mirror of https://github.com/CGAL/cgal
25 lines
1.4 KiB
TeX
25 lines
1.4 KiB
TeX
This chapter describes \cgal's interpolation package which implements
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natural neighbor coordinate functions as well as different
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methods for scattered data interpolation most of which are based on
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natural neighbor coordinates. The functions for computing natural neighbor
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coordinates in Euclidean space are described in
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Section~\ref{sec:coordinates},
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the functions concerning the coordinate and neighbor
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computation on surfaces are discussed in Section~\ref{sec:surface}.
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In Section~\ref{sec:interpolation}, we describe the different interpolation
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functions.
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Scattered data interpolation solves the following problem: given
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measures of a function on a set of discrete data points, the task is
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to interpolate this function on an arbitrary query point.
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More formally, let $\mathcal{P}=\{\mathbf{p_1},\ldots ,\mathbf{p_n}\}$ be a set of
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$n$ points in $\mathbb{R}^2$ or $\mathbb{R}^3$ and $\Phi$ be a scalar
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function defined inside the convex hull of $\mathcal{P}$. We assume that
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the function values are known at the points of $\mathcal{P}$, i.e. to
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each $\mathbf{p_i} \in \mathcal{P}$, we associate $z_i =
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\Phi(\mathbf{p_i})$. Sometimes, the gradient of $\Phi$ is also known
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at $\mathbf{p_i}$. It is denoted $\mathbf{g_i}= \nabla
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\Phi(\mathbf{p_i})$. The interpolation is carried out for an arbitrary query point
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$\mathbf{x}$. Except for interpolation on surfaces, $\mathbf{x}$ must lie
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inside the convex hull of $\mathcal{P}$.
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