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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 horn cyclides with 2 imaginary double points and 2 real double points. The horn cyclide is a cyclide formed by the 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.
The 2 real double points connect components, that are not lying on each other.
Title model and translation
Vier Formen der Dupin'schen Cyclide. Horncyclide; zwei reelle Knotenpunkte vereinigen zwei auseinander liegende Flächenmäntel.
Four forms of the cyclide of Dupin. Horn cyclide; two real double points join the two sheets of the surface that lying apart.
Text on Sticker and translation
Cyclide of Dupin.
Dr. P. Vogel under the supervision of Dr. A. Brill
München in 1880
Martin Schilling in Halle a.S.
Schilling, M., Catalog mathematischer Modelle für den höheren mathematischen Unterricht, Leipzig: Verlag von Martin Schilling, 1911