Geometric Hermite Interpolation

A smooth planar curve with non-vanishing curvature can be approximated by cubic splines with order $ 6$, rather than order $ 4$ as expected in view of Taylor's theorem. The figure illustrates how a spline segment $ p$ can be constructed with the aid of the Bézier representation

$\displaystyle t\mapsto p(t) = \sum_{k=0}^3 c_k\,\binom{3}{k}(1-t)^{3-k}t^k,\quad t\in[0,1] \,, $

by interpolating position, tangent direction, and curvature at the endpoints.

\includegraphics[width=10cm]{HermiteFig.eps}

It is conjectured that spline curves of degree $ \le n$ approximate smooth curves in $ \mathbb{R}^d$ with order

$\displaystyle O(h^\alpha),\quad \alpha = n+1+\left\lfloor\frac{n-1}{d-1}\right\rfloor \,, $

as the maximal length $ h$ of the spline segments tends to zero.


$ \square$ with C. de Boor and M. Sabin, CAGD 4 (1987), 269-278.

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