#### Overview of this book

Animation is one of the most important parts of any game. Modern animation systems work directly with track-driven animation and provide support for advanced techniques such as inverse kinematics (IK), blend trees, and dual quaternion skinning. This book will walk you through everything you need to get an optimized, production-ready animation system up and running, and contains all the code required to build the animation system. You’ll start by learning the basic principles, and then delve into the core topics of animation programming by building a curve-based skinned animation system. You’ll implement different skinning techniques and explore advanced animation topics such as IK, animation blending, dual quaternion skinning, and crowd rendering. The animation system you will build following this book can be easily integrated into your next game development project. The book is intended to be read from start to finish, although each chapter is self-contained and can be read independently as well. By the end of this book, you’ll have implemented a modern animation system and got to grips with optimization concepts and advanced animation techniques.
Table of Contents (17 chapters)
Preface
Chapter 1: Creating a Game Window
Free Chapter
Chapter 2: Implementing Vectors
Chapter 3: Implementing Matrices
Chapter 4: Implementing Quaternions
Chapter 5: Implementing Transforms
Chapter 6: Building an Abstract Renderer
Chapter 7: Exploring the glTF File Format
Chapter 8: Creating Curves, Frames, and Tracks
Chapter 9: Implementing Animation Clips
Chapter 10: Mesh Skinning
Chapter 11: Optimizing the Animation Pipeline
Chapter 12: Blending between Animations
Chapter 13: Implementing Inverse Kinematics
Chapter 14: Using Dual Quaternions for Skinning
Chapter 15: Rendering Instanced Crowds
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# Inverting a matrix

Multiplying a matrix by its inverse will always result in the identity matrix. An inverted matrix has the opposite mapping of the non-inverted matrix. Not all matrices have an inverse. Only matrices with a non-zero determinant can be inverted.

Inverting matrices is an important operation; the view matrix that is used to transform three-dimensional objects to be displayed on-screen is the inverse of the camera's position and rotation. Another place where inverted matrices become important is skinning, which will be covered in Chapter 10, Mesh Skinning.

Finding the inverse of a matrix is rather complicated as it needs other support functions (such as transpose and adjugate). In this section, you will build these support functions first and then build the inverse function after they have all been built. So firstly, we need to transpose the matrix.

## Transpose

To transpose a matrix, flip every element of the matrix across its main diagonal. For example...