"Bald Eagle" <cre### [at] netscapenet> wrote:
> "Bald Eagle" <cre### [at] netscapenet> wrote:
> > ... I'd say that this demonstrates that a solution isn't
> > possible, given the parameters.
> >
>
> > So I think that some of the constraints are mutually exclusive.
>
> However; depending on if the design parameters can be modified and still provide
> an acceptable solution to the intended goal:
>
> Perhaps an elliptical  "torus" that is in reality a Dupin cyclide would be just
> the thing:
> https://en.wikipedia.org/wiki/Dupin_cyclide#Elliptic_cyclides
>
> - BW

So, for whatever reason, my brain revisited this, and I wanted a quick
evaluation of if my intuition about using a cyclide in place of an actual torus
would work.

enjoy.

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A section of a Dupin cyclide (specifically a ring cyclide) can play the


Below is the detailed geometric reasoning.

Why an elliptically‑centered circular torus is impossible
A standard torus is defined as the surface generated by revolving a circle of

rR>r. Its defining property is:

The tube centerline is a perfect circle of radius RRR.



A circle of radius rrr cannot remain perpendicular to an elliptical trajectory
while maintaining constant distance from the ellipse.
As the tube center attempts to follow the ellipse, the required orientation
twists inconsistently.
The surface self‑intersects unless the cross‑section distorts.
A constant‑radius normal field cannot exist along a plane ellipse.

Thus:

A torus with constant circular cross‑section and an elliptical major
radius cannot be embedded in R3\mathbb{R}^3R3 without distortion.

This is a classical result from differential geometry: a tube of constant radius
can only be swept around a curve whose curvature κ(s)\kappa(s)κ(s)
satisfies
r<1max⁡κ(s)r < \frac{1}{\max \kappa(s)}r<maxκ(s)1​
and whose normal vector rotates consistently. A planar ellipse does not satisfy
this for a fixed radius circle.

How a Dupin cyclide solves this


Its cross sections by suitable families of spheres are circles, and the
curvature lines are circles.

That means:

The surface bends around in a general way not limited to being the
parallel‑offset of a single circle.
The curvature lines are Villarceau circles and orthogonal circular families,
giving a structure very similar to a torus but with more flexibility.



Why a cyclide can work:
A general ring cyclide is obtained from a torus by inversion in a sphere:
Cyclide=Inversion(Torus)\text{Cyclide} = \text{Inversion} ( \text{Torus}
)Cyclide=Inversion(Torus)
Inversion preserves:

circles → circles,
spheres → spheres,
constant‑curvature structure in a generalized way.

But inversion does not preserve distances, so the resulting surface can bend and

elliptical‑centerline torus.
Thus a Dupin cyclide:

has circular curvature lines (like a torus),

maintains circular cross‑sections in a transformed sense,
can be fitted to curves (directrices) that are not circles.





A partial ring cyclide can provide:

an elongated toroidal shape,
smooth curvature without singularities,
circular fiber curves (Villarceau circles),


This is used in:

CAD modeling systems,
sweep surfaces,
transition surfaces (Fillets/Blends),
quadric‑based modeling applications.


Practical interpretation:
If you want:

a circular tube,
bending around an ellipse,
without changing the tube radius,

then a cyclide patch is the mathematically correct surface.

structures. A selected region of a cyclide can approximate a torus stretched
around an ellipse, because the inversion mapping allows shapes impossible in
Euclidean rigid sweeps.

Summary







cyclides in geometric modeling.

If you want, I can:

derive the exact cyclide equation needed for a specific ellipse,
construct Villarceau circle fibrations along that cyclide,
generate POV‑Ray macros for a cyclide section that approximates a torus
swept along an ellipse,
create a PDF with diagrams and derivations,
provide a .pov scene with curvature‑coloring and isosurface rendering.

Would you like the full construction?

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