It is a well-known result of instantaneous spherical kinematics that the locus of those points of the moving sphere, whose paths have a vanishing geodesic curvature, is a curve w on a cubic cone Ω with vertex in the center O of the sphere. In this paper we give a simple construction of the inflection curve w using the following theorem: The intersecting curve l of the inflection cone Ω with a sphere κ, which is centered on the pole-axis p and contains the point O, lies on a cylinder of revolution. This cylinder contains the inflection circle of that planar motion in the tangent plane τ of κ in the pole P = p ∩ κ (P ≠ O), whose relationship between the points and centers of curvature of their paths is induced in τ by the spherical motion. Furthermore we use this result to draw some geometrical conclusions on the set of the ∞1 inflection curves belonging to a given canonical frame. In a special case the inflection curve is a spherical trochoid.
ASJC Scopus subject areas
- Mechanics of Materials
- Mechanical Engineering
- Computer Science Applications