The hypo-cycloid 2:1 construction

By rolling the blue circle with radius 1 inside the black circle with radius 2 the fixed red point on the blue circle traces out the the green diameter in the black circle. If red point and the green diameter is offset by the same amount we obtain a red circle moving back and forth in the green canal.

The construction above is lifted to horizontal planes z = constant. The blue circle is rolled a distance proportional to the height z. Observe that the blue circles no longer are above each other. By rotating the picture in each horizontal plane the blue circles can be positioned over each other such as to form a blue cylinder inside a black cylinder. The motion generated by letting the blue cylinder roll inside the black cylinder is in each horzontal plane the same as before.

A series of pump chambers are formed and they move upwards as rigid bodies under the motion generated by the letting the blue cylinder roll inside the black cylinder.