# Serie IX, Nr 7

## Surface of order four, model of a Cyclide of Dupin.

Mathematical description
The Dupin cyclides are characterized by the property that all their lines of curvature are pieces of circles or straight lines. The standard examples are: the ring cyclides, the spindle cyclides and the horn cyclides.
The model shows the case of the ring cyclides with 4 imaginary double points. The ring Cyclide is a cyclide formed by inversion of a ring torus. The ring torus is one of the three standard tori and it is given by the parametric equations:

x = (c + a cos(v)) cos(u)
y = (c + a cos(v)) sin(u)
z = a sin(v),
with c > a. This is the torus which is generally meant when the term “torus” is used without qualification. Inversion is the process of transforming points P to their inverse points P'. In this case, with respect to a sphere. This, together with Serie IX, nr 8, is the same model as Serie V, nr 5a, only with another ratio. Several tangential planes and the intersections of those planes are drawn on the model.

Title model and translation
Fläche vierter Ordnung. Modell der Dupin'schen Cyclide.
Surface of order four, model of a Cyclide of Dupin.

Text on Sticker and translation
Dupin'sche Cyclide.
Cyclide of Dupin.

Designer
Dr. Kummer
Berlin in 1883

Company