Equiform kinematics interpreted via dual-space geometry

Ali Abdela Ali; Anas A. M. Arafa; Fahad Abdulaziz Alsidrani; Ali Abdela Ali; Anas A. M. Arafa; Fahad Abdulaziz Alsidrani

doi:10.3934/math.20261182

AIMS Mathematics

2026, Volume 11, Issue 9: 29796-29815. doi: 10.3934/math.20261182

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Research article

Equiform kinematics interpreted via dual-space geometry

Department of Mathematics, College of Science, Qassim University, Buraydah, 52571, Saudi Arabia

Received: 26 April 2026 Revised: 30 August 2026 Accepted: 03 September 2026 Published: 15 September 2026
MSC : 53A17, 53A25, 70B10, 70E15

This paper investigates equiform spatial kinematics using dual-vector line geometry. Oriented lines in Euclidean three-space are represented by unit dual vectors or equivalently by normalized Plücker coordinates consisting of a unit direction vector and an orthogonal moment vector. A consistent moving-frame formulation is developed for transformations combining a dual orthogonal motion with a time-dependent uniform scale factor. The condition under which the tangential component of acceleration vanishes is derived while retaining all derivatives of the scale factor. The corresponding line loci are then characterized in Plücker space, with particular attention to Bresse complexes, line congruences, and the associated ruled surfaces. Explicit parametric expressions are obtained for a selected dual spherical motion, and the rigid-body formulation is recovered as the special case of a constant unit scale. The geometric significance of the selected equiform motion is demonstrated through the resulting Bresse complexes, line congruences, and ruled-surface families. Reproducible computational visualizations illustrate the resulting geometric structures. The developed framework provides a line-geometric approach to equiform and rigid-body kinematics with potential applications in mechanism theory, robotics, and computer-aided geometric design.
- computer simulation,
- dual-vector calculus,
- equiform motion,
- Bresse complex,
- Plücker coordinates,
- line congruence,
- ruled surface,
- spatial kinematics
Citation: Ali Abdela Ali, Anas A. M. Arafa, Fahad Abdulaziz Alsidrani. Equiform kinematics interpreted via dual-space geometry[J]. AIMS Mathematics, 2026, 11(9): 29796-29815. doi: 10.3934/math.20261182

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Abstract

This paper investigates equiform spatial kinematics using dual-vector line geometry. Oriented lines in Euclidean three-space are represented by unit dual vectors or equivalently by normalized Plücker coordinates consisting of a unit direction vector and an orthogonal moment vector. A consistent moving-frame formulation is developed for transformations combining a dual orthogonal motion with a time-dependent uniform scale factor. The condition under which the tangential component of acceleration vanishes is derived while retaining all derivatives of the scale factor. The corresponding line loci are then characterized in Plücker space, with particular attention to Bresse complexes, line congruences, and the associated ruled surfaces. Explicit parametric expressions are obtained for a selected dual spherical motion, and the rigid-body formulation is recovered as the special case of a constant unit scale. The geometric significance of the selected equiform motion is demonstrated through the resulting Bresse complexes, line congruences, and ruled-surface families. Reproducible computational visualizations illustrate the resulting geometric structures. The developed framework provides a line-geometric approach to equiform and rigid-body kinematics with potential applications in mechanism theory, robotics, and computer-aided geometric design.

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