Stanford CS248B

Resources

References and learning tools for physics-based animation and simulation. General material comes first, followed by readings for each lecture, which are added after the lecture is given. For a specific offering's schedule, slides, and assignments, see the list of course offerings.

General

Free course notes and books

• Andrew Witkin and David Baraff. Physically Based Modeling: Principles and Practice. SIGGRAPH 1997 course notes.
  The classic introduction, with short chapters on differential equations, particle dynamics, implicit methods, constrained dynamics, and rigid bodies.
• Daniel Shiffman. The Nature of Code: Simulating Natural Systems with JavaScript. No Starch Press, 2024. Free to read online.
  All examples are in p5.js, the library we use. Chapters on vectors, forces, oscillation, particle systems, and physics libraries.
• Justin Solomon. Numerical Algorithms. A K Peters/CRC Press, 2015. Free PDF from the author.
  Chapter 15 of the PDF, Ordinary Differential Equations, covers the integrators used in this course.
• Theodore Kim and David Eberle. Dynamic Deformables: Implementation and Production Practicalities. SIGGRAPH 2020 and 2022 Courses (ACM: 2020, 2022).
  Notes and code for deformable simulation as practiced at Pixar.
• Eftychios Sifakis and Jernej Barbič. FEM Simulation of 3D Deformable Solids: A Practitioner's Guide to Theory, Discretization and Model Reduction. SIGGRAPH 2012 Courses (ACM).
• Robert Bridson and Matthias Müller-Fischer. Fluid Simulation for Computer Animation. SIGGRAPH 2007 course notes.
  Free notes that later grew into Bridson's book, listed below.

Books

• Christer Ericson. Real-Time Collision Detection. CRC Press, 2004.
  The standard practical reference on bounding volumes, spatial partitioning, intersection tests, and robustness.
• Robert Bridson. Fluid Simulation for Computer Graphics, 2nd edition. A K Peters/CRC Press, 2015.

Surveys and tutorials

On publisher sites; these may need Stanford library access.

• Adam W. Bargteil and Tamar Shinar. An Introduction to Physics-Based Animation. SIGGRAPH 2019 Courses.
• Andrew Nealen, Matthias Müller, Richard Keiser, Eddy Boxerman, and Mark Carlson. Physically Based Deformable Models in Computer Graphics. Computer Graphics Forum, 2006.
• Jan Bender, Matthias Müller, Miguel A. Otaduy, Matthias Teschner, and Miles Macklin. A Survey on Position-Based Simulation Methods in Computer Graphics. Computer Graphics Forum, 2014.
• Matthias Müller, Bruno Heidelberger, Marcus Hennix, and John Ratcliff. Position Based Dynamics. Journal of Visual Communication and Image Representation, 2007.

Numerical linear algebra

Background for solving the large sparse linear systems in this course, such as the pressure Poisson equation in fluids and the systems in implicit integration. If scientific computing is new to you, start with the first two.

• Jonathan Richard Shewchuk. An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Carnegie Mellon University, 1994. Free PDF.
  Builds conjugate gradients step by step from quadratic forms and steepest descent, with many illustrations, and covers preconditioning. A gentle first read on the method.
• Solomon, Numerical Algorithms (listed above): Chapter 3 (Linear Systems and the LU Decomposition), Chapter 4 (Designing and Analyzing Linear Systems), and Chapter 11 (Iterative Linear Solvers).
• Lloyd N. Trefethen and David Bau III. Numerical Linear Algebra. SIAM, 1997; twenty-fifth anniversary edition, 2022.
  A standard introduction to classical numerical linear algebra.
• Gene H. Golub and Charles F. Van Loan. Matrix Computations, 4th edition. Johns Hopkins University Press, 2013.
  A comprehensive reference, including direct factorizations.
• Timothy A. Davis. Direct Methods for Sparse Linear Systems. SIAM, 2006.
  Sparse Cholesky and LU factorization, fill-reducing orderings, and the CSparse code.
• Richard Barrett, Michael Berry, Tony F. Chan, James Demmel, June Donato, Jack Dongarra, Victor Eijkhout, Roldan Pozo, Charles Romine, and Henk van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, 1994. Free electronic version on Netlib.
  Short, practical descriptions of the standard iterative methods, including conjugate gradients, and of preconditioners such as Jacobi, SSOR, and incomplete factorizations.
• Yousef Saad. Iterative Methods for Sparse Linear Systems, 2nd edition. SIAM, 2003. Free PDF from the author.
  Krylov subspace methods and preconditioning in depth, with a chapter on multigrid.
• William L. Briggs, Van Emden Henson, and Steve F. McCormick. A Multigrid Tutorial, 2nd edition. SIAM, 2000.
  An introduction to multigrid, through algebraic multigrid.

Websites, videos, and talks

• Christopher Batty. Physics-Based Animation blog.
  Lists new papers by conference, such as SIGGRAPH and SIGGRAPH Asia, and catalogues people and software in the field. Good for browsing what is current in research.
• Matthias Müller. Ten Minute Physics (YouTube).
  Short videos, each with a small JavaScript demo that runs in the browser.
• Erin Catto. Box2D publications.
  GDC talks by the author of the Box2D physics engine, with slides and some source code, on integration, constraints, contact, GJK distance, continuous collision, and bounding volume hierarchies.

Programming tools

• p5.js and its reference documentation.
• OpenProcessing.
• Shadertoy, co-founded by Inigo Quilez and Pol Jeremias, a site for creating and sharing real-time graphics programs. Many SDF and ray-marching examples run live in the browser.
• Graphtoy, Inigo Quilez's browser tool for graphing functions. Handy for trying out SDF formulas and boolean combinations.

By Lecture (Fall 2026)

Lecture 2: Particle Systems and ODEs

• Witkin and Baraff, Physically Based Modeling: the chapters "Differential Equation Basics", "Particle Dynamics", and "Implicit Methods".
• Solomon, Numerical Algorithms: Chapter 15 of the PDF, Ordinary Differential Equations.
• Uri M. Ascher and Linda R. Petzold. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, 1998.
  Chapters 2 to 4 (On Problem Stability; Basic Methods, Basic Concepts; One-Step Methods) are the most relevant to this lecture.
• Shiffman, The Nature of Code: Chapter 2 (Forces) and Chapter 4 (Particle Systems), in p5.js.
• John Denker. Basic Properties of a Symplectic Integrator.
  A visual explanation of why a symplectic integrator behaves better than a non-symplectic one.
• Erin Catto, "Numerical Integration" (GDC 2009) and "Numerical Methods" (GDC 2015), on the Box2D publications page.
• William T. Reeves. Particle Systems—a Technique for Modeling a Class of Fuzzy Objects. ACM Transactions on Graphics, 1983.
• Andrew P. Witkin and Paul S. Heckbert. Using Particles to Sample and Control Implicit Surfaces. SIGGRAPH '94.

Lecture 3: Resolving Particle Collisions

• Robert Bridson, Ronald Fedkiw, and John Anderson. Robust Treatment of Collisions, Contact and Friction for Cloth Animation. ACM Transactions on Graphics, 2002.
• Inigo Quilez on signed distance functions:
  • 2D primitives, with live demos of all of them in his Shadertoy playlist.
  • 3D primitives, with formulas for combining them, and a Shadertoy playlist of the primitives.
  • Ray marching SDFs.
• John C. Hart. Sphere Tracing: A Geometric Method for the Antialiased Ray Tracing of Implicit Surfaces. The Visual Computer, 1996.
  The basis of the ray-marched ball collisions in P1.
• Matthias Müller, Ten Minute Physics, How to write a pinball simulation (video, code).
  Ball-capsule collisions, collisions against boundaries made of line segments, and touch and mouse input. A useful companion to P1.

Lecture 4: Collision Detection I

• Ericson, Real-Time Collision Detection.
• Real-Time Rendering, Object/Object Intersection.
  A grid of intersection tests for common shapes, pointing to code and references.
• Jonathan Richard Shewchuk. Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates. Discrete & Computational Geometry, 1997.
  The exact orientation tests mentioned in lecture. Code and an overview.
• Stefan Gottschalk, Ming C. Lin, and Dinesh Manocha. OBBTree: A Hierarchical Structure for Rapid Interference Detection. SIGGRAPH 96.
  The source of the 15-axis separating axis test for OBBs.
• David Eberly. Dynamic Collision Detection using Oriented Bounding Boxes. Geometric Tools, 1999 (revised 2008). Free PDF.
• J. T. Klosowski, M. Held, J. S. B. Mitchell, H. Sowizral, and K. Zikan. Efficient Collision Detection Using Bounding Volume Hierarchies of k-DOPs. IEEE Transactions on Visualization and Computer Graphics, 1998.
• Gino van den Bergen. Efficient Collision Detection of Complex Deformable Models using AABB Trees. Journal of Graphics Tools, 1997.
• Philip M. Hubbard. Approximating Polyhedra with Spheres for Time-Critical Collision Detection. ACM Transactions on Graphics, 1996.

Lecture 5: Collision Detection II

• Jonathan D. Cohen, Ming C. Lin, Dinesh Manocha, and Madhav Ponamgi. I-COLLIDE: An Interactive and Exact Collision Detection System for Large-Scale Environments. Symposium on Interactive 3D Graphics, 1995.
  Sweep and prune.
• David Baraff. Dynamic Simulation of Non-Penetrating Rigid Bodies. Ph.D. thesis, Cornell University, 1992 (Cornell CS Technical Report 92-1275). Listed on the author's page.
• Scott Le Grand. Broad-Phase Collision Detection with CUDA. GPU Gems 3, Chapter 32. Free online from NVIDIA.
• Matthias Teschner, Bruno Heidelberger, Matthias Müller, Danat Pomeranets, and Markus Gross. Optimized Spatial Hashing for Collision Detection of Deformable Objects. Vision, Modeling, and Visualization (VMV) 2003.
• Matthias Müller, Ten Minute Physics episodes, each with a browser demo and PDF notes:
  • Spatial hashing: video, demo, notes.
  • Sweep and prune: video, demo, notes.
  • Bounding volume hierarchies with Morton codes: video, demo, notes.
• Erin Catto, "Dynamic Bounding Volume Hierarchies" (GDC 2019), on the Box2D publications page.
• M. Teschner, S. Kimmerle, B. Heidelberger, G. Zachmann, L. Raghupathi, A. Fuhrmann, M.-P. Cani, F. Faure, N. Magnenat-Thalmann, W. Strasser, and P. Volino. Collision Detection for Deformable Objects. Computer Graphics Forum, 2005.
  A survey.
• Doug L. James and Dinesh K. Pai. BD-Tree: Output-Sensitive Collision Detection for Reduced Deformable Models. ACM Transactions on Graphics, 2004.
• Danny M. Kaufman, Rasmus Tamstorf, Breannan Smith, Jean-Marie Aubry, and Eitan Grinspun. Adaptive Nonlinearity for Collisions in Complex Rod Assemblies. ACM Transactions on Graphics, 2014 (project page).
• Huamin Wang. Defending Continuous Collision Detection against Errors. ACM Transactions on Graphics, 2014.

Deformable Models (Lectures 6 and 7)

• Andrew Witkin and David Baraff. Physically Based Modeling, online SIGGRAPH 2001 course notes. Cited in lecture. The Pixar address in the slides now redirects to a general Pixar page, so use the Internet Archive copy of the course index. Its "Cloth and Fur Energy Functions" slides are the source of the hair bending model and the length-dependent bending stiffness used in lecture. The course page allows the lecture-note chapters, but not the slide sets, to be freely duplicated and distributed.
  The 1997 Witkin and Baraff notes listed under General are the live companion, but they do not include the cloth and fur material.
• David Baraff and Andrew Witkin. Large Steps in Cloth Simulation. SIGGRAPH 98. Author's PDF.
  Where the energy, constraint function, and Jacobian formulation used in these lectures comes from, together with the implicit integration that makes stiff cloth practical.
• Rasmus Tamstorf and Eitan Grinspun. Discrete Bending Forces and Their Jacobians. Graphical Models, 2013. Disney Animation page, with the PDF.
  Compact closed-form bending derivatives, and the answer to the lecture's "what went wrong" with naive symbolic differentiation.
• Eitan Grinspun, Anil N. Hirani, Mathieu Desbrun, and Peter Schröder. Discrete Shells. Eurographics/SIGGRAPH Symposium on Computer Animation, 2003 (Eurographics Digital Library).
  The dihedral-angle bending model behind the cloth and shell bending slides.
• Naive Mathematica codegen example, linked from the lecture slides.
  A cautionary tale: differentiating a hinge bending energy symbolically, with no structure imposed, explodes into an unusable number of terms.
• Also listed under General, and worth reading for these lectures:
  • Kim and Eberle, Dynamic Deformables: how this is done in production.
  • Sifakis and Barbič, FEM Simulation of 3D Deformable Solids: where mass-spring models give way to continuum mechanics.
  • Nealen et al., Physically Based Deformable Models in Computer Graphics: a survey of the whole area.
  • Müller et al., Position Based Dynamics, and Bender et al., A Survey on Position-Based Simulation Methods in Computer Graphics: the position-level relatives of the area springs used in P2.

Deformable Collision Processing (Lectures 8 and 9)

• Cem Yuksel. High-Performance Polynomial Root Finding for Graphics. Proceedings of the ACM on Computer Graphics and Interactive Techniques, 2022. Author's page, with code.
  Cited in lecture. Why the textbook root formulas are not good enough when collision detection generates billions of polynomials.
• Xavier Provot. Collision and Self-Collision Handling in Cloth Model Dedicated to Design Garments. Computer Animation and Simulation '97, Eurographics.
  The classic cloth collision pipeline these lectures descend from.
• David Harmon, Etienne Vouga, Rasmus Tamstorf, and Eitan Grinspun. Robust Treatment of Simultaneous Collisions. ACM Transactions on Graphics, 2008.
  What to do when the iterative impulse loop will not converge, and why the order collisions are processed in matters.
• Minchen Li, Zachary Ferguson, Teseo Schneider, Timothy Langlois, Denis Zorin, Daniele Panozzo, Chenfanfu Jiang, and Danny M. Kaufman. Incremental Potential Contact: Intersection- and Inversion-free, Large-Deformation Dynamics. ACM Transactions on Graphics, 2020.
  A modern alternative that guarantees no interpenetration, rather than detecting and repairing it.
• Also listed above, and central to these lectures:
  • Bridson, Fedkiw, and Anderson, Robust Treatment of Collisions, Contact and Friction for Cloth Animation (under Lecture 3): the reference for combining impulses, repulsion forces, and a fallback for the hard cases.
  • Wang, Defending Continuous Collision Detection against Errors, and Kaufman et al., Adaptive Nonlinearity for Collisions in Complex Rod Assemblies (under Lecture 5): floating-point robustness in exactly these tests.
  • Baraff and Witkin, Large Steps in Cloth Simulation (under Deformable Models): the implicit integration that closes the lecture, and the constraint filtering that inverse-mass filtering descends from.
  • Ascher and Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations (under Lecture 2): stiffness and implicit methods in general.

Fluids (Lecture 10)

• Robert Bridson and Matthias Müller-Fischer. Fluid Simulation, SIGGRAPH 2007 course notes (course page).
  The main reference for this lecture. The course page also has public-domain code for conjugate gradients with modified incomplete Cholesky preconditioning, for the pressure equations.
• Robert Bridson. Fluid Simulation for Computer Graphics, 2nd edition (listed above): the book these notes grew into.
• Smoothed particle hydrodynamics (SPH) and particle fluids
  • SPH began in astrophysics: L. B. Lucy, A Numerical Approach to the Testing of the Fission Hypothesis, The Astronomical Journal, 1977; and R. A. Gingold and J. J. Monaghan, Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars, Monthly Notices of the Royal Astronomical Society, 1977. For a review, see J. J. Monaghan, Smoothed Particle Hydrodynamics, Annual Review of Astronomy and Astrophysics, 1992.
  • Matthias Müller, David Charypar, and Markus Gross. Particle-Based Fluid Simulation for Interactive Applications. Symposium on Computer Animation, 2003. Author's PDF.
  • Barbara Solenthaler and Renato Pajarola. Predictive-Corrective Incompressible SPH. ACM Transactions on Graphics, 2009.
  • Miles Macklin and Matthias Müller. Position Based Fluids. ACM Transactions on Graphics, 2013.
  • Nadir Akinci, Markus Ihmsen, Gizem Akinci, Barbara Solenthaler, and Matthias Teschner. Versatile Rigid-Fluid Coupling for Incompressible SPH. ACM Transactions on Graphics, 2012.
  • Fernando de Goes, Corentin Wallez, Jin Huang, Dmitry Pavlov, and Mathieu Desbrun. Power Particles: An Incompressible Fluid Solver Based on Power Diagrams. ACM Transactions on Graphics, 2015.
  • Jihun Yu and Greg Turk. Reconstructing Surfaces of Particle-Based Fluids Using Anisotropic Kernels. ACM Transactions on Graphics, 2013; first presented at the Symposium on Computer Animation, 2010.
    Turning particles into a smooth fluid surface: each particle's smoothing kernel is stretched into an ellipsoid along the directions found by principal component analysis of its neighbors, which captures thin streams and sharp features.
• Grid-based (Eulerian) fluids
  • Jos Stam. Stable Fluids. SIGGRAPH '99.
  • Ronald Fedkiw, Jos Stam, and Henrik Wann Jensen. Visual Simulation of Smoke. SIGGRAPH 2001.
  • Matthias Müller, Ten Minute Physics, how to write an Eulerian fluid simulator: video, demo, notes.
• Hybrid particle-grid methods
  • Yongning Zhu and Robert Bridson. Animating Sand as a Fluid. ACM Transactions on Graphics, 2005.
    Cited in lecture for FLIP.
  • Matthias Müller, Ten Minute Physics, how to write a FLIP water simulator: video, demo, notes.
  • Chenfanfu Jiang, Craig Schroeder, Andrew Selle, Joseph Teran, and Alexey Stomakhin. The Affine Particle-in-Cell Method. ACM Transactions on Graphics, 2015.
  • Alexey Stomakhin, Craig Schroeder, Lawrence Chai, Joseph Teran, and Andrew Selle. A Material Point Method for Snow Simulation. ACM Transactions on Graphics, 2013.
• Solving the pressure Poisson equation
  • Start with the Numerical linear algebra group under General, especially Shewchuk's conjugate gradient notes, and with Bridson's notes and book above.
  • Aleka McAdams, Eftychios Sifakis, and Joseph Teran. A Parallel Multigrid Poisson Solver for Fluids Simulation on Large Grids. Symposium on Computer Animation, 2010.

Forward Kinematics and Articulated Characters (Lecture 11)

• Kevin M. Lynch and Frank C. Park. Modern Robotics: Mechanics, Planning, and Control. Cambridge University Press, 2017. Free preprint from the authors.
  Chapter 3 (Rigid-Body Motions) and Chapter 4 (Forward Kinematics). Chapter 6 covers inverse kinematics.
• Steven M. LaValle. Planning Algorithms. Cambridge University Press, 2006. Free online from the author.
  Chapter 3 (Geometric Representations and Transformations): rigid-body transformations, 3D rotations, and transforming kinematic chains and trees.
• Ijaz Akhter and Michael J. Black. Pose-Conditioned Joint Angle Limits for 3D Human Pose Reconstruction. IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2015. Open access PDF.
  Cited in lecture on how complex human joint limits are.
• Yifeng Jiang and C. Karen Liu. Data-Driven Approach to Simulating Realistic Human Joint Constraints. IEEE International Conference on Robotics and Automation (ICRA), 2018. arXiv version.
  The learned, pose-dependent joint limits shown in lecture.
• Fernando de Goes, William Sheffler, and Kurt Fleischer. Character Articulation through Profile Curves. ACM Transactions on Graphics, 2022.
  CurveNets, cited in lecture as an example of how film character rigs go beyond a skeleton.

Inverse Kinematics (Lecture 12)

• Samuel R. Buss. Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares Methods. University of California, San Diego. Free PDF from the author.
  A short introduction to the Jacobian methods the lecture builds on: the Jacobian transpose, the pseudoinverse, and damped least squares, which handles near-singular poses.
• Andreas Aristidou and Joan Lasenby. FABRIK: A Fast, Iterative Solver for the Inverse Kinematics Problem. Graphical Models, 2011. Author's FABRIK page and PDF, both linked from the slides.
  The heuristic alternative shown in lecture, which finds each joint position by locating a point on a line.
• Ladislav Kavan, Steven Collins, Jiří Žára, and Carol O'Sullivan. Skinning with Dual Quaternions. Symposium on Interactive 3D Graphics and Games, 2007.
  A widely used alternative to linear blend skinning that avoids the twisting (candy-wrapper) artifact the lecture demonstrates.
• Matthew Loper, Naureen Mahmood, Javier Romero, Gerard Pons-Moll, and Michael J. Black. SMPL: A Skinned Multi-Person Linear Model. ACM Transactions on Graphics, 2015. Project page.
  Cited in lecture as an example of learning-based skinning.
• Yiming Xie, Varun Jampani, Lei Zhong, Deqing Sun, and Huaizu Jiang. OmniControl: Control Any Joint at Any Time for Human Motion Generation. International Conference on Learning Representations, 2024. arXiv version.
  The text-to-motion example cited in lecture, which also lets users constrain any joint at any time.
• Setareh Cohan, Guy Tevet, Daniele Reda, Xue Bin Peng, and Michiel van de Panne. Flexible Motion In-betweening with Diffusion Models. SIGGRAPH 2024.
  The in-betweening example that closes the lecture.
• Also listed above: Lynch and Park, Modern Robotics (under Lecture 11), whose Chapter 6 is a full treatment of inverse kinematics, including the Newton-Raphson method.

Keyframe Interpolation (Lecture 13)

• Rick Parent and Andrew Hogue. Basic Animation Techniques, in Computer Animation. Elsevier, 2026.
  A textbook chapter on basic animation techniques.
• Doris H. U. Kochanek and Richard H. Bartels. Interpolating Splines with Local Tension, Continuity, and Bias Control. SIGGRAPH 84.
  TCB splines: Catmull-Rom-style interpolation with tension, continuity, and bias controls at each key.
• John Lasseter. Principles of Traditional Animation Applied to 3D Computer Animation. SIGGRAPH 87.
  The principles of traditional animation, such as timing and slow in and slow out, applied to computer animation.
• Freya Holmér. The Continuity of Splines. Video.
  A long (about 74-minute) animated video on splines and their continuity.
• Also listed under General: Solomon, Numerical Algorithms, for the tridiagonal linear system that the natural C2 interpolating spline solves for its tangents.

3D Orientation (Lecture 14)

• Stanley W. Shepperd. Quaternion from Rotation Matrix. Journal of Guidance and Control, 1978.
  The numerically stable conversion cited in lecture, which picks a formula based on which term is largest.
• Yi Zhou, Connelly Barnes, Jingwan Lu, Jimei Yang, and Hao Li. On the Continuity of Rotation Representations in Neural Networks. CVPR, 2019.
  The two-column representation covered in Part 5.
• Ken Shoemake. Animating Rotation with Quaternion Curves. SIGGRAPH 85.
  The paper that introduced slerp, the spherical linear interpolation of quaternions used in lecture.
• F. Sebastian Grassia. Practical Parameterization of Rotations Using the Exponential Map. Journal of Graphics Tools, 1998.
  On using the exponential map, the axis-angle parameterization of the lecture, in practice.
• Doug L. James and Christopher D. Twigg. Skinning Mesh Animations. ACM Transactions on Graphics, 2005. Project page.
  Cited in lecture as an example of clustering rotation sequences.
• Also listed under Lecture 11: Lynch and Park, Modern Robotics, whose Chapter 3 covers rotation matrices and exponential coordinates for rotation.