 Shadow tracing for a subdivision curve

For a point light at $L$, the shadow of a curve point is the first surface intersection on its ray beyond that point. Certified patch bounds and subdivision isolate intersections; regular hits are corrected with a three-variable Newton method solve. Differentiating the displayed equation gives $[S_u,S_v,-(C-L)](u^{\prime },v^{\prime },{\lambda }^{\prime }{)}^T=\lambda C^{\prime }$, a predictor for continuation along the shadow. Surface boundaries and grazing rays require clipping and reseeding; every visible component must be isolated, rather than assumed detectable from a fixed sample of the casting curve.