Dupin Cyclides

This page is a sub-page of our page on Surfaces.

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Related KMR-pages:

Canal Surfaces
Developable Surfaces
Focal Surfaces

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Other related sources of information:

Images of Dupin cyclides
Dupin Cyclide (without focal curves) at Wikipedia
Clifford torus at Wikipedia
Torus at Wikipedia
Dupin Cyclide (with focal curves) at MathCurve.com
Charles Dupin at Wikipedia
Cyclides (as inversions of tori, cones and cylinders) by Alain Esculier
Cyclide at Wolfram MathWorld
Paraboloid at Wikipedia
Paraboloidal coordinates at Wikipedia

John Pickering (artist)

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The interactive simulations on this page can be navigated with the Free Viewer of the Graphing Calculator.

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Dupin Cyclides – The closest relatives to the Sphere:

Interactive simulation of a Dupin Cyclide with its Principal Net.
Change the parameters a, b, c in order to change the shape of the surface, and change the parameters k, m in order to move the principal curves. For a Dupin Cyclide, both families of principal curves are circles. The formulas are reached by pulling down at the top of the frame.

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Dupin Cyclide – PN1:

The interactive simulation that created this movie.

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Dupin Cyclide – PN2:

The interactive simulation that created this movie.

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Dupin Cyclide – PN1 lines:

The interactive simulation that created this movie.

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Dupin Cyclide – PN2 lines:

The interactive simulation that created this movie.

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Dupin Cyclide – WaveFronts:

The interactive simulation that created this movie.

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Dupin Cyclide Wave-front 2:

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Inversion of a Dupin cyclide (pn1 lines):

The interactive simulation that created this movie

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Inversion of parallel Dupin cyclides

The interactive simulation that created this movie

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Dupin Cyclides as Inversions of Right Circular Cones:

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Inversion of a right circular cone with a moving tangent plane:

The interactive simulation that created this movie.

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Inversion of a right circular cone 1:

The interactive simulation that created this movie.

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Inversion of a right circular cone 2:

The interactive simulation that created this movie.

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Inversion of a right circular cone 3:

The interactive simulation that created this movie.

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Inversion of a rotating right circular cone:

The interactive simulation that created this movie.

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A circular cylinder can be regarded as a right circular cone with its vertex at infinity.

Inversion of a circular cylinder with a moving tangent plane:

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Inversion of a rotating circular cylinder 2:

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Inversion of a rotating circular cylinder 3:

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Inversion of a rotating circular cylinder 4:

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Confocal Paraboloids acting as mirrors to a planar wavefront

Confocal Paraboloids – reflected Dupin cyclide.gcf

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Reflection of a planar wavefront in a saddle surface
turns the wavefront into a Dupin cyclide
:

Confocal-paraboloids-planar-wavefront-into-cyclide-reflection.gcf

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A set of (horizontally positioned) confocal paraboloids give the same (Dupine cyclide) reflection of a planar wave front that is incident along the vertical axis:

The interactive simulation that created this movie.

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The same variation of the confocal paraboloid mirrors is shown here without the reflected wavefront (which is the same for all the paraboloids among the confocal family:

Confocal Paraboloids – reflect vertical rays into Dupin cyclide.gcf

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One thought on “Dupin Cyclides

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