Toroid

TOROID

Curve studied by Cauchy in 1841, Catalan and Breton des Champs (who named it) in 1844.
The name toroid comes from torus.

Cartesian parametrization: .
for the ellipse: and a distance d from the ellipse.
Biquartic.

The toroids are the parallel curves of the ellipse, hence the involutes of the evolute of the ellipse.
The name comes from the fact that the toroids are none other than the visible outlines of the torus.

The toroid is the visible outline of the torus

The toroids are, in general, composed of two ovals, except when the distance is the distance between the extrema of the radius of curvature of the ellipse , in which case one of the components has four cusps located on the evolute of the ellipse.
Opposite, evolution of the toroid, as the distance d to the ellipse increases. In green, the evolute of the ellipse.

Opposite, the toroid (in green) of the blue ellipse, in the case d = 2a, length that also is the diameter of the ellipse.
With the curve traced in red, we get a curve with the same diameter as the ellipse and which is a curve with constant width, looking like the Reulaux triangle, and different from the obvious circumscribed circle of the ellipse.

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The toroids are, in general, composed of two ovals, except when the distance is the distance between the extrema of the radius of curvature of the ellipse , in which case one of the components has four cusps located on the evolute of the ellipse. Opposite, evolution of the toroid, as the distance d to the ellipse increases. In green, the evolute of the ellipse.
Opposite, the toroid (in green) of the blue ellipse, in the case d = 2a, length that also is the diameter of the ellipse. With the curve traced in red, we get a curve with the same diameter as the ellipse and which is a curve with constant width, looking like the Reulaux triangle, and different from the obvious circumscribed circle of the ellipse.