The Forward Kinematics of Rolling Contact of Timelike Surfaces With Spacelike Trajectory Curves

Mehmet Aydinalp, Mustafa Kazaz, Hüseyin Uğurlu


In this paper, we investigate the forward kinematics of spin-rolling motion without sliding of one timelike surface on another timelike surface along the spacelike contact trajectory curves of the surfaces in Lorentzian 3-space. A Darboux frame method is adopted to develop instantaneous kinematics of spin-rolling motion, which occurs in a nonholonomic system. Then, new kinematic formulations of spin-rolling motion of timelike moving surface with regards to contravariant vectors, rolling velocity, and geometric invariants are obtained. Namely, the translational velocity formulation of an arbitrary point and the equation of the angular velocity formulation on the timelike moving surface are derived. The equation, which is represented with geometric invariants, can be easily applied to arbitrary spacelike parametric surface and spacelike contact trajectory curve and can be differentiated to any order. The influence of the relative curvatures and torsion on spin-rolling kinematics is clearly presented.


Lorentzian 3-Space; Darboux frame; Forward kinematics; Pure-rolling; Rolling contact; Spin-rolling

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