COURSE NOTES · 2013Crane, de Goes, Desbrun & Schröder
Digital Geometry Processing with Discrete Exterior Calculus
The canonical computational path from cochains and incidence matrices through metric Hodge stars and geometry-processing algorithms.
PAPER ↔ OPERATOR ↔ EXPERIMENT
Follow the idea
back to its source.
This is a selective reading path, not an exhaustive bibliography. Start with the notes beside the lesson you are studying; follow the research papers when you need to distinguish a teaching model from a solver architecture.
Discrete exterior calculus and Hodge decomposition
Read these when an incidence matrix, Hodge star, adjoint, boundary condition, or harmonic component feels like a convention rather than a mathematical choice.
COURSE NOTES · 2016de Goes, Desbrun & Tong
Vector Field Processing on Triangle Meshes
A representation-aware account of face-, edge-, and vertex-based fields, with their differential operators and Hodge decompositions.
READ THESE BESIDE LESSON 1
Two gentler entries—and the design principle behind this site.
The course notes explain the smooth language; the browser tutorial makes the discrete matrices concrete; the essay explains why the calculation should remain manipulable.
SIGGRAPH COURSE · 2023Wang, Nabizadeh & Chern
Exterior Calculus in Graphics
An advanced-undergraduate bridge from vectors and covectors to differential forms, continuum mechanics, and fluid dynamics. This is the conceptual reference for the early Clebsch lessons.
INTERACTIVE TUTORIALGeometry Collective
Discrete Exterior Calculus
A focused implementation exercise for building incidence matrices, Hodge stars, and the Laplace–Beltrami operator on a triangle mesh.
DESIGN ESSAY · 2011/2024Bret Victor
Explorable Explanations
The pedagogical north star: a mathematical claim should be accompanied by a model whose assumptions, state, and consequences can be inspected.
Representations, connections, and field design
These works separate where a field is stored from how vectors in different tangent planes are compared.
CGF · 2010Crane, Desbrun & Schröder
Trivial Connections on Discrete Surfaces
Builds smooth discrete connections with controlled singularities—the intrinsic reference behind connection-based field comparison.
ACM TOG · 2013Knöppel, Crane, Pinkall & Schröder
Globally Optimal Direction Fields
Turns smooth direction-field design and singularity placement into a sparse global eigenproblem.
REFERENCE LIBRARYVaxman et al.
Directional tutorial
A practical taxonomy and C++ implementation of tangent bundles, symmetries, representations, matching, and field processing.
ACM TOG · 2023Feng, Gillespie & Crane
Winding Numbers on Discrete Surfaces
A clean representation-first example: encode difficult curve data by differential one-forms, use Hodge decomposition to isolate topology, then reconstruct the geometric answer.
Integrability is not one operator
Closest exact fields, curl penalties, hard constraints, unit-length fields, and parameterization-aligned fields solve related but different problems.
ACM TOG · 2015Diamanti, Vaxman, Panozzo & Sorkine-Hornung
Integrable PolyVector Fields
Optimizes tangent fields that both respect design constraints and correspond exactly to local parameterization gradients.
ACM TOG · 2016Liu, Tong, de Goes & Desbrun
Discrete Connection and Covariant Derivative
Develops a principled discrete connection for analysis and design, clarifying what it means to differentiate vertex-based tangent vectors.
From direction fields to stripe patterns
The lab’s complex phase is a compact teaching model of this lineage. The important leap is from a local direction to a globally continuous periodic function.
ACM TOG · 2015Knöppel, Crane, Pinkall & Schröder
Stripe Patterns on Surfaces
Introduces globally continuous, equally spaced surface stripes as the smallest eigenvector of a sparse connection energy.
ACM TOG · 2019Vekhter, Zhuo, Gil Fandino, Huang & Vouga
Weaving Geodesic Foliations
Separates designing an approximately geodesic unit field from integrating it into a periodic foliation, including relaxed global integrability.
ACM TOG · 2023Montes Maestre et al.
Differentiable Stripe Patterns for Inverse Design
Extends stripe patterns into an end-to-end differentiable design space and addresses non-uniqueness in the original formulation.
Sparse derivatives without hand-written Hessians
The browser exposes the local-element idea; these systems show two routes into larger compiled research code.
CGF · 2022Schmidt, Born, Bommes, Campen & Kobbelt
TinyAD: Automatic Differentiation in Geometry Processing Made Simple
Differentiates small mesh elements through second order and assembles their sparse global gradient and Hessian.
COMPILER ADMoses, Churavy et al.
Enzyme
Differentiates optimized LLVM IR across several source languages—a complementary path when research code should remain ordinary compiled code.
Fluids on surfaces: a deliberate reading order
These papers do not describe one interchangeable method. They progressively change the stored state, the preserved structure, the topology, and finally the governing equations.
1 · TRANSPORT + TRANSFER
How does information cross a time step?
These works answer a different question from Clebsch representation. Semi-Lagrangian methods backtrace from grid samples; APIC moves particles and enriches their transfers to a force-solving grid. Either idea can be combined with a separately chosen fluid state.
SIGGRAPH · 1999Jos Stam
Stable Fluids
Uses semi-Lagrangian advection and implicit solves to permit large, visually stable timesteps, followed by pressure projection for incompressibility. Its interpolation is robust but can smooth fine vortical structure.
ACM TOG · 2015Jiang, Schroeder, Selle, Teran & Stomakhin
The Affine Particle-In-Cell Method
Augments particles with a locally affine velocity description, reducing the information loss of PIC transfers while retaining a filtering property and conserving angular momentum across the method’s transfers.
ORIENTATION NOTEBOOKDirection Field Lab
How Should a Fluid Remember?
Places representation, transport, forces, and constraints on separate axes, then switches one live periodic surface between prescribed random kinematics and Clebsch shallow-water dynamics.
2 · REPRESENTATION
What should the solver carry?
Read these after the gentle Clebsch introduction. They explain why circulation is naturally a one-form measurement, and why Clebsch variables are a factorization of that one-form rather than a new physical fluid.
ACM TOG · 2017Chern, Knöppel, Pinkall & Schröder
Inside Fluids: Clebsch Maps for Visualization and Processing
Uses Clebsch maps to expose vortex structure and make a three-dimensional velocity field accessible through transported scalar data.
ACM TOG · 2022Nabizadeh, Wang, Ramamoorthi & Chern
Covector Fluids
Rewrites Euler advection in covector form so the numerical update follows circulation and Kelvin’s theorem more directly than ordinary velocity interpolation.
ACM TOG · 2023Yin, Nabizadeh, Wu, Wang & Chern
Fluid Cohomology
Derives the missing evolution of harmonic velocity components on domains with holes, periodic boundaries, or genus. This is the research foundation for the torus lesson.
3 · VARIATIONAL STRUCTURE
Which invariants survive discretization?
The older work establishes the discrete-diffeomorphism and Lagrange–d’Alembert lineage. The new vakonomic formulation makes a different choice for nonholonomic constraints and supplies the clearest research destination for low-rank Clebsch state.
PHYSICA D · 2011Pavlov, Mullen, Tong, Kanso, Marsden & Desbrun
Structure-Preserving Discretization of Incompressible Fluids
Approximates volume-preserving diffeomorphisms by a finite-dimensional Lie group and derives a discrete Eulerian integrator with a discrete Kelvin theorem.
PREPRINT · 2026Roy-Chowdhury, Nabizadeh, Gross, Gruber & Chern
Vakonomic Fluids
Treats the discrete velocity constraint vakonomically, yielding Lie–Poisson dynamics, a discrete relabeling symmetry, Kelvin circulation, and a low-rank momentum-map representation analogous to Clebsch variables.
DCDS · 2014Desbrun, Gawlik, Gay-Balmaz & Zeitlin
Variational Discretization for Rotating Stratified Fluids
Extends the geometric integrator viewpoint to advected density and rotation, preserving a discrete Kelvin theorem and long-term energy behavior. It is a useful bridge toward geophysical shallow-water structure.
4 · TOPOLOGY
Vorticity is local; periods are global.
The annulus lab introduces one missing circulation. The flat-torus lab exposes two. Return to Fluid Cohomology above for the equations that evolve those harmonic coefficients rather than freezing them.
5 · SURFACE PHYSICS
What changes when the domain bends—or the depth evolves?
Shallow water replaces the incompressible horizontal constraint by depth-weighted continuity. Viscous surface flow adds a separate geometric burden: curvature, boundary conditions, and harmonic-mode dynamics enter the dissipation.
SCA · 2007Wang, Miller & Turk
Solving General Shallow Wave Equations on Surfaces
A graphics baseline for shallow water directly on curved surfaces without requiring one global parameterization.
ACM TOG · 2025Zhu, Yin & Chern
Viscous Vortex Dynamics on Surfaces
Derives curvature-dependent viscous terms and the evolution of harmonic components on surfaces of arbitrary topology, including non-orientable domains and several boundary conditions.
COURSE SYNTHESISDirection Field Lab
Clebsch Shallow Water
The course endpoint combines transported labels, covector assembly, metric conversion, depth flux, and harmonic bookkeeping. It is a teaching architecture—not yet an implementation of the vakonomic solver.
FOR STUDENT REPORTS
Cite the method you actually use.
This page is an orientation map. Follow the project or DOI link for the authors’ preferred BibTeX and version of record; record the version date for preprints; and distinguish an inspiration, a teaching analogue, and an implemented method.