Geometry & streak rule
N threads share one torus, evenly phased around the tube by 2π/N. Thread k is
C_k(u) = ((R + a cos w_k) cos u, (R + a cos w_k) sin u, a sin w_k)
w_k = ωu + 2πk/N
Tangent streaks. A head travels along u, wrapping the torus forever. At each tick position it computes the curve tangent T_k(u) = dC_k/du, normalises it, and lays down a short segment C_k(u) ± L·T̂_k(u). Because the curve sits on the surface, the segment lies in the torus's tangent plane — it touches the surface without cutting into it.
Breaks. Streaks are laid in series of 10. After the tenth the head keeps travelling for a short gap distance while drawing nothing, then the next series of 10 begins. Older streaks fade behind the head, leaving a dashed tangent trail.