Transversal surface intersection tracing

For a parametric surface $F(u,v)$ and a scalar implicit equation $f(P)=0$, set $g=f\circ F$. At a regular transversal intersection, $\nabla g\ne 0$, and $(-g_v,g_u)$ is a parameter tangent vector. Normalize its image under $DF$ to choose physical arc length steps. A predictor followed by Newton method correction of $g=0$ and a transverse plane equation gives a local continuation method. Certified patch bounds are useful for finding seeds; corner signs alone do not detect a closed intersection entirely inside a patch.

At regular transverse intersections, the displayed map has a rank-three $3\times 4$ Jacobian matrix. Its one-dimensional kernel is the parameter tangent, while the physical tangent is parallel to the cross product of the surface normals. A continuation method predicts along that tangent and corrects $F=0$ together with a transverse section equation using Newton method. Patch bounds and component isolation supply seeds; one seed alone cannot discover every disconnected curve or closed loop. Tangency or coincident surfaces violate the regular-curve assumptions and require separate analysis.