= Solution
Work first in the parameter rectangle of the <parametric surface>. Set $g(u,v)=f(F(u,v))$. With $F_u,F_v$ linearly independent, the chain rule gives
$$
g_u=\nabla f(F)\cdot F_u,\qquad g_v=\nabla f(F)\cdot F_v.
$$
The intersection is transversal precisely when the implicit <normal vector> $\nabla f(F)$ and the parametric <normal vector> $F_u\times F_v$ are not parallel. Consequently $(g_u,g_v)\ne(0,0)$, and the <implicit function theorem> makes $g=0$ a locally regular parameter curve. This is the setting for <transversal surface intersection tracing> by a predictor-corrector <continuation method>.
Find one seed on the required component. Subdivide parameter patches, enclosing their images by <bounding volumes> and testing whether a certified range of $f$ excludes zero. For a positive-weight <NURBS>, the active control-point <convex hull> supplies a convenient enclosure; <interval arithmetic> can tighten the range of the composed function. Search the surviving patches for a bracketed root or a local corrected seed. Merely checking signs at four corners is inadequate: a small closed intersection can lie wholly inside a patch with the same corner signs. An adaptive or certified patch search, or a supplied seed, avoids this failure. For a <sphere> centered at $C$, use $f(P)=\|P-C\|^2-r^2$ and $\nabla f=2(P-C)$; for a <torus>, use its implicit polynomial and its differentiated polynomial in the same algorithm.
At a corrected seed $\eta_k=(u_k,v_k)$, an unnormalized parameter <tangent vector> and a unit spatial <tangent vector> are
$$
q_0=(-g_v,g_u),\qquad T_k={F_u(q_0)_1+F_v(q_0)_2\over\|F_u(q_0)_1+F_v(q_0)_2\|}.
$$
Choose the sign consistently with the previous <tangent vector>. Normalize $q=q_0/\|DFq_0\|$ and predict $\eta_p=\eta_k+hq$, so $h$ is approximately a physical <arc length> step. Correct the predictor by <Newton method> applied to
$$
g(\eta)=0,\qquad T_k\cdot(F(\eta)-F(\eta_k))-h=0.
$$
The Jacobian has rows $(g_u,g_v)$ and $(T_k\cdot F_u,T_k\cdot F_v)$. At the starting point its first row vanishes along $q$, whereas its second row applied to $q$ is one. The two rows are therefore independent. The transverse-plane condition prevents the corrector from sliding arbitrarily along the intersection. Restrict correction to a neighborhood of the predictor, use damping, and reduce $h$ when correction fails or its displacement becomes excessive; this prevents jumps to another nearby branch.
To obtain a sufficiently dense sequence, impose a physical point-spacing bound, a bound on tangent turning, and a chord-error estimate. For local <curvature> $\kappa$, the small-step sagitta is approximately $\kappa h^2/8$; estimate curvature from successive <tangent vectors> and reduce $h$ until this and the desired spacing are acceptable. The correction residual should be small relative to the drawing tolerance. Uniform steps in $u$ or $v$ alone do not ensure spatial density because the surface metric can vary considerably.
For an open component, trace in both tangent directions from the seed. On reaching a parameter boundary, locate the endpoint by solving $g=0$ together with that boundary's parameter equation, rather than accepting an overshoot. For a closed component, stop only after nontrivial travel and return to the initial parameter location with compatible tangent orientation; near-coincidence in three-dimensional space alone can mistake a self-intersection or neighboring branch for closure. Cross periodic seams using the appropriate chart identification. \b[Adaptive physical stepping, local correction, and component-aware termination give the required point sequence.] A vanishing parameter gradient signals a tangency or singularity outside the transversal hypothesis and must be reported rather than passed through by this regular algorithm.
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