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Efficient generation of random rotation matrices in four dimensions.

Jakob Tómas Bullerjahn1, Balázs Fábián1, Gerhard Hummer1,2

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This study presents an efficient algorithm for generating random four-dimensional (4D) rotation matrices. This method ensures unbiased, uniform sampling of the SO(4) group, aiding in molecular simulations and robotics.

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Area of Science:

  • Mathematics
  • Physics
  • Computer Science

Background:

  • Four-dimensional (4D) rotations are crucial in robotics, computer vision, and rigid-body mechanics.
  • Transforming between equimomental systems of point masses is a key application in mechanics.
  • Efficient sampling of the SO(4) group is needed for various computational tasks.

Purpose of the Study:

  • To develop an efficient algorithm for generating random 4D rotation matrices.
  • To cover an arbitrary, predefined range of rotation angles.
  • To enable unbiased and uniform sampling over the SO(4) group.

Main Methods:

  • Algorithm development for random 4D rotation matrix generation.
  • Integration with Monte Carlo methods for SO(4) group sampling.
  • Construction of matrices for unbiased and uniform sampling.

Main Results:

  • An efficient algorithm for generating random 4D rotation matrices is provided.
  • The algorithm generates matrices covering a specified range of rotation angles.
  • Repeated application of the generated matrices leads to uniform sampling over SO(4).

Conclusions:

  • The developed algorithm efficiently generates random 4D rotations for applications in robotics and mechanics.
  • It facilitates optimized mass partitioning in coarse-grained molecular simulation models.
  • Stable time integration in molecular dynamics is improved through this method.