Über zykloidale radlinien und planetenbewegungen

Abstract

In his fundamental article “Hohere Radlinien” W. WUNDERLICH has shown, that for planetary motions the polodes, the paths of points and the envelopes of straight lines are trochoidal curves without exception. The last ones are called cycloidal trochoids and among all trochoidal curves they have the characteristic property, that the motion of the canonical frame is a planetary motion (Theorem 1). This well known theorem, for which a new proof is given, is fundamental for the following new statements. If parallel rays are reflected on a given cycloidal trochoid h of degree n, they envelope a cycloidal trochoid k of degree 2n (Theorem 2). Figure 1 shows a nephroid h with the given rays parallel to the common tangent in the cusps of h. Figure 2 shows a STEINER-cycloid h, the given rays have a general direction. In Section 4 first is shown, that any given trochoid can be used as moving or fixed polode of an one-parametric family of planetary motions (Theorem 3). Further it is proven, that either no one of the two polodes of a planetary motion is cycloidal or both of them (Theorem 4). Therefore one can distinguish between “not cycloidal planetary motions” and “cycloidal planetary motions.” It is surprising, that among all planetary motions the cycloidal ones can be characterized by a constant proportion between the curvatures of the polodes in the respective poles (Theorem 5). The ordinary planetary motions with the polecircles represent a trivial special case.