Cast from a, in the mathematical institute of the royal university of technology in Munich manufactured, original."

These parabolic ring cyclides have the remarkable feature, that it breaks down the 3-space in two congruent parts: The hollow form, that was used for the manufacturing of the plaster model, is congruent to the model.

With the parabolic ring cyclides, all junctures are imaginary, but the connecting straight from both pairs, that wholly lies on the surface, is real. Except those, two other intersecting straights and a infinite distant straight also exist on the surface .

The carved circles are curvature lines. The Dupin cyclides are generally of the fourth order. The parabolic cylinder though contain the absolute conic section (infinity distant imaginary sphere circle) only once, the infinity distant plane separates itself as a component. Therefore the plane is a third-order surface. An extensive depiction of the Dupin cyclides by Ulrich Pinkanll can be found in Fischer, 1986, Kommentarband p. 30 ff. In particular remarkable is, that every Dupin cyclide can be obtained by an inversion of a torus. The Dupin ring cyclides for example can be obtained, when the centre point of the inversion lies on the surface area of the ring torus. Every image of a Dupin cyclide under a inversion is once more a Dupin cyclide.