--- a/doc/interpreter/geometry.txi	Fri Aug 24 20:45:00 2007 +0000
+++ b/doc/interpreter/geometry.txi	Sat Aug 25 00:35:43 2007 +0000
@@ -6,17 +6,34 @@
 @node Geometry
 @chapter Geometry
 
+Much of geometry code in Octave is based on the QHull @footnote{Barber,
+C.B., Dobkin, D.P., and Huhdanpaa, H.T., "The Quickhull algorithm for
+convex hulls," ACM Trans. on Mathematical Software, 22(4):469-483, Dec
+1996, @url{http://www.qhull.org}}. Some of the documentation for Qhull,
+particularly for the options that can be passed to @code{delaunay},
+@code{voronoi} and @code{convhull}, etc, is relevant to Octave users.
+
 @menu
 * Delaunay Triangulation::
 * Voronoi Diagrams::
 * Convex Hull::
-* Plotting the Triangulation::
 * Interpolation on Scattered Data::
 @end menu
 
 @node Delaunay Triangulation
 @section Delaunay Triangulation
 
+The Delaunay triangulation is constructed from a set of
+circum-circles. These circum-circles are chosen so that there are at
+least three of the points in the set to triangulation on the
+circumference of the circum-circle. None of the points in the set of
+points falls within any of the circum-circles.
+
+In general there are only three points on the circumference of any
+circum-circle. However, in the some cases, and in particular for the
+case of a regular grid, 4 or more points can be on a single
+circum-circle. In this case the Delaunay triangulation is not unique. 
+
 @DOCSTRING(delaunay)
 
 The 3- and N-dimensional extension of the Delaunay triangulation are
@@ -30,10 +47,72 @@
 
 @DOCSTRING(delaunayn)
 
+An example of a Delaunay triangulation of a set of points is
+
+@example
+@group
+rand ("state", 2);
+x = rand (10, 1);
+y = rand (10, 1);
+T = delaunay (x, y);
+X = [ x(T(:,1)); x(T(:,2)); x(T(:,3)); x(T(:,1)) ];
+Y = [ y(T(:,1)); y(T(:,2)); y(T(:,3)); y(T(:,1)) ];
+axis ([0, 1, 0, 1]);
+plot(X, Y, "b", x, y, "r*");
+@end group
+@end example
+
+@ifnotinfo
+@noindent
+The result of which can be seen in @ref{fig:delaunay}.
+
+@float Figure,fig:delaunay
+@image{delaunay,8cm}
+@caption{Delaunay triangulation of a random set of points}
+@end float
+@end ifnotinfo
+
 @menu
+* Plotting the Triangulation::
 * Identifying points in Triangulation::
 @end menu
 
+@node Plotting the Triangulation
+@subsection Plotting the Triangulation
+
+Octave has the functions @code{triplot} and @code{trimesh} to plot the
+Delaunay triangulation of a 2-dimensional set of points.
+
+@DOCSTRING(triplot)
+
+@DOCSTRING(trimesh)
+
+The difference between @code{triplot} and @code{trimesh} is that the
+former only plots the 2-dimensional triangulation itself, whereas the
+second plots the value of some function @code{f (@var{x}, @var{y})}.
+An example of the use of the @code{triplot} function is
+
+@example
+@group
+rand ("state", 2)
+x = rand (20, 1);
+y = rand (20, 1);
+tri = delaunay (x, y);
+triplot (tri, x, y);
+@end group
+@end example
+
+that plot the Delaunay triangulation of a set of random points in
+2-dimensions.
+@ifnotinfo
+The output of the above can be seen in @ref{fig:triplot}.
+
+@float Figure,fig:triplot
+@image{triplot,8cm}
+@caption{Delaunay triangulation of a random set of points}
+@end float
+@end ifnotinfo
+
 @node Identifying points in Triangulation
 @subsection Identifying points in Triangulation
 
@@ -143,7 +222,7 @@
 The @code{dsearch} and @code{dsearchn} find the closest point in a
 tessellation to the desired point. The desired point does not
 necessarily have to be in the tessellation, and even if it the returned
-point of the tessellation does not have to be one of the vertices of the
+point of the tessellation does not have to be one of the vertexes of the
 N-simplex within which the desired point is found.
 
 @DOCSTRING(dsearch)
@@ -183,33 +262,138 @@
 such that all points in @code{@var{v}(@var{p})}, a partitions of the
 tessellation where @var{p} is a member of @var{s}, are closer to @var{p}
 than any other point in @var{s}.  The Voronoi diagram is related to the
-Delaunay triangulation of a set of points.
+Delaunay triangulation of a set of points, in that the vertexes of the
+Voronoi tessellation are the center's of the circum-circles of the
+simplicies of the Delaunay tessellation. 
 
 @DOCSTRING(voronoi)
 
 @DOCSTRING(voronoin)
 
+An example of the use of @code{voronoi} is
+
+@example
+@group
+rand("state",9);
+x = rand(10,1);
+y = rand(10,1);
+tri = delaunay (x, y);
+[vx, vy] = voronoi (x, y, tri);
+triplot (tri, x, y, "b");
+hold on;
+plot (vx, vy, "r");
+@end group
+@end example
+
+@ifnotinfo
+@noindent
+The result of which can be seen in @ref{fig:voronoi}. Note that the
+circum-circle of one of the triangles has been added to this figure, to
+make the relationship between the Delaunay tessellation and the Voronoi
+diagram clearer.
+
+@float Figure,fig:voronoi
+@image{voronoi,8cm}
+@caption{Delaunay triangulation and Voronoi diagram of a random set of points}
+@end float
+@end ifnotinfo
+
+Additional information about the size of the facets of a Voronoi diagram
+can be had with the @code{polyarea} function.
+
+@DOCSTRING(polyarea)
+
+An example of the use of @code{polyarea} might be 
+
+@example
+@group
+rand ("state", 2);
+x = rand (10, 1);
+y = rand (10, 1);
+[c, f] = voronoin ([x, y]);
+af = zeros (size(f));
+for i = 1 : length (f)
+  af(i) = polyarea (c (f @{i, :@}, 1), c (f @{i, :@}, 2));
+endfor
+@end group
+@end example
+
+Facets of the Voronoi diagram with a vertex at infinity have infinity area.
+
 @node Convex Hull
 @section Convex Hull
 
+The convex hull of a set of points, is the minimum convex envelope
+containing all of the points. Octave has the functions @code{convhull}
+and @code{convhulln} to calculate the convec hull of 2-dimensional and
+N-dimensional sets of points.
+
 @DOCSTRING(convhull)
 
 @DOCSTRING(convhulln)
 
-@node Plotting the Triangulation
-@section Plotting the Triangulation
+An example of the use of @code{convhull} is
 
-@DOCSTRING(triplot)
+@example
+@group
+x = -3:0.05:3;
+y = abs (sin (x));
+k = convhull (x, y);
+plot (x(k), y(k), "r-", x, y, "b+");
+axis ([-3.05, 3.05, -0.05, 1.05]);
+@end group
+@end example
 
-@DOCSTRING(trimesh)
+@ifnotinfo
+@noindent
+The output of the above can be seen in @ref{fig:convhull}.
 
-@DOCSTRING(polyarea)
+@float Figure,fig:convhull
+@image{convhull,8cm}
+@caption{The convex hull of a simple set of points}
+@end float
+@end ifnotinfo
 
 @node Interpolation on Scattered Data
 @section Interpolation on Scattered Data
 
+An important use of the Delaunay tessellation is that it can be used to
+interpolate from scattered data to an arbitrary set of points. To do
+this the N-simplex of the known set of points is calculated with
+@code{delaunay}, @code{delaunay3} or @code{delaunayn}. Then the
+simplicies in to which the desired points are found are
+identified. Finally the vertices of the simplicies are used to
+interpolate to the desired points. The functions that perform this
+interpolation are @code{griddata}, @code{griddata3} and
+@code{griddatan}.
+
 @DOCSTRING(griddata)
 
 @DOCSTRING(griddata3)
 
 @DOCSTRING(griddatan)
+
+An example of the use of the @code{griddata} function is
+
+@example
+@group
+rand("state",1);
+x=2*rand(1000,1)-1;
+y=2*rand(size(x))-1;
+z=sin(2*(x.^2+y.^2));
+[xx,yy]=meshgrid(linspace(-1,1,32));
+griddata(x,y,z,xx,yy);
+@end group
+@end example
+
+@noindent
+that interpolates from a random scattering of points, to a uniform
+grid. 
+@ifnotinfo
+The output of the above can be seen in @ref{fig:griddata}.
+
+@float Figure,fig:griddata
+@image{griddata,8cm}
+@caption{Interpolation from a scattered data to a regular grid}
+@end float
+@end ifnotinfo