@c Copyright (C) 2007-2019 John W. Eaton and David Bateman
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@node Geometry
@chapter Geometry

Much of the geometry code in Octave is based on the Qhull
library@footnote{@nospell{Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T.},
@cite{The Quickhull Algorithm for Convex Hulls}, @nospell{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::
* 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 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)

For 3-D inputs @code{delaunay} returns a set of tetrahedra that satisfy the
Delaunay circum-circle criteria.  Similarly, @code{delaunayn} returns the
N-dimensional simplex satisfying the Delaunay circum-circle criteria.
The N-dimensional extension of a triangulation is called a tessellation.

@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
@center @image{delaunay,4in}
@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}, @code{trimesh}, and @code{trisurf}
to plot the Delaunay triangulation of a 2-dimensional set of points.
@code{tetramesh} will plot the triangulation of a 3-dimensional set of points.

@DOCSTRING(triplot)

@DOCSTRING(trimesh)

@DOCSTRING(trisurf)

@DOCSTRING(tetramesh)

The difference between @code{triplot}, and @code{trimesh} or @code{trisurf},
is that the former only plots the 2-dimensional triangulation itself, whereas
the second two plot the value of a 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

@noindent
which plots 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
@center @image{triplot,4in}
@caption{Delaunay triangulation of a random set of points}
@end float
@end ifnotinfo

@node Identifying Points in Triangulation
@subsection Identifying Points in Triangulation

It is often necessary to identify whether a particular point in the
N-dimensional space is within the Delaunay tessellation of a set of
points in this N-dimensional space, and if so which N-simplex contains
the point and which point in the tessellation is closest to the desired
point.  The functions @code{tsearch} and @code{dsearch} perform this
function in a triangulation, and @code{tsearchn} and @code{dsearchn} in
an N-dimensional tessellation.

To identify whether a particular point represented by a vector @var{p}
falls within one of the simplices of an N-simplex, we can write the
Cartesian coordinates of the point in a parametric form with respect to
the N-simplex.  This parametric form is called the Barycentric
Coordinates of the point.  If the points defining the N-simplex are given
by @var{N} + 1 vectors @code{@var{t}(@var{i},:)}, then the Barycentric
coordinates defining the point @var{p} are given by

@example
@var{p} = @var{beta} * @var{t}
@end example

@noindent
where @var{beta} contains @var{N} + 1 values that together as a vector
represent the Barycentric coordinates of the point @var{p}.  To ensure a unique
solution for the values of @var{beta} an additional criteria of

@example
sum (@var{beta}) == 1
@end example

@noindent
is imposed, and we can therefore write the above as

@example
@group
@var{p} - @var{t}(end, :) = @var{beta}(1:end-1) * (@var{t}(1:end-1, :)
                - ones (@var{N}, 1) * @var{t}(end, :)
@end group
@end example

@noindent
Solving for @var{beta} we can then write

@example
@group
@var{beta}(1:end-1) = (@var{p} - @var{t}(end, :)) /
                (@var{t}(1:end-1, :) - ones (@var{N}, 1) * @var{t}(end, :))
@var{beta}(end) = sum (@var{beta}(1:end-1))
@end group
@end example

@noindent
which gives the formula for the conversion of the Cartesian coordinates
of the point @var{p} to the Barycentric coordinates @var{beta}.  An
important property of the Barycentric coordinates is that for all points
in the N-simplex

@example
0 <= @var{beta}(@var{i}) <= 1
@end example

@noindent
Therefore, the test in @code{tsearch} and @code{tsearchn} essentially
only needs to express each point in terms of the Barycentric coordinates
of each of the simplices of the N-simplex and test the values of
@var{beta}.  This is exactly the implementation used in
@code{tsearchn}.  @code{tsearch} is optimized for 2-dimensions and the
Barycentric coordinates are not explicitly formed.

@DOCSTRING(tsearch)

@DOCSTRING(tsearchn)

An example of the use of @code{tsearch} can be seen with the simple
triangulation

@example
@group
@var{x} = [-1; -1; 1; 1];
@var{y} = [-1; 1; -1; 1];
@var{tri} = [1, 2, 3; 2, 3, 4];
@end group
@end example

@noindent
consisting of two triangles defined by @var{tri}.  We can then identify
which triangle a point falls in like

@example
@group
tsearch (@var{x}, @var{y}, @var{tri}, -0.5, -0.5)
@result{} 1
tsearch (@var{x}, @var{y}, @var{tri}, 0.5, 0.5)
@result{} 2
@end group
@end example

@noindent
and we can confirm that a point doesn't lie within one of the triangles like

@example
@group
tsearch (@var{x}, @var{y}, @var{tri}, 2, 2)
@result{} NaN
@end group
@end example

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 vertexes of the
N-simplex within which the desired point is found.

@DOCSTRING(dsearch)

@DOCSTRING(dsearchn)

An example of the use of @code{dsearch}, using the above values of
@var{x}, @var{y} and @var{tri} is

@example
@group
dsearch (@var{x}, @var{y}, @var{tri}, -2, -2)
@result{} 1
@end group
@end example

If you wish the points that are outside the tessellation to be flagged,
then @code{dsearchn} can be used as

@example
@group
dsearchn ([@var{x}, @var{y}], @var{tri}, [-2, -2], NaN)
@result{} NaN
dsearchn ([@var{x}, @var{y}], @var{tri}, [-0.5, -0.5], NaN)
@result{} 1
@end group
@end example

@noindent
where the point outside the tessellation are then flagged with @code{NaN}.

@node Voronoi Diagrams
@section Voronoi Diagrams

A Voronoi diagram or Voronoi tessellation of a set of points @var{s} in
an N-dimensional space, is the tessellation of the N-dimensional space
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, in that the vertexes of the
Voronoi tessellation are the centers of the circum-circles of the
simplices 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
@center @image{voronoi,4in}
@caption{Delaunay triangulation (blue lines) and Voronoi diagram (red lines)
of a random set of points}
@end float
@end ifnotinfo

Additional information about the size of the facets of a Voronoi
diagram, and which points of a set of points is in a polygon can be had
with the @code{polyarea} and @code{inpolygon} functions respectively.

@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.  A simplified version of @code{polyarea} for rectangles is
available with @code{rectint}

@DOCSTRING(rectint)

@DOCSTRING(inpolygon)

An example of the use of @code{inpolygon} might be

@example
@group
randn ("state", 2);
x = randn (100, 1);
y = randn (100, 1);
vx = cos (pi * [-1 : 0.1: 1]);
vy = sin (pi * [-1 : 0.1 : 1]);
in = inpolygon (x, y, vx, vy);
plot (vx, vy, x(in), y(in), "r+", x(!in), y(!in), "bo");
axis ([-2, 2, -2, 2]);
@end group
@end example

@ifnotinfo
@noindent
The result of which can be seen in @ref{fig:inpolygon}.

@float Figure,fig:inpolygon
@center @image{inpolygon,4in}
@caption{Demonstration of the @code{inpolygon} function to determine the
points inside a polygon}
@end float
@end ifnotinfo

@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 convex hull of 2-dimensional and
N-dimensional sets of points.

@DOCSTRING(convhull)

@DOCSTRING(convhulln)

An example of the use of @code{convhull} is

@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

@ifnotinfo
@noindent
The output of the above can be seen in @ref{fig:convhull}.

@float Figure,fig:convhull
@center @image{convhull,4in}
@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} or @code{delaunayn}.  Then the simplices in to which the
desired points are found are identified.  Finally the vertices of the simplices
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));
zz = griddata (x, y, z, xx, yy);
mesh (xx, yy, zz);
@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
@center @image{griddata,4in}
@caption{Interpolation from a scattered data to a regular grid}
@end float
@end ifnotinfo
