This course includes:
(updated Oct 2021)

35 hours of ondemand video

Downloadable resources and exercises

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Is this course for you?
This course is a complete immersion into the world of 2D game physics. You'll learn all the required math, going from basic vector math, to matrices, until we touch some calculus to solve physics constraints. This course is aimed at students that already know some programming, but want to learn how a simple 2D physics engine (like Box2D or Chipmunk2D) work under the hood.
Prerequisites
Before taking this course, you must have:

A basic knowledge of coding (variables, ifelse, loops, functions, OOP)

Some experience with either Linux, Windows, or macOS
Particle Physics
We'll begin our journey looking at simple particle physics. We'll start by learning what simulation via software really is, how it works, review the required math, and look at the fundamentals of movement.
RigidBody Physics
Once we get the basics out of the way, we'll start looking at rigidbody physics. This means we'll learn about different shapes, handling angular movement, and also looking at collision detection and collision resolution. This is a super important topic and we'll have some fun poking rigidbodies around.
Constraints
We'll also take a deep look at constrained rigidbody simulation, which will help us add some different types of objects to our engine (like joints, anchor points, and ragdolls). Ultimately, constraints will help improve the stability of our engine, and they are an opportunity for us to look at some interesting math topics, including a little bit of calculus.
Course Content

1
Introduction

2
Vectors
 Vector & Scalar Quantities FREE PREVIEW
 Using the P5js Web Editor
 Visualizing Errors in the Web Editor
 Coding a Vec2 Class
 Quiz: Scalars & Vectors

3
Vector Operations
 Vector Magnitude
 Vector Addition & Subtraction
 Methods for Vector Addition & Subtraction
 Vector Equality
 Scaling Vectors
 Static Methods
 Applications of Vector Addition & Subtraction
 Quiz: Vector Arithmetic

4
Dot Product & Cross Product
 Dot Product
 Is the Dot Product Commutative?
 Dot Product Length
 Quiz: Dot Product
 Cross Product
 Coding the Cross Product Method
 Exercise: Perpendicular 2D Vector
 Perpendicular 2D Vector
 Exercise: Vec3 Methods
 Vec3 Methods
 Applications of Dot Product & Cross Product
 Quiz: Cross Product

5
Normalizing Vectors
 Vector Normalization
 Coding the Normalization Method

6
Vector Transformations
 Scaling, Translating, and Rotating Vectors
 Quick Review of Sine & Cosine
 Vector Rotation Proof (xcomponent)
 Vector Rotation Proof (ycomponent)
 Coding the Vec2 Rotate Method
 Concluding our JavaScript Vector Class

7
Vectors in C++
 Vec2 C++ Header File
 A Quick Look at C++ Vec2 Syntax
 Vec2 Operator Overloading

8
Starting our C++ Project
 Technologies & Dependencies
 Folder Structure
 Initial Project Files
 Compiling using GCC & Linux
 Makefile
 Configuring Visual Studio on Windows

9
Particle Physics
 Introduction to Particle Physics
 Particle Class
 Particle Velocity
 Exercise: Operator + to Add Vectors
 Using the + Operator to Add Vectors
 Controlling our Framerate
 Framerate Independent Movement
 Clamping Invalid DeltaTime Values
 Quiz: Framerate & DeltaTime
 Moving in a Constant Velocity
 What About Mass?
 Changing the Particle's Velocity
 Keeping the Particle Inside the Window

10
Simulating Movement
 Constant Acceleration
 Discrete vs. Continuous
 Integration & Movement Simulation
 Different Notations for Differentiation
 Integral as Continuous Sum
 Different Integration Methods
 Particle Integrate Function
 Quiz: Integration

11
Applying Forces
 Applying Forces to Particles
 Kinematics vs. Kinetics
 Function to Add Force
 Particles with Different Mass
 The Weight Force
 Inverse of the Mass
 Quiz: Forces & Mass
 Applying Forces with the Keyboard

12
Drag & Friction Forces
 Drag Force
 Drag Force Function
 Handling Mouse Clicks with SDL
 Unexpected Drag Behavior
 Friction Force
 Friction Force Function

13
Gravitational Attraction
 Gravitational Attraction Force
 Gravitational Attraction Force Function

14
Spring Forces
 Spring Force
 Spring Force Function
 Exercise: Spring Forces
 Multiple Particles: Chain
 Multiple Particles: Soft body
 Soft Bodies & Verlet Integration

15
RigidBody Physics
 RigidBodies
 Shapes
 Shape Class
 Shape Class Implementation
 Circle Shape
 Angular Velocity & Angular Acceleration
 Torque & Moment of Inertia
 Different Shapes & Moment of Inertia
 Circle Shape Angular Motion
 Angular Velocity DIrection
 Quiz: Angular Motion

16
Shapes & Vertices
 Box Vertices
 Local Space vs. World Space
 Body Update Function
 Why Not a Shape Draw Function?
 No Draw Method in the Shape Class

17
Colliding Circles
 CircleCircle Collision Detection
 CircleCircle Collision Class
 CircleCircle Collision Implementation
 Collision Contact Information
 Collision Information Code
 Broad Phase & Narrow Phase
 The Projection Method
 Objects with Infinite Mass
 FloatingPoint Comparison

18
Impulse & Momentum
 Impulse Method & Momentum
 Impulse
 Linear Impulse
 Coding the Linear Impulse Method
 Is Linear Collision Response Enough?
 Quiz: Collisions

19
Colliding Boxes
 AABB Collision Detection
 SAT: Separating Axis Theorem
 Finding Minimum Separation with SAT
 PolygonPolygon Collision Code
 Code to Find SAT Minimum Separation
 Refactoring the SAT Separation Function
 Finding Extra Collision Information with SAT
 PolygonPolygon Collision Information

20
Angular Collision Resolution
 Linear & Angular Velocity At Point
 PostCollision Velocity At Point
 Computing Linear & Angular Impulse
 Collision Distance Vectors RaRb
 2D Cross Product
 Coding the Impulse Along Normal
 Exercise: Impulse Along Tangent
 Friction Impulse Along Tangent
 Removing Window Boundaries Check
 Collision Resolution in 3D

21
Colliding Circles & Polygons
 CirclePolygon Collision Detection
 Finding Polygon's Nearest Edge with Circle
 Exercise: CirclePolygon Edge Regions
 CirclePolygon Collision Information
 CirclePolygon Collision Resolution
 Exercise: Polygons with Multiple Vertices
 Polygon with Multiple Vertices

22
Rendering Textures
 Loading SDL Textures
 Rendering Circle Texture
 Textures Inside RigidBodies?

23
The World Object
 World Class
 Implementing World Functions

24
Collision Instability
 Refactoring Function to Update Vertices
 Local Solvers vs. Global Solvers
 A Naive Iterative Positional Correction

25
Constraints
 Constrained RigidBody Physics
 Position vs. Velocity Constraints
 Example Velocity Constraint & Bias Factor
 Example Distance Constraint & Bias Factor
 Constraint Forces & Constrained Movement
 ForceBased vs. ImpulseBased Constraints
 The Constraint Class
 Position vs. Velocity Constraint

26
Matrices
 VecN Class
 Implementing VecN Functions
 VecN Operator Overloading
 Matrices
 MatMN Class
 Matrix Transpose
 Matrix Multiplication
 Matrix Multiplication Function
 Seeing Beyond the Matrix

27
Distance Constraint
 Generalized Velocity Constraint
 Solving Violated Velocity Constraints
 Constraint Class Inheritance
 Distance Constraint
 Joint Constraint Class
 Joint Constraint vs. Distance Constraint
 Converting World Space to Local Space
 Inverse Rotation Matrix
 World List of Constraints
 Refactoring Body Update
 Deriving the Distance Jacobian
 Populating the Distance Jacobian
 Solving System of Equations (Ax=b)
 GaussSeidel Method
 Constrained Pendulum
 Quiz: Distance Constraints

28
System of Constraints
 Solving System of Constraints Iteratively
 Warm Starting
 Adding the Bias Term
 Exercise: Ragdoll
 Ragdoll with Joint Constraints
 Preventing NaN Errors

29
Penetration Constraint
 Penetration Constraint
 Deriving the Penetration Jacobian
 Penetration Constraint Class
 Solving Penetration Constraints
 Linear Complementary Problem
 Penetration Warm Starting
 A Quick Note on Penetration Warm Starting
 Penetration Constraint Friction
 Clamping Friction Magnitude Values
 Penetration Constraint Bounciness

30
Stack of Boxes
 Unstable Stack of Boxes
 Allowing for Multiple Contact Points
 Reference & Incident Edges
 Finding Incident Edge
 Getting Ready for Clipping
 Clipping Function
 Testing MultiContact Boxes
 Testing Multiple Objects & Constraints

31
Optimization Techniques
 Contact Caching
 Continuous Collision Detection
 Computing the Moment of Inertia of Polygons
 Broad & Narrow Split

32
Conclusion
 Conclusion & Next Steps
 Copyright & Content Distribution
 Moving Forward
Instructor

Senior Lecturer
Gustavo Pezzi
Gustavo teaches computer science and mathematics at BPP University, London. He studies how teaching game programming can help enhance awareness and understanding of basic mathematics and physics. He is also a professional programmer with several years of experience in the computing industry. His academic path includes institutions such as Pittsburg State University, City University of London, and University of Oxford.