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<?xml version="1.0" encoding="UTF-8"?>
<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns5="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="splin3d" xml:lang="en">
<refnamediv>
<refname>splin3d</refname>
<refpurpose>spline gridded 3d interpolation</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>tl = splin3d(x, y, z, v, [order])</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>x,y,z</term>
<listitem>
<para>strictly increasing row vectors (each with at least 3
components) defining the 3d interpolation grid
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>v</term>
<listitem>
<para>nx x ny x nz hypermatrix (nx, ny, nz being the length of
<literal>x</literal>, <literal>y</literal> and
<literal>z</literal>)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>order</term>
<listitem>
<para>(optional) a 1x3 vector [kx,ky,kz] given the order of the
tensor spline in each direction (default [4,4,4], i.e. tricubic
spline)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>tl</term>
<listitem>
<para>a tlist of type splin3d defining the spline</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
This function computes a 3d tensor spline <emphasis>s</emphasis>
which interpolates the <emphasis>(xi,yj,zk,vijk)</emphasis> points, ie, we
have <emphasis>s(xi,yj,zk)=vijk</emphasis> for all
<emphasis>i=1,..,nx</emphasis>, <emphasis>j=1,..,ny</emphasis> and
<emphasis>k=1,..,nz</emphasis>. The resulting spline
<emphasis>s</emphasis> is defined by <literal>tl</literal> which consists
in a B-spline-tensor representation of <emphasis>s</emphasis>. The
evaluation of <emphasis>s</emphasis> at some points must be done by the
<link linkend="interp3d">interp3d</link> function (to compute
<emphasis>s</emphasis> and its first derivatives) or by the <link linkend="bsplin3val">bsplin3val</link> function (to compute an arbitrary
derivative of <emphasis>s</emphasis>) . Several kind of splines may be
computed by selecting the order of the spline in each direction
<literal>order=[kx,ky,kz]</literal>.
</para>
</refsection>
<refsection>
<title>Remark</title>
<para>This function works under the conditions:</para>
<informalequation>
<mediaobject>
<imageobject>
<imagedata align="center" fileref="../mml/splin3d_equation1.mml"/>
</imageobject>
</mediaobject>
</informalequation>
<para>an error being issued when they are not respected.</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
// example 1
// =============================================================================
func = "v=cos(2*%pi*x).*sin(2*%pi*y).*cos(2*%pi*z)";
deff("v=f(x,y,z)",func);
n = 10; // n x n x n interpolation points
x = linspace(0,1,n); y=x; z=x; // interpolation grid
[X,Y,Z] = ndgrid(x,y,z);
V = f(X,Y,Z);
tl = splin3d(x,y,z,V,[5 5 5]);
m = 10000;
// compute an approximated error
xp = grand(m,1,"def"); yp = grand(m,1,"def"); zp = grand(m,1,"def");
vp_exact = f(xp,yp,zp);
vp_interp = interp3d(xp,yp,zp, tl);
er = max(abs(vp_exact - vp_interp))
// now retry with n=20 and see the error
]]></programlisting>
<programlisting role="example"><![CDATA[
// example 2 (see linear_interpn help page which have the
// same example with trilinear interpolation)
// =============================================================================
exec("SCI/modules/interpolation/demos/interp_demo.sci")
func = "v=(x-0.5).^2 + (y-0.5).^3 + (z-0.5).^2";
deff("v=f(x,y,z)",func);
n = 5;
x = linspace(0,1,n); y=x; z=x;
[X,Y,Z] = ndgrid(x,y,z);
V = f(X,Y,Z);
tl = splin3d(x,y,z,V);
// compute (and display) the 3d spline interpolant on some slices
m = 41;
direction = ["z=" "z=" "z=" "x=" "y="];
val = [ 0.1 0.5 0.9 0.5 0.5];
ebox = [0 1 0 1 0 1];
XF=[]; YF=[]; ZF=[]; VF=[];
for i = 1:length(val)
[Xm,Xp,Ym,Yp,Zm,Zp] = slice_parallelepiped(direction(i), val(i), ebox, m, m, m);
Vm = interp3d(Xm,Ym,Zm, tl);
[xf,yf,zf,vf] = nf3dq(Xm,Ym,Zm,Vm,1);
XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; VF = [VF vf];
Vp = interp3d(Xp,Yp,Zp, tl);
[xf,yf,zf,vf] = nf3dq(Xp,Yp,Zp,Vp,1);
XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; VF = [VF vf];
end
nb_col = 128;
vmin = min(VF); vmax = max(VF);
color_example = dsearch(VF,linspace(vmin,vmax,nb_col+1));
xset("colormap",jetcolormap(nb_col));
clf(); xset("hidden3d",xget("background"));
colorbar(vmin,vmax)
plot3d(XF, YF, list(ZF,color_example), flag=[-1 6 4])
xtitle("3d spline interpolation of "+func)
show_window()
]]></programlisting>
<scilab:image localized="true">
exec("SCI/modules/interpolation/demos/interp_demo.sci")
func = "v=(x-0.5).^2 + (y-0.5).^3 + (z-0.5).^2";
deff("v=f(x,y,z)",func);
n = 5;
x = linspace(0,1,n); y=x; z=x;
[X,Y,Z] = ndgrid(x,y,z);
V = f(X,Y,Z);
tl = splin3d(x,y,z,V);
m = 41;
direction = ["z=" "z=" "z=" "x=" "y="];
val = [ 0.1 0.5 0.9 0.5 0.5];
ebox = [0 1 0 1 0 1];
XF=[]; YF=[]; ZF=[]; VF=[];
for i = 1:length(val)
[Xm,Xp,Ym,Yp,Zm,Zp] = slice_parallelepiped(direction(i), val(i), ebox, m, m, m);
Vm = interp3d(Xm,Ym,Zm, tl);
[xf,yf,zf,vf] = nf3dq(Xm,Ym,Zm,Vm,1);
XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; VF = [VF vf];
Vp = interp3d(Xp,Yp,Zp, tl);
[xf,yf,zf,vf] = nf3dq(Xp,Yp,Zp,Vp,1);
XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; VF = [VF vf];
end
nb_col = 128;
vmin = min(VF); vmax = max(VF);
color_example = dsearch(VF,linspace(vmin,vmax,nb_col+1));
xset("colormap",jetcolormap(nb_col));
clf(); xset("hidden3d",xget("background"));
colorbar(vmin,vmax)
plot3d(XF, YF, list(ZF,color_example), flag=[-1 6 4])
xtitle("3d spline interpolation of "+func)
</scilab:image>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="linear_interpn">linear_interpn</link>
</member>
<member>
<link linkend="interp3d">interp3d</link>
</member>
<member>
<link linkend="bsplin3val">bsplin3val</link>
</member>
</simplelist>
</refsection>
<refsection>
<title>History</title>
<revhistory>
<revision>
<revnumber>5.4.0</revnumber>
<revremark>previously, imaginary part of input arguments were implicitly ignored.</revremark>
</revision>
</revhistory>
</refsection>
</refentry>
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