Ribbon-wrap torus
Geometry
A plain matte ring is wound with one narrow metallic ribbon. The body stays quiet: most of its surface is untouched, and the wrap supplies the highlight.
X(u) = ((R + r_b cos(wu)) cos u, (R + r_b cos(wu)) sin u, r_b sin(wu))
The ribbon centreline X(u) is a helix that rides exactly on the tube surface, making w turns per loop. At each point the frame is the surface normal, the travel direction and their cross product:
n̂ = (cos u cos(wu), sin u cos(wu), sin(wu))t̂ = normalize(dX/du)b̂ = n̂ × t̂
The cross-section is a flat ellipse lying on the surface: half-width W along the tangent b̂, half-thickness T along the normal, its underside raised h off the body — P(u,s) = X(u) + n̂(h + T(1 + cos s)) + b̂(W sin s). Wide and thin is what reads as a ribbon rather than a round tube, and keeping the section above the surface stops the band sinking into the body.
Body and wrap are shaded together but with different materials: the dark body takes a weak, broad specular (25) while the ribbon takes a tight, strong one (140, exponent 60) so it flashes as metal against the matte ground.