According to W. WUNDERLICH the product of n rotations with constant velocities about different points in the Euclidean plane is called a planetary motion of degree n. If at the same the center of this motion runs on a fixed straight line with constant velocity, we get the cycloid motion of degree n. The polodes of this motion and the point-paths, the so-called cycloids of degree n have been studied by H. HORNINGER. This article is concerned with the envelopes of straight lines. It is shown that these curves are cycloids of degree 2n. They constitute the class of G-cycloids within the great family of cycloids. The motion of the canonical frame along a G-cycloid is a cycloid-motion again. Therefore also the evolute and all evolutoids of G-cycloids are G-cycloids. With the result, the parallel rays reflected rays on a given G-cycloid, a theoem of W. JANK is generalized. Also the Mannheim-curve of a G-cycloid can be proven to be a cycloid. Finally a theorem dealing with the equiform motion defined by the radius of curvature along a G-cycloid is presented.
ASJC Scopus subject areas
- Mechanics of Materials
- Mechanical Engineering
- Computer Science Applications