Twisted oval-tube torus

A circular centreline carries an elliptical tube, and that ellipse rotates as it travels around the ring. The body reads as a ribbon of even width whose broad face keeps turning.

T = (−sin u, cos u, 0), N = (cos u, sin u, 0), B = (0,0,1)
t = twist · u, d1 = cos t N + sin t B, d2 = −sin t N + cos t B
P(u,v) = (R cos u, R sin u, 0) + p cos v d1 + q sin v d2

p and q are the cross-section semi-axes. Because the broad face keeps rotating toward the light, the highlight spreads into a wide flowing band instead of snapping to a thin line; the sheen slider sets that band's strength (a softer specular exponent than the other studies).

With a half-integer twist the cross-section comes back to the start reversed — turning the ellipse inside out once around the loop, a Möbius-like seam. Whole-number twists close seamlessly.