= Certified curve-plane intersection
{title2=$F(t)=n\cdot C(t)-d$}
Intersecting a <parametric curve> with a plane reduces to isolating zeros of a scalar function. A certified range for $F$ rejects intervals that do not contain zero. On an interval whose <derivative> range excludes zero, a sign-changing endpoint bracket gives exactly one root, which <bisection method> or a safeguarded <Newton root-finding iteration> refines. Other intervals are subdivided conservatively; tangencies and coplanar pieces need multiplicity or identity tests supported by the representation. An unresolved small interval is a possible contact, not a proof of an intersection. With finitely many transverse intersections and convergent enclosures, isolation terminates to any prescribed tolerance.
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