Enveloping the Oloid
The oloid is the convex hull of two unit circles that lie in perpendicular planes where each circle contains the center of the other; you can show these circles. This Demonstration visualizes how the oloid develops from the inversion of a rectangular kaleidocycle, the center part of the evertible cube from Paul Schatz. The red line connects two vertices of the tetrahedra that form the kaleidocycle. This line is always tangent to the surface of the oloid, so when the line is moved by inverting the kaleidocycle it envelops the oloid. Some of the tetrahedra are hidden by the oloid; to make them visible, decrease the opacity of the oloid.
To describe the oloid parametrically, we calculate the coordinates of the two points and that are the endpoints of the red line. Let be a point on the circle that lies in the - plane with center at the origin, so that .
is the intersection of the tangent to at with the axis. is the point that lies on the circle perpendicular to with center . The tangent to at has the same intersection , so the points and are coupled through their conjugate . With , , and (the signs correspond to the upper and lower hemispheres). Interestingly, the length of is always equal to .
is also the set of points that touch the ground when the oloid lies on a plane, so when the plane is inclined, the contact points move over the surface of the oloid like the red line. This means that every point of the surface of the oloid touches with the plane as it moves forward. The movement is somewhat staggering but follows a straight line.
The parametric description of the oloid is , with the parameters and . The parameter is restricted because we must have . Paul Schatz found the oloid in 1933 by imagining its shape while he studied the movement of the kaleidocycle. See http://www.paul-schatz.ch/.
 H. Dirnböck and H. Stachel, "The Development of the Oloid," Journal for Geometry and Graphics, 1(2), 1997 pp. 105–118.