# 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