Curvature Theory on Contact and Transfer Characteristics of Enveloping Curves

Abstract

Abstract In differential geometry, a curve is characterized by the curvature properties and so is a point trajectory in curvature theory. However, due to the rolling and sliding between contact curves, the characterization of enveloping curves embedded on rigid bodies in relative motion is not complete without the transfer rate of the contact point. This paper presents the new perspectives on not only the curvature characteristics but also the transfer characteristics between enveloping curves embedded on rigid bodies. The paper contains three parts. First, a pair of enveloping curves can be described by a point trajectory which traces two curves simultaneously. Second, the paper treated instant center as a tracing point, which traces the moving centrode and the fixed centrode on each of the bodies. This characterizes the rolling between centrodes and the transfer rate of the instant center on each of centrodes. The treatment can extend the instant center transfer velocity to any higher order. The third part is about the rolling and sliding between enveloping curves embedded on rigid bodies motion. A kinematic model to describe the transfer rate of contact point based on the motion of the instant center is proposed. The detailed curvature and transfer properties of line- and circle-envelopes are presented. Due to the simple treatment, this might be the first paper that successfully achieved the curvature properties of circle-envelopes. The method offers the vital kinematic aspect of enveloping curves distinctly different from the conventional curvature theory and enveloping theory.