Physics-based Animation 4860-1081 2025S

Lecture at graduate school of information science and technology in the university of Tokyo, spring semester, 2025

ITC-LMS

For Zoom link, Slack, and GitHub Classroom invitations

• https://utol.ecc.u-tokyo.ac.jp/lms/course?idnumber=2025_4886_4860-1081_01

Instructor

Dr. Nobuyuki Umetani

• email: n.umetani@gmail.com

• url: http://www.nobuyuki-umetani.com/

• lab's website: https://cgenglab.github.io/en/

• Teaching assistants: Kenji Tojo (https://kenji-tojo.github.io/), Yuhan Wu (https://yoharol.github.io/pages/myself/)

Time

• Monday 2rd period, 10:25pm - 12:10pm

Course Description

Computer-generated images are everywhere in movies, video games, and VR. This course is an introduction to the techniques to animate objects in computer graphics based on the law of physics. The aim of the course is to get familiar with applied mathematics such as linear algebra, vector analysis, partial differential equations, tensor mechanics, variational principle, optimization, and numerical analysis through the animation techniques for particle systems, rigid bodies, and elastic bodies. There are C#/Rust/Python programming assignments to acquire research-oriented graphics programming skills. The students also learn basics use of the computer graphics software Unity and Blender.

Topics:

• mass-spring simulation
• rigid body simulation
• elastic body simulation
• cloth and hair modeling & simulation
• collision-detection using spatial hashing
• finite boundary method

Lecture Schedule

Day	Topic	Assignment	Slide
(1) Apr. 7	Introduction Data structure for simulation, Explicit/implicit shape representation		[1,2]
(2) Apr. 21	Programming 1 Introduction to Rust language, GitHub	task00	[3,4]
(3) Apr. 28	Time Integration Backward & forward Euler method, particle system, Barnes–Hut method	task01	[5,6]
(4) May 8	Newtonian Mechanics	task02	[7]
(5) May 12	Collision Detection principal component analysis sort & sweep method, bounding volume hierarchy	task03	[8,9]
(6) May 19	Linear System matrix data structure, Conjugate gradient method	task04	[10,12]
(7) May 26	Simple Elastic Energy Newton-Raphson method mass-spring system	task05	[11,13]
(8) June 9	Programming 2 Introduction to Unity & C#	task06	[14,15,16]
(9) June 16	Dynamic Simulation Variational time integration	task07	[17,18]
(10) June 23	Optimization with Constraint Lagrange multiplier method	task08	[19]
(11) June 30	Rotation Rotation representation	task09	[20]
(12) July 7	Rigid Body Dynamics inertia tensor, angular velocity	task10	[21,22]
(13) July 14	Continuum Mechanics tensor, finite element method		[23,24,25]

Slides

• [1] Introduction
• [2] Data Structure
• [3] Rust
• [4] Git & GitHub
• [5] Time Integration
• [6] Floating Point Number
• [7] Newtonian Mechanics
• [8] Jagged Array
• [9] Collision Detection
• [10] Matrix Data Structure
• [11] Linear System Solver
• [12] Jacobian & Hessian
• [13] Grid & Mesh Interpolation
• [14] Unity
• [15] Mass Spring System
• [16] Vector Differentiation
• [17] Optimization
• [18] Variational Integration
• [19] Optimization with Constraints
• [20] Rotation
• [21] Angular Velocity
• [22] Rigid Body Approximation
• [23] Tensor
• [24] Finite Element Method
• [25] Geometric Deformation

Grading

• 20% lecture attendance
  • Attendance is counted based on writing a secret keyword on LMS. The keyword is announced for each lecture.
• 80% small assignments
  • see below

Assignments

There are many small programming assignments. To do the assignments, you need to create your own copy of this repository through GitHub Classroom. These assignements needs to be submitted using pull request functionality of the GitHub. Look at the following document.

How to Submit the Assignments

The final submission deadline is at 15:00 pm on July 17th, 2025. The late submission won't be accepted after this final submission deadline.

Task ID	Language	Title (tentative)	Thumbnail
task00	Rust	Building Rust project & submitting GitHub pull request
task01	Rust	Implicit Time Integration
task02	Rust	Linear Momentum Conservation
task03	Rust	Acceleration of N-body Simulation
task04	Rust	Accelerated nearest search using Kd-Tree
task05	Rust	Solving Laplace equation with Gauss-Seidel Method
task06	C#	Gradient Descent for Mass-Spring Simulation
task07	C#	Dynamic Mass-spring System using Variational Euler Time Integration
task08	C#	Controlling Volume of a Mesh using Lagrange-Multiplier Method
task09	C#	Rotation and Energy Minimization
task10	C#	Simulation of Rigid Body Precession

Policy

• Do the assignment by yourself. Don't share the assignments with others.
• Don't post the answers of the assignment on Slack
• Late submission of an assignment is subject to grade deduction
• Score each assignment will not be open soon (instructor needs to adjust weights of the score later)
• You can use AIs tool for coding.

Reading Material

• Ten Min Physics (Youtube channel)
• Physically Based Modeling: Principles and Practice, Siggraph '97 Course notes by Dr. Baraff
• Physics-Based Animation by Kenny Erleben et al. (free textobook about rigid body dynamics)
• Dynamic Deformables: Implementation and Production Practicalities, SIGGRAPH 2020 Courses
• Awesome Computer Graphics (GitHub)
• Skinning: Real-time Shape Deformation SIGGRAPH 2014 Course

My Past Lectures

• Applied Computer Graphics 2024S
• Physics-based Animation 2023S
• Applied Computer Graphics 2022S
• Physics-based Animation 2021S