Bishop curves and orthogonal trajectories

In a geometry seminar at the University of Illinois in March 2010, we presented the cubic quadrarc as the intersection of the cylinders x2 + y2 = 1 and x2 + z2 = 1, and also as the intersection of the sphere x2 + y2 + z2 = 2 and the cube having vertices (±1,±1,±1). During the discussion, Professor Richard Bishop pointed out that this curve, consisting of four arcs, is only 1-smooth at the joints of arcs. He suggested an intersection of elliptic cylinders, and by varying them we obtain a family of Bishop curves which are everywhere infinitely smooth. By an “elliptic cylinder” we mean a cylinder whose base is an ellipse. Figures 2 and 3 indicate that for each pair of intersecting elliptic cylinders, one is parallel to the x-axis, and the other, to the y-axis. In order to tell more about these cylinders (at the end of this section) we begin with the parametric equations given by Professor Bishop. Let S denote the sphere x2 + y2 + z2 = 2. The T -Bishop curve on S, for any T in [−1, 1], is the union of four arcs, the first given by