric quantities on a mesh. For simplicity, you may assume that the mesh has no boundary.

Getting Started
Please implement the following routines in core/geometry.js:

1. angle
2. dihedralAngle
3. vertexNormalAngleWeighted
4. vertexNormalSphereInscribed
5. vertexNormalAreaWeighted
6. vertexNormalGaussianCurvature
7. vertexNormalMeanCurvature
8. angleDefect
9. totalAngleDefect
10. scalarMeanCurvature
11. circumcentricDualArea
12. principalCurvatures

Notes

1. The dihedral angle between the normals $N_{ijk}$ and $N_{ijl}$ of two adjacent faces $ijk$ and $ijl$ (respectively) is given by
$$\theta_{ij} := \text{atan2}\left(\frac{e_{ij}}{\|e_{ij}\|} \cdot \left(N_{ijk} \times N_{jil}\right), N_{ijk} \cdot N_{jil}\right)$$

where $e_{ij}$ is the vector from vertex $i$ to vertex $j$.

2. The formulas for the angle weighted normal, sphere inscribed normal, area weighted normal, discrete Gaussian curvature normal and discrete mean curvature normal at vertex $i$ are
\begin{aligned} N_i^\phi &:= \sum_{ijk \in F} \phi_i^{jk}N_{ijk}\\ N_i^S &:= \sum_{ijk \in F} \frac{e_{ij} \times e_{ik}} {\|e_{ij}\|^2\|e_{ik}\|^2}\\ N_i^A &:= \sum_{ijk \in F} A_{ijk}N_{ijk}\\ KN_i &= \frac 12 \sum_{ij \in E} \frac{\theta_{ij}}{\|e_{ij}\|}e_{ij}\\ HN_i &= \frac 12 \sum_{ij \in E}\left(\cot\left(\alpha_k^{ij}\right) + \cot\left(\beta_l^{ij}\right)\right)e_{ij} \end{aligned}

where $\phi_i^{jk}$ is the interior angle between edges $e_{ij}$ and $e_{ik}$, and $A_{ijk}$ is the area of face $ijk$. Note that sums are taken only over elements (edges or faces) containing vertex $i$. Normalize the final value of all your normal vectors before returning them.

3. The circumcentric dual area at vertex $i$ is given by
$A_i := \frac 18 \sum_{ijk \in F} \|e_{ik}\|^2\cot\left(\alpha_j^{ki}\right) + \|e_{ij}\|^2\cot\left(\beta_k^{ij}\right)$

4. The discrete scalar Gaussian curvature (also known as angle defect) and discrete scalar mean curvature at vertex $i$ are given by
\begin{aligned} K_i &:= 2\pi – \sum_{ijk \in F} \phi_i^{jk}\\ H_i &:= \frac 12 \sum_{ij \in E} \theta_{ij}\|e_{ij}\| \end{aligned}

Note that these quantities are discrete 2-forms, i.e., they represent the total Gaussian and mean curvature integrated over a neighborhood of a vertex. 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).

5. You are required to derive expressions for the principal curvatures $\kappa_1$ and $\kappa_2$ in exercise 4 of the written assignment. Your implementation of principalCurvatures should return the (pointwise) minimum and maximum principal curvature values at a vertex (in that order).

Submission Instructions

Please rename your geometry.js file to geometry.txt and put it in a single zip file called solution.zip. This file and your solution to the written exercises should be submitted together in a single email to Geometry.Collective@gmail.com with the subject line DDG19A2.

## Reading 5—Curves and Surfaces (due 3/7)

Your next reading complements our in-class discussion of the geometry of curves and surfaces. In particular, you should read Chapter 3 of the course notes, pages 28–44. This reading is due next Thursday, March 7.

Handin instructions are described on the Assignments Page.

Since these notes just barely scratch the surface (literally), I am often asked for recommendations on books that provide a deeper discussion of surfaces. The honest answer is, “I don’t know; I mostly didn’t 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:

## Lecture 11—Discrete Curves

This lecture presents the discrete counterpart of the previous lecture on smooth curves. Here we also arrive at a discrete version of the fundamental theorem for plane curves: a discrete curve is completely determined by its discrete parameterization (a.k.a. edge lengths) and its discrete curvature (a.k.a. exterior angles). Can you come up with a discrete version of the fundamental theorem for space curves? If we think of torsion as the rate at which the binormal is changing, then a natural analogue might be to (i) associate a binormal $B_i$ with each vertex, equal to the normal of the plane containing $f_{i-1}$, $f_{i}$, and $f_{i+1}$, and (ii) associate a torsion $\tau_{ij}$ to each edge $ij$, equal to the angle between $B_i$ and $B_{i+1}$. Using this data, can you recover a discrete space curve from edge lengths $\ell_{ij}$, exterior angles $\kappa_i$ at vertices, and torsions $\tau_{ij}$ associated with edges? What’s the actual algorithm? (If you find this problem intriguing, leave a comment in the notes! It’s not required for class credit.)

## Lecture 10—Smooth Curves

After spending a great deal of time understanding some basic algebraic and analytic tools (exterior algebra and exterior calculus), we’ll finally start talking about geometry in earnest, starting with smooth plane and space curves. Even low-dimensional geometry like curves reveal a lot of the phenomena that arise when studying curved manifolds in general. 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. 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. Next lecture, for instance, we’ll see some examples of algorithms for curvature flow, which naturally play well with representations based on curvature!

1. Here’s a piece of advice for assignment 1 which I forgot to put in the original post: 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 don’t have to do the exercise for this homework).
2. I just emailed out grades for assignment 0. If you have questions or concerns about your graded assignment, or if you did not receive a graded assignment, please feel free to email me or to talk to me during office hours.
3. Very few people have been attending my office hours on Fridays, so I am moving my office hours to be 2:30pm-4pm on Monday afternoons. They will still be held outside of Smith 232.
4. If you would like to see solutions for assignment 0, we will bring some printed copies to class on Tuesday. I can also print you solutions if you stop by during office hours.

## Slides—Discrete Exterior Calculus

This lecture wraps up our discussion of discrete exterior calculus, which will provide the basis for many of the algorithms we’ll develop in this class. Here we’ll encounter the same operations as in the smooth setting (Hodge star, wedge product, exterior derivative, etc.), which in the discrete setting are encoded by simple matrices that translate problems involving differential forms into ordinary linear algebra problems.

## Assignment 1 (Coding): Exterior Calculus (Due 2/26)

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 $d_1$. 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. The diagram shown above will be updated to indicate what kind of differential k-form is currently displayed. 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!) should help you build further intuition about what these operators mean and how they work

Getting Started

• For this assignment, you need to implement the following routines:
1. in core/geometry.js
1. cotan
2. barycentricDualArea
2. in core/discrete-exterior-calculus.js
1. buildHodgeStar0Form
2. buildHodgeStar1Form
3. buildHodgeStar2Form
4. buildExteriorDerivative0Form
5. buildExteriorDerivative1Form

In practice, a simple and efficient way to compute the cotangent of the angle $\theta$ between two vectors $u$ and $v$ is to use the cross product and the dot product rather than calling any trigonometric functions directly; we ask that you implement your solution this way. (Hint: how are the dot and cross product of two vectors related to the cosine and sine of the angle between them?)

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 $i$.

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 don’t have to do the exercise for this homework).

Submission Instructions

Please rename your geometry.js and discrete-exterior-calculus.js files to geometry.txt and discrete-exterior-calculus.txt (respectively) and submit them in a single zip file called solution.zip by email to Geometry.Collective@gmail.com.

## Assignment 1 (Written): Exterior Calculus (Due 2/26)

The written portion of assignment 1 is now available (below), which covers some of the fundamental tools we’ll be using in our class. Initially this assignment may look a bit intimidating, but keep a few things in mind:

• 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.
• Moreover, note that you are required to complete only three problems from each section.

Finally, don’t be shy about asking us questions here in the comments, via email, or during office hours.  We want to help you succeed on this assignment, so that you can enjoy all the adventures yet to come…

This assignment is due on Tuesday, February 26.

## Slides—Discrete Differential Forms

In this lecture, we turn smooth differential $k$-forms into discrete objects that we can actually compute with. The basic idea is actually quite simple: to capture some information about a differential $k$-form, we integrate it over each oriented $k$-simplex of a mesh. The resulting values are just ordinary numbers that give us some sense of what the original $k$-form must have looked like.

## Slides—Exterior Calculus II: Integration

Our first lecture on exterior calculus covered differentiation; our second lecture completes the picture by discussing integration of differential forms. The relationship between integration and differentiation is encapsulated by Stokes’ theorem, which generalizes the fundamental theorem of calculus, as well as many other important theorems from vector calculus and complex analysis (divergence theorem, Green’s theorem, Cauchy’s integral formula, etc.). 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.

## Reading 4: Exterior Calculus — due 2/14

The next reading assignment will wrap up our discussion of exterior calculus, both smooth and discrete. 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. In particular, the basic data we will work with in the computational setting are “integrals of derivatives,” which amount to simple scalar quantities we can associate with the vertices, edges, faces, etc. of a simplicial mesh. These tools will provide the basis for the algorithms we’ll explore throughout the rest of the semester.

The reading is the remainder of Chapter 4 from the course notes, “A Quick and Dirty Introduction to Exterior Calculus”, Sections 4.6 through 4.8 (pages 67–83). Note that you just have to read these sections; you do not have to do the written exercises; a different set of written problems will be posted later on. The reading is due Thursday, February 14 at 10am. See the assignments page for handin instructions.

## Slides—Exterior Calculus I: Differentiation

Ordinary calculus provides tools for understanding rates of change (via derivatives), total quantities (via integration), and the total change (via the fundamental theorem of calculus). Exterior calculus generalizes these ideas to $n$-dimensional quantities that arise throughout geometry and physics. 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.

## Slides—Differential Forms in $R^n$

A differential $k$-form describes a $k$-dimensional measurement at each point of space. For instance, the area 2-form on a sphere tells us how much each little piece of the sphere should contribute to an integral over the sphere. In this lecture we give some basic facts and definitions about differential $k$-forms, and how to work with them in coordinates. Ultimately differential $k$-forms will pave the way to a general notion of integration, which in turn will be our basic mechanism for turning smooth equations into discrete ones (by integrating over elements of a mesh).

## Reading 3: Exterior Algebra and k-Forms (due 2/7)

Your next reading assignment will help you review the concepts we’ve 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. 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.

The reading is Chapter 4, “A Quick and Dirty Introduction to Exterior Calculus”, up through section 4.5.1 (pages 45–65). It will be due Thursday, February 7 at 10am Eastern time. See the assignments page for handin instructions.

Your next homework will give you some hands-on practice with differential forms; just take this time to get familiar with the basic concepts.

## Office Hours + Homework Typo

I will hold office hours tomorrow (Friday Feb. 1) in the Smith Hall 2nd floor open space from 3pm to 4pm. Next week, since the homework is due on Thursday, I will hold office hours on Tuesday from 3pm to 4pm in the same place. After that, I will go back to having office hours on Fridays.

Sorry for the short notice. If these times don’t work for you, feel free to email me and we can arrange to meet another time.

Also, somebody pointed out a typo in the coding assignment during recitation – the vertex-edge and edge-face adjacency matrices should be sparse matrices, not dense like it says in the comments/documentation.

## Slides—$k$-Forms

Today we continue our journey toward building up (discrete) exterior calculus by talking about how to measure little k-dimensional volumes. 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. Such measurements will ultimately allow us to talk about integration over curved spaces; in the discrete setting, these measurements will be the basic data we associate with the elements of mesh.

## Recitation

We will have a recitation on Tuesday at 4:30 in Gates 4102 to go over the Javascript framework that we use in this class. I will review the halfedge data structure, and I’ll go over some examples of using our framework. I will also talk about sparse matrices and solving linear systems of equations.

Recitation Slides

## Slides—Exterior Algebra

Today’s lecture will cover one of the basic tools we’ll use throughout the rest of the course: exterior algebra. The basic idea is to add a couple new operations to our usual list of vector operations (dot product, cross product, etc.) that make it easy to talk about volumes rather than just vectors. If you felt ok working with things like the cross product and the determinant in your linear algebra/vector calculus courses, this shouldn’t be too big of a leap. (If not, could be a good moment for a review!)

## Assignment 0 (Written): Combinatorial Surfaces — due 2/7

For the written part of your first homework you will do some exercises that will help familiarize you with basic descriptions and representations of combinatorial surfaces (simplicial surfaces, adjacency matrices, halfedge meshes), which will help prepare you to work with such surfaces as we continue through the course. (If any of this stuff seems abstract right now, don’t worry: we’ll use it over and over again to implement “real” algorithms starting in just a couple weeks!)

You must complete 8 out of 15 exercises in the Written Exercises section of Chapter 2 of the course notes. You may choose any set of 8 exercises you like, but if you do more than 8, please mark clearly on your submission which ones you would like us to grade.

The assignment is due on February 7, 2019 at 5:59:59pm Eastern (not at midnight!). Further hand-in instructions can be found on this page.

## Assignment 0 (Coding): Combinatorial Surfaces — due 2/7

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. Once implemented, you will be able to select simplices and apply these operations to them by clicking the appropriate buttons in the viewer (shown above).

Getting Started

• Please download or clone the files in this repository. It contains a fast and flexible framework for 3D geometry processing implemented in Javascript. Over the course of the semester, you will implement all of your coding assignments here.
• For this assignment, you will need to implement the following routines in projects/simplicial-complex-operators/simplicial-complex-operators.js:
• assignElementIndices
• buildVertexVector
• buildEdgeVector
• buildFaceVector
• star
• closure
• isComplex
• isPureComplex
• boundary

Notes

• This assignment comes with a viewer projects/simplicial-complex-operators/index.html which lets you apply your operators to simplices of meshes and visualize the results.
• Selecting simplices will not work until you fill in the assignElementIndices function.
• The assignment also comes with a test script tests/simplicial-complex-operators/text.html which you can use to verify the correctness of your operators.
• The code framework is implemented in Javascript, which means no compilation or installation is necessary on any platform. You can simply get started by opening the index.html file in projects/discrete-exterior-calculus/ in a web browser. We recommend using Chrome or Firefox. Safari has poor WebGL performance.
• If you do not have prior experience with Javascript, do not worry! You should be able to get a handle on Javascript syntax by reading through some of the code in the framework (a good place to start might be core/geometry.js). The framework also contains extensive documentation (see docs/index.html).
• All browsers come with tools for debugging (for instance the JavaScript Console in Chrome).

Submission Instructions

Please rename your simplicial-complex-operators.js file to simplicial-complex-operators.txt and submit it in a zip file called solution.zip to Geometry.Collective@gmail.com.

## Reading 2: Combinatorial Surfaces (Due 1/29)

Your next reading will take a dive into purely combinatorial descriptions of surfaces, i.e., those that capture connectivity, but not geometry.  These descriptions and data structures will provide the foundation for all the geometry and algorithms we’ll build up in this class.  (The reading also provides the essential background for your first written and coding assignments!)

The reading is Chapter 2, pages 7–20 of our course notes, which can always be accessed from the link above.

Your short 2-3 sentence summary is due by 10am Eastern on January 29, 2019.  Handin instructions can be found on the assignment page.

## Reading 1: Overview of DDG (Due 1/22)

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):

1. (pages 1–3) Crane & Wardetzky, “A Glimpse into Discrete Differential Geometry”.
2. (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.
3. (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!

## Assignment -1: Favorite Formula

Part of your course grade is determined by participation, which can include both in-class participation as well as discussion here on the course webpage.  Therefore, your first assignment is to:

1. create an account, and
To make things interesting, your comment should include a description of your favorite mathematical formula typeset in $\LaTeX$.  If you don’t know how to use $\LaTeX$ this is a great opportunity to learn — a very basic introduction can be found here.  (And if you don’t have a favorite mathematical formula, this is a great time to pick one!)

(P.S. Anyone interested in hearing about some cool “favorite theorems” should check out this podcast.)

## Welcome to Discrete Differential Geometry! (Spring 2019)

Welcome to the website for 15-458/858 (Discrete Differential Geometry).  Here you’ll find course notes, lecture slides, and homework (see links on the right).

If you are a student in the class, register now by clicking here!

To participate in the class, you must register using your Andrew (CMU) email address.

A few things to note:

• You can include mathematical notation in your questions using standard $\LaTeX$ syntax.  For instance, when enclosed in a pair of dollar signs, an expression like \int_M K dA = 2\pi\chi gets typeset as $\int_M K dA = 2\pi\chi$.