# The hypo- epi-cycloid 2:1 construction

 When the black circle with radius 1 rolls on the outside of the blue circle with radius 2 the fixed red point on the small circle traces out an arc of the epi-cycloid E2 and when it rolls on the inside the blue circle it traces out an arc of the hypo-cycloid H2 (a diameter). The two arcs form the curve A1. When the circle with radius 1 rolls on the outside of the circle with radius 4 the fixed green point on the small circle traces out arcs of the epi-cycloid E4 and when it rolls on the inside it traces out arcs of hypo-cycloid H4. The four arcs form the curve A2. When the blue circle with radius 2 rolls inside the black circle with radius 4 the moving red curve A1 is enveloped by the green curve A2. 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.