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# Serie V, Nr 5a

## Four forms of the cyclide of Dupin. Ring cyclide with imaginary double points.

**Mathematical description**

The Dupin's 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.
Inversion is the process of transforming points *P* to their inverse points *P'*.
The ring torus is one of the three standard tori and is given by the parametrizatoin:
**
**

** ***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.
**Title model and translation**

Vier Formen der Dupin'schen Cyclide. Ringcyclide mit imaginären Knotenpunkten.

Four forms of the cyclide of Dupin. Ring cyclide with imaginary double points.

**Text on Sticker and translation**

Dupin'sche Cyclide.

Cyclide of Dupin.

**Designer**

Dr. P. Vogel under the supervision of Dr. A. Brill

München in 1880

**Company**

Martin Schilling in Halle a.S.

**Original price**

Mark 10,50

**Dimension**

7-14-14 cm

**Material**

Plaster

**Literature**

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