The Image Curve of the Planet in a Spherical Epicyclic Gear Train

This paper uses the Euler parameters of the motion of the planet of a spherical epicyclic gear train to obtain a curve on the surface of a hypersphere in four dimensions. This curve, called the image curve, represents the rotational motion of the planet as it rolls without slipping on the fixed gear. Two image curves are obtained for two different choices of moving and fixed reference frames and it is shown that they are related by an orthogonal transformation in four dimensional space. The differential properties of the image curve are computed and it is found that the curvature and torsion are constant. A reference position is chosen and the canonical frame and instantaneous invariants of the motion are determined in terms of the dimensions of the gear train.