\documentclass{article}
\title{
Control net approximation to splines}

\author{
J\"org Peters, University of Florida \\
joint work with David Lutterkort}

\begin{document}
\maketitle
The distance between a spline $s= b N = \sum_{i=0}^d b_i N^d_i$
with Greville abscissae $t^*_i$ and its control polygon $\ell$
on the interval $[t^*_k, t^*_{k+1}]$
is expressed in the nonnegative, convex basis functions $\beta_{ki}$
and second differences $\Delta_2 b_i$ as
$$
        s-\ell = (\Delta_2 b) \beta_k = \sum_{i=0}^d (\Delta_2 b_i) \beta_{ki}.
$$
Application of the $L^r$ norm, $| \cdot |_r$,
and H\"older's inequality for vectors with $1/p+1/q=1$
yields the families of bounds
$$
        |s-\ell|_r \le \|(\Delta_2 b)||_p |\| \beta_k\|_q |_r.
$$
Tighter bounds follow by separating $(\Delta_2 b)$ into the vector of
nonnegative second differences $(\Delta^+_2 b)$
with $\Delta^+_2 b_i = \max \{\Delta_2 b_i, 0\}$ and the vector of
nonpositive second differences $(\Delta^-_2 b)$:
$$
        (\Delta^-_2 b)\beta_k \le s-\ell \le (\Delta^+_2 b)\beta_k.
$$
In particular, since $\ell + (\Delta^+_2 b)\beta_k$ is convex,
it can be bounded above by $\overline e$, its linear interpolant
at the Greville abscissae $t^*_k$ and $t^*_{k+1}$,  and
below by the analogous $\underline e$.
This yields a piecewise linear \emph{envelope}  consisting of
$$
        \underline e \le s \le \overline e
$$
on the interval $[t^*_k, t^*_{k+1}]$.
Such envelopes are the basis for reducing a continuous nonlinear
feasibility problem of 1-sided and 2-sided spline approximation
to a linear program. Examples illustrate the effectiveness of
this approach for computing smooth spline paths
that stay within a given polygonal channel.
\end{document}