A torus hides a secret: slice it the obvious ways and you get ovals or pairs of ovals — but tilt the knife to just the right angle (a plane tangent to the tube on BOTH sides) and out fall two perfect circles, each as big as the ring itself. Discovered by Yvon Villarceau in 1848. Slide the cut from flat up to the magic tilt and watch the ovals snap into true circles.
For a torus of major radius R and tube radius r, the bitangent (Villarceau) plane makes angle α with the equator where sinα = r/R. Cut at exactly that angle and the intersection is not an oval but TWO circles, each of radius R — the same radius as the ring’s own centre-line. The instrument sweeps the tilt and measures the radius of the slice curve; a fail-loud self-check throws unless, at the Villarceau angle, that radius is constant and equals R (a genuine circle). R=3, r=1, so α = arcsin(1/3) ≈ 19.47°. ◆ real topology, node-verified.
A clean geometric demonstration for one torus (R=3, r=1); the two Villarceau circles plus the two obvious circles (the equator and a meridian) are the four families of circles that lie fully on a torus — the exact, classical result.