General rule product matrices?

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SUMMARY

The discussion focuses on the mathematical properties of the product of two rotation matrices within the special orthogonal group SO(3). Specifically, it addresses how to express the product R(φ₁)R(φ₂) as a new rotation matrix, detailing the resultant axis and angle of rotation. The topological model of the rotation group is utilized, emphasizing the representation of rotations by an angle φ and an axis n. The reference to quaternions and spatial rotation provides additional context for understanding these transformations.

PREREQUISITES
  • Understanding of rotation matrices in 3D space
  • Familiarity with the special orthogonal group SO(3)
  • Knowledge of quaternion mathematics
  • Basic principles of linear algebra
NEXT STEPS
  • Research the properties of SO(3) and its applications in robotics
  • Learn about quaternion representation of rotations
  • Explore the derivation of the rotation matrix product formula
  • Study the geometric interpretation of rotation matrices in 3D
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Mathematicians, physicists, computer graphics developers, and robotics engineers interested in the mathematical foundations of 3D rotations and transformations.

Yoran91
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Hi everyone,

I'm trying to find a general rule that expresses the product of two rotation matrices as a new matrix.
I'm adopting the topological model of the rotation group, so any rotation which is specified by an angle \phi and an axis \hat{n} is written R(\hat{n}\phi)= R(\vec{\phi}).

So, given two rotation matrices R(\vec{\phi}_1) , R(\vec{\phi}_2) \in SO(3), what can be said about the product R(\vec{\phi}_1)R(\vec{\phi}_2)?

What is are the axis and the amount of degrees that specifies this rotation matrix?
 
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