Curvature Singularity of Space Curves and Its Relationship to Computational Mechanics

Abstract

Curve geometry plays a fundamental role in many aspects of analytical and computational mechanics, particularly in developing new data-driven science (DDS) approaches. Furthermore, curvature and torsion of space curves serve as deformation measures that need to be properly interpreted, shedding light on the significance of relationship between differential-geometry curve framing methods and computational-mechanics motion description. Alternate space-curve framing methods were proposed to address the existence of Frenet frame at isolated zero-curvature points. In this paper, both mechanics and differential-geometry approaches are used to establish Frenet-frame continuity and the existence of Serret-Frenet equations at curvature-vanishing points for curves with arbitrary parameterization. Frenet–Euler angles, referred to for brevity as Frenet angles, are used to define curve geometry, with particular attention given to the definition of Frenet bank angle used to prove the existence of curve normal and binormal vectors at curvature-vanishing points. Solving curvature-singularity problem and using mechanics description based on Frenet angles contributes to successful development and computer implementation of new DDS approaches based on analysis of recorded motion trajectories (RMT). Centrifugal-inertia force is always in direction of curve normal vector, and in most applications, this force is continuous and approaches zero value as curve curvature approaches zero. Discontinuity in definition of Frenet frame can negatively impact the quality of numerical results that define RMT curves. The study also demonstrates that Frenet-frame curvature singularity can be solved without need for integrating curve torsion, which is not, in general, an exact differential.