Rigid body kinematics in an intuitive group-theoretic approach, as completely as possible: Part I Rotational phenomena

TL;DR

This paper advances a complete, intuitive group theoretic treatment of rigid body rotations within the $O(3)$ and $SO(3)$ framework, emphasizing an active viewpoint. It provides a systematic development of rotation representations including Euler and Tait-Bryan angles, axis angle, Rodrigues formula, and the links to Lie algebras and exponential mappings, complemented by the SU(2) description and spinor perspective. Key contributions include explicit relations among active and passive representations, transformation rules between bases, and a thorough treatment of continuous rotations and gimbal lock, all aimed at building a rigorous foundation for Part II on general rigid body motion and manifold structures. The work has practical impact for researchers requiring a transparent, derivation-based understanding of rotations and their representations in physics and engineering, with broad connections to topology and quantum spin systems.

Abstract

This paper focuses on rotational phenomena of rigid body kinematics. It discusses them in a group-theoretic approach as completely as possible, using methods and notations as intuitive as possible. With a review of current literature, this article also covers some original parts that remain largely unexplored.

Figures (5)

• Figure 1: Euler angles.
• Figure 2: Tait-Bryan angles.
• Figure 3: A gyroscope.
• Figure 4: Composition of rotations.
• Figure 5: The non-commutative property.

Theorems & Definitions (79)

• Theorem 1
• proof
• Definition 1
• Theorem 2
• proof
• Definition 2
• Definition 3
• Theorem 3
• proof
• Theorem 4