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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 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
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