Curvature theory on contact and transfer characteristics of enveloping curves

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 (or shifting) characteristics of the contact point. This paper presents the new perspectives and the first comprehensive theory on not only the curvature characteristics but also the transfer characteristics between enveloping curves embedded on rigid bodies. The paper contains three parts. In the first part, a point traces a curve on the moving body and consequently traces a curve on the fixed body. Both generated curves form a pair of enveloping curves. This part establishes the foundation of the paper. Because each enveloping curve is treated as a point trajectory. One may examine all aspects of the enveloping process. Essentially this unmasks the veil that has hindered further understanding and observation of the enveloping behavior beyond the fundamental curvature. It represents a significant advancement on envelope theory. In the second part, the moving point is the instant center, which traces the moving centrode on the moving body and the fixed centrode on the fixed body. It characterizes the rolling between centrodes and the transfer characteristics of the instant center on each centrode. It not only offers a simple way to treat the instant center transfer (shifting) velocity but also successfully extends it to any order of motion. The third part is about the rolling and sliding of between enveloping curves embedded on rigid bodies in relative motion. It addresses the transfer characteristics of the contact on each of the contact curves for the first time. The transfer characteristics are functions of the rigid body motion characteristics. This part offers the vital kinematic aspect of enveloping curves distinctly different from the conventional curvature theory that addresses an individual curve. The proposed enveloping curvature theory offers an important model to account for all aspects of the contact and removes the veil that blurs the contact behavior caused by the traditional envelope definition of F(λ, x, y) =^∂F _∂λ (λ, , ) = 0. This is a kinematic solution for envelopes. The proposed theory is illustrated with an example of two rolling cylinders.