Quaternions: History, Theory, and Implementation
Explore the fascinating world of quaternions, from their historical roots to their powerful applications in rotations, interpolation, and more.
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Introduction to Quaternions
Unit 1: The Genesis of Quaternions
Hamilton's Inspiration
Hamilton's Quaternion Leap
Early Quaternion Applications
The Quaternion Legacy
Quaternions vs Vectors
Unit 2: Quaternion Foundations
Defining Quaternions
Quaternion Algebra
Quaternion Multiplication
Conjugate and Norm
Inverse of a Quaternion
Unit 3: Quaternions and Complex Numbers
Complex Numbers Review
Quaternions as Extension
Complex Subalgebra
Rotations in 2D vs 3D
Applications Compared
Quaternion Arithmetic and Properties
Unit 1: Introduction to Quaternion Arithmetic
What are Quaternions?
Quaternion Addition
Quaternion Subtraction
Scalar Multiplication
Coding Add/Subtract
Unit 2: Quaternion Multiplication and Division
Quaternion Multiplication
Multiplication Examples
Quaternion Division
Division Examples
Coding Multiplication
Unit 3: Norms, Conjugates, and Unit Quaternions
Quaternion Conjugate
Quaternion Norm
Unit Quaternions
Properties and Uses
Coding Norm/Conjugate
Representing Rotations with Quaternions
Unit 1: Quaternion Fundamentals
Intro to Quaternions
Quaternion Algebra
Quaternion Multiplication
Quaternion Conjugate
Unit Quaternions
Unit 2: Quaternion-Rotation Conversion
Axis-Angle to Quat
Quat to Axis-Angle
Rotation Matrix to Quat
Quat to Rotation Matrix
Euler Angles to Quat
Unit 3: Advantages and Avoiding Pitfalls
Avoiding Gimbal Lock
Quat vs. Other Reps
Quaternion Interpolation
Numerical Stability
Double Cover of SO(3)
Quaternion Interpolation and Applications
Unit 1: Introduction to Quaternion Interpolation
Why Interpolate?
Linear Interpolation
Unit Quaternion Review
Dot Product Reminder
Shortest Path
Unit 2: Spherical Linear Interpolation (Slerp)
Slerp Introduction
Slerp Formula
Slerp Implementation
Slerp Benefits
Slerp Limitations
Unit 3: Advanced Interpolation Techniques
Squad Overview
Tangent Quaternions
Squad Formula
Implementing Squad
Benefits of Squad
Unit 4: Applications and Libraries
Graphics Interpolation
Robotics Interpolation
Aerospace Use
Using Libraries
Optimizing Interpolation
Advanced Quaternion Topics and Optimization
Unit 1: Dual Quaternions: Representation and Algebra
Intro to Dual Quaternions
Dual Quaternion Algebra
Normalization
Conversion
DQ to Transformation Matrix
Unit 2: Applications of Dual Quaternions
Skinning
Implementation
Blending
Distance Queries
Kinematics
Unit 3: Advanced Quaternion Topics
Hypercomplex Numbers
Special Quaternions
Rotor Group
Geometric Algebra
Spinors
Unit 4: Quaternion Optimization Techniques
SIMD Intrinsics
Approximation Methods
Precomputation
Data Alignment
Code Profiling