Mathematics for Game Development

5.

Advanced Rotation: Quaternions

5.1.

Limitations of Matrix Rotations

5.1.1.

5.1.1.1.

Definition and Causes

5.1.1.2.

Euler Angle Problems

5.1.1.3.

Loss of Degrees of Freedom

5.1.2.

Interpolation Issues

5.1.2.1.

Linear Interpolation Problems

5.1.2.2.

Non-Smooth Rotation Paths

5.1.3.

Storage and Computation Costs

5.1.3.1.

5.1.3.2.

Computational Overhead

5.2.

Quaternion Fundamentals

5.2.1.

Quaternion Structure

5.2.1.1.

Four-Component Representation

5.2.1.2.

Scalar and Vector Parts

5.2.1.3.

Mathematical Notation

5.2.2.

Quaternion Interpretation

5.2.2.1.

Unit Quaternions

5.2.2.2.

Rotation Representation

5.2.2.3.

Relationship to Complex Numbers

5.2.3.

Axis-Angle Representation

5.2.3.1.

5.2.3.2.

5.2.3.3.

Conversion to Quaternion

5.2.4.

Comparison with Euler Angles

5.2.4.1.

Advantages of Quaternions

5.2.4.2.

Disadvantages of Quaternions

5.3.

Quaternion Operations

5.3.1.

Basic Quaternion Arithmetic

5.3.1.1.

Addition and Subtraction

5.3.1.2.

Scalar Multiplication

5.3.1.3.

Magnitude Calculation

5.3.2.

Quaternion Multiplication

5.3.2.1.

Hamilton Product

5.3.2.2.

Non-Commutativity

5.3.2.3.

Rotation Composition

5.3.3.

Quaternion Normalization

5.3.3.1.

Unit Quaternion Requirement

5.3.3.2.

Normalization Process

5.3.4.

Quaternion Conjugate

5.3.4.1.

Inverse Rotation

5.3.5.

Quaternion Inverse

5.3.5.1.

5.3.6.

Vector Rotation with Quaternions

5.3.6.1.

Rotation Formula

5.3.6.2.

5.4.

Quaternion Interpolation

5.4.1.

Linear Interpolation Limitations

5.4.1.1.

Non-Constant Angular Velocity

5.4.1.2.

5.4.2.

Spherical Linear Interpolation

5.4.2.1.

SLERP Algorithm

5.4.2.2.

Constant Angular Velocity

5.4.2.3.

Smooth Rotation Paths

5.4.3.

5.4.3.1.

Animation Systems

5.4.3.2.

Camera Movement

5.4.3.3.

Character Rotation

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6. Trigonometry in Games