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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 spindle cycloid with 2 imaginary double points and 2 real double points. The spindle cyclide is a cyclide formed by the inversion of a spindle torus. The spindle 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.
Inversion is the process of transforming points P to their inverse points P'.
The 2 real double points connect components that are lying on each other.
Title model and translation
Vier Formen der Dupin'schen Cyclide. Spindelcyclide; zwei reelle Knotenpunkte vereinigen zwei ineinander liegende Flächenmäntel.
Four forms of the cyclide of Dupin. Spindle cyclide; one sheet of the surface lying on another and joint by two real 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