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# Serie IX, Nr 8

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

**Explanation**

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 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 is next to Serie IX, nr 7 the same model as
Serie V, nr 5a, only with another ratio.
Drawn on the model are several tangential planes and intersections of those planes.
**Title model**

Fläche vierter Ordnung. Modelle der Dupin'schen Cyclide.

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

**Information sticker**

Dupin'sche Cyclide.

Cyclide of Dupin.

**Designer**

Dr. Kummer

Berlin in 1883

**Company**

Martin Schilling in Halle a.S.

**Original price**

Mark 14

**Dimension**

9-9-6 cm

**Material**

Plaster

**Literature**

Schilling, M., Catalog mathematischer Modelle für den höheren
mathematischen Unterricht, Leipzig: Verlag von Martin Schilling, 1911