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Make the concept of $C^0$ clearer.
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fixes #57
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hugoledoux committed Dec 9, 2024
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6 changes: 3 additions & 3 deletions interpol/interpol.tex
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Expand Up @@ -56,12 +56,12 @@ \section{What is a good interpolation method for terrains?}[Good interpolation?]
\item \textbf{computationally efficient}: it should be possible to implement the method and get an efficient result. Efficiency is of course subjective. For a student doing this course, efficiency might mean that the method generates a result in matter of minutes or an hour on a laptop, for the homework dataset. For a mapping agency, running a process for a day on a supercomputer for a whole country might be efficient. Observe that the complexity of the algorithm is measured not only on the number $n$ of points in the dataset, but how many neighbours $k$ are used to perform one location estimation.
\item \textbf{automatic}: the method must require as little input as possible from the user, \ie\ it should not rely on user-defined parameters that require \emph{a priori} knowledge of the dataset.
\end{enumerate}
\begin{figure}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{figs/continuity}
\caption{$C^0$ interpolant is a function that is continuous but the first derivative is not possible at certain locations; $C^1$ interpolant has its first derivative possible everywhere; $C^2$ interpolant has its second derivative possible everywhere (this one is more difficult to draw).}%
\caption{Continuity of an interpolant. \textbf{(a)} an interpolant that is not continous. \textbf{(b)} a $C^0$ interpolant is a function that is continuous but the first derivative is not possible at certain locations. \textbf{(c)} a $C^1$ interpolant has its first derivative possible everywhere. \textbf{(d)} a $C^2$ interpolant has its second derivative possible everywhere (this one is more difficult to draw).}%
\labfig{fig:continuity}
\end{figure}
\end{figure*}



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