= Shadow tracing for a parametric curve
{title2=$S(u,v)=P+\alpha(C(t)-P),\quad\alpha>1$}
From a point light source $P$, a caster point $C(t)$ projects onto a receiving surface along the displayed ray. On a regular transverse branch, differentiating gives $[S_u,S_v,-(C-P)](u',v',\alpha')^T=\alpha C'$. <Newton method> correction and certified patch/curve bounds support continuation and isolation of all visible branches. If the shadow's second <derivative> is bounded by $M$ on an interval of width $h$, its chord error is at most $Mh^2/8$. A fixed sample grid can miss entire shadow components; surface boundaries, occlusion changes and grazing rays require explicit treatment.
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