#### 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.
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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# Transforming vectors

To multiply a vector and a quaternion, you first have to turn the vector into a pure quaternion. What is a pure quaternion? It's a quaternion whose `W` component is `0` and the vector part is normalized. Assume you have a quaternion, q, and a vector, v. First, turn v into a pure quaternion, represented as v':

Next, multiply q by v' then multiply the result by the inverse of q. The result of this multiplication is a pure quaternion whose vector part contains the rotated vector. The quaternion becomes the following:

Why is v' multiplied by q and then q-1? Multiplying by q will rotate the vector twice as much as the rotation of q. Multiplying by q-1 brings the vector back into the expected range. This formula can be simplified further.

Deriving this formula is outside the scope of this book. Given a quaternion, q, and

a vector, v, the simplified vector quaternion multiplication formula is listed as follows.

qv refers to...