Abstract:

First, we define three kinds of measures of angles in order to handle rolling and rotation.

In Part One, we give the relations among them:

\(d \theta = d \varphi – d\phi\)

And the accumulated exterior angles \(\varphi\) and \(\phi\) are highly depended on the geometric features of the objects concerned ( i.e. curvatures of the boundaries of objects):

\(dD = R(\varphi)d\varphi\) and \( dD = -r(\phi)d\phi.\)

In Part Two, we further study the length of the trajectory formed by certain point on the rolling object. The result is:

\(dT_p = L_p(\phi)|d\theta|.\)

Also we will give some applications of these results under the circumstance that the curvatures are uniform (The curvatures can be positive, zero and negative).

FULL Article