= Solution
A <parametric curve interrogation> interface first supplies the parameter domain and the point $C(t)$. At regular parameters, it also supplies $C'(t)$ and, when needed, higher <derivatives>. The unit <tangent vector>, speed and <curvature> are then obtained from
$$
\frac{C'}{\|C'\|},\qquad \|C'\|,\qquad
\kappa=\frac{\|C'\times C''\|}{\|C'\|^3}
$$
in three dimensions. The formulas require $C'\ne0$; a Frenet normal additionally requires nonzero <curvature>. A practical interface can restrict the curve to a parameter interval, provide endpoints, and give a certified <bounding volume> or a derivative-range bound. Arc length, closest-point and intersection queries are higher-level operations built from these elementary enquiries rather than consequences of point evaluation alone.
For a plane $n\cdot x=d$ with $n\ne0$, define
$$
F(t)=n\cdot C(t)-d,\qquad F'(t)=n\cdot C'(t).
$$
This reduces <certified curve-plane intersection> to scalar root isolation. A <subdivision curve> can expose the parametric enquiries by evaluating its limit at a parameter address: refine the corresponding piece until its certified positional enclosure reaches the requested accuracy, or use its exact local <spline> representation. <Derivative> evaluation similarly needs the <derivative> refinement rule and its error bound. A finite control vertex is not automatically a point of the limit curve.
One reliable algorithm is \b[range-based isolation followed by safeguarded Newton iteration]:
* Split the domain at representation breaks. For each interval $I$, enclose $F(I)$ by projecting its certified curve bound onto $n$. Reject $I$ if this interval excludes zero.
* If the <derivative> enclosure excludes zero and the endpoint values have opposite signs, there is exactly one intersection in $I$. Record any endpoint zero separately and avoid counting it twice.
* Within such a bracket, propose $t_{\rm new}=t-F(t)/F'(t)$ using the <Newton root-finding iteration>. Accept only a well-conditioned step inside a suitably smaller portion of the bracket; otherwise use the <bisection method>. Update the sign bracket and stop when its parameter width and the curve's positional error meet the tolerances.
* Subdivide every other undecided interval. Use representation-supported tests for tangent roots and coplanar pieces; do not reject an interval merely because its endpoint signs agree.
For a transverse root, the <derivative> is nonzero nearby, so sufficiently fine intervals have monotone $F$ and the sign bracket isolates it. Bisection provides a global error bound; near a simple root Newton iteration gives quadratic convergence. Good enclosures discard most of the curve before expensive limit evaluation, and the method does not require a specific subdivision mask.
The weaknesses are equally important. Near tangency $F'$ is small and Newton steps are ill-conditioned; same-sign endpoints may conceal a tangent root or two crossings. An arc lying in the plane has infinitely many intersections and should be returned as an interval, not a point. Finite-precision evaluations require error enclosures before a sign is certified. For an arbitrary smooth point-evaluation oracle, \b[no finite sampling scheme can certify all intersections]: a smooth bump between sampled parameters can create extra crossings without changing any sampled values or <derivatives>. With finitely many isolated transverse intersections and convergent certified enclosures, the algorithm terminates to any prescribed tolerance. For tangencies it either uses stronger representation tests or reports unresolved possible-contact intervals, rather than silently omitting them.
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