Quaternion-Based Representation of 3D Space

Quaternions serve as an efficient mathematical tool for 3D space representation, widely applied in computer graphics, robotics, and game development. Compared to traditional Euler angles or rotation matrices, quaternions offer distinct advantages.

Compact Expression Quaternions consist of a real component and three imaginary components, typically represented as q = w + xi + yj + zk, where w denotes the real part and (x, y, z) represent the imaginary components. This compact structure makes quaternions more efficient for storage and computation. In code implementation, quaternion operations often leverage optimized libraries like Eigen or GLM for vectorized calculations.

High Computational Precision Unlike Euler angles that may suffer from gimbal lock issues, quaternions maintain numerical stability during continuous rotations without singularities. Additionally, quaternion interpolation techniques like SLERP (Spherical Linear Interpolation) excel in animation and smooth pose transitions. A typical SLERP implementation involves normalizing quaternions and calculating the interpolation parameter t using trigonometric functions for smooth rotation paths.

Reduced Storage Requirements Quaternions require only 4 floating-point numbers to fully describe 3D rotations, while rotation matrices need 9 elements. This gives quaternions advantages in memory usage and computational efficiency, particularly beneficial for real-time systems and large-scale 3D scene processing. In programming, quaternion classes often include methods for conversion to/from rotation matrices and Euler angles to facilitate integration with existing pipelines.

These characteristics make quaternions an ideal choice for 3D rotation representation, extensively used in virtual reality, pose estimation, and motion control applications. Common implementations include quaternion multiplication for combining rotations and conjugate operations for inverse transformations.