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# Serie V, Nr 5d

## Four forms of the cyclide of Dupin. Parabolic cyclide with two real double points;
spreads out to infinite with a single sheet of the surface.

**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 parabolic horn cycloid with 2 imaginary double points and 2 real double
points.
Parabolic horn cyclide is a cyclide formed by inversion of a horn torus.
The horn torus is one of the three standard tori and given by the parametric equations:
**
**

** ***x* = a(1+cos(*v*))cos(*u*)
*y* = a(1+cos(*v*))sin(*u*)
*z* = a sin(*v*)

**
**
Inversion is the process of transforming points *P* to their inverse points *P'*.
In this case, with respect to a sphere, which center lies on the horn torus.
The real double points are connected with a straight line on the surface.
The surface is of pure order three.
**Title model and translation**

Vier Formen der Dupin'schen Cyclide. Parabolische Cyclide mit zwei reellen Knotenpunkten; erstreckt
sich mit einem unpaaren Flächenmäntel ins Unendliche.

Four forms of the cyclide of Dupin. Parabolic cyclide with two real double points; spreads out to
infinite with a single sheet of the surface.

**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**

L. Brill in Darmstadt.

**Original price**

Mark 13,50

**Dimension**

12-15 cm

**Material**

Plaster

**Literature**

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