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The Dupin 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. The ring torus is one of the three standard tori and given by the parametric equations:
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
Inversion is the process of transforming points P to their inverse points P'.
In this case, with respect to a sphere.
This is next to Serie IX, nr 7 the same model as
Serie V, nr 5a, only with another ratio.
Drawn on the model are several tangential planes and intersections of those planes.
Fläche vierter Ordnung. Modelle der Dupin'schen Cyclide.
Surface of order four, model of a Cyclide of Dupin.
Cyclide of Dupin.
Berlin in 1883
Martin Schilling in Halle a.S.
Schilling, M., Catalog mathematischer Modelle für den höheren mathematischen Unterricht, Leipzig: Verlag von Martin Schilling, 1911