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<H2><A NAME="SECTION00522000000000000000">
3.2.2 Gridding with Splines in Tension</A>
</H2>
<P>
As an alternative, we may use a global procedure to grid our data.
This approach, implemented in the program <A NAME="tex2html221"
HREF="../surface.html"><I><B>surface</B></I></A><A NAME="3004"></A>, represents
an improvement over standard minimum curvature algorithms by allowing
users to introduce some tension into the surface.
Physically, we are trying to force a thin elastic plate to go through
all our data points; the values of this surface at the grid points
become the gridded data. Mathematically, we want to find the function
<IMG
WIDTH="43" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img56.gif"
ALT="$z(x, y)$"> that satisfies the following constraints:
<BR><A NAME="1295"></A>
<P>
<!-- MATH
$\begin{array}{ll}
z(x_k, y_k) = z_k, & \mbox{for all data $(x_k, y_k, z_k), k =1,n$} \\
(1-t)\nabla^4 z - t \nabla^2 z = 0 & \mbox{elsewhere}
\end{array}$
-->
<IMG
WIDTH="367" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
SRC="img57.gif"
ALT="\( \begin{array}{ll}
z(x_k, y_k) = z_k, & \mbox{for all data $(x_k, y_k, z_k), k =1,n$} \\
(1-t)\nabla^4 z - t \nabla^2 z = 0 & \mbox{elsewhere}
\end{array} \)">
<BR>
<P>
where <IMG
WIDTH="9" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img58.gif"
ALT="$t$"> is the ``tension'', <!-- MATH
$0 \leq t \leq 1$
-->
<IMG
WIDTH="64" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img59.gif"
ALT="$0 \leq t \leq 1$">. Basically, as
<!-- MATH
$t \rightarrow 0$
-->
<IMG
WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img60.gif"
ALT="$t \rightarrow 0$"> we obtain the minimum curvature solution, while as
<!-- MATH
$t \rightarrow \infty$
-->
<IMG
WIDTH="44" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img61.gif"
ALT="$t \rightarrow \infty$"> we go towards a harmonic solution (which is linear
in cross-section). The theory behind all this is quite involved
and we do not have the time to explain it all here, please see
<I>Smith and Wessel</I> [1990] for details. Some of the most important
switches for this program are indicated in Table <A HREF="node64.html#tbl:surface">3.3</A><A NAME="tex2html217"
HREF="footnode.html#foot1746"><SUP>3.1</SUP></A>.
<P>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1749"></A>
<TABLE>
<CAPTION><STRONG>Table 3.3:</STRONG>
Some of the options in <A NAME="tex2html218"
HREF="../surface.html"><I><B>surface</B></I></A><A NAME="3017"></A></CAPTION>
<TR><TD>
<DIV ALIGN="CENTER"><TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1">
<I>Option</I></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> <I>Purpose</I></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT><FONT SIZE="-1"><B>-A</B></FONT><FONT SIZE="-1"><I>aspect</I> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> Sets aspect ratio for anisotropic grids. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT><FONT SIZE="-1"><B>-C</B></FONT><FONT SIZE="-1"><I>limit</I> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> Sets convergence limit. Default is 1/1000 of data range. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT><FONT SIZE="-1"><B>-T</B></FONT><FONT SIZE="-1"><I>tension</I> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> Sets the tension [Default is 0] </FONT></TD>
</TR>
</TABLE><FONT SIZE="-1">
<A NAME="tbl:surface"></A></FONT></DIV></TD></TR>
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<ADDRESS>
Paul Wessel
2001-04-18
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