Asymptotic approximation by quadratic spline curves (Q2714175)

A quadratic spline curve \(Q_n\) associated with a plane curve \(C\) consists of \(n\) pieces of parabolas with common tangents at their common points, all lying on \(C\), and having the same endpoints as \(C\). Denote by \({\mathcal Q}_n(C)\) the set of all these spline curves and, for \(Q_n\in {\mathcal Q}_n(C)\), let \(\delta (C,Q_n)\) denote the area of the region between \(C\) and \(Q_n\). The paper is concerned with the asymptotic behavior of the quantity \(\delta (C,{\mathcal Q}_n) = \inf \{\delta (C, Q_n): Q_n \in {\mathcal Q}_n(C)\},\) for \(n\to \infty\). The author proves that if \(C\) is a plane curve of differentiability class \({\mathcal C}^4\) and with positive (or negative) affine curvature \(\kappa (s)\) then NEWLINE\[NEWLINE \delta (C,{\mathcal Q}_n) \sim \frac{1}{240}\left (\int _0^\lambda |\kappa^{1/5} (s)|ds\right) ^5 \frac{1}{n^4} NEWLINE\]NEWLINE as \(n\to \infty\), where \(\lambda\) is the affine length of \(C\) and \(s, 0\leq s\leq \lambda, \) the affine arc length. NEWLINENEWLINENEWLINEThe asymptotic formula for the approximation of a plane convex curve \(C\) by convex polygons with \(n\) knots, all lying on \(C\) and having the same endpoints as \(C\), was found by \textit{L. Fejes Tóth} [Bull. Am. Math. Soc. 4, 139-146 (1948; Zbl 0031.27901)].