(octave.info)Voronoi Diagrams
30.2 Voronoi Diagrams
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A Voronoi diagram or Voronoi tessellation of a set of points S in an
N-dimensional space, is the tessellation of the N-dimensional space such
that all points in ‘V(P)’, a partitions of the tessellation where P is a
member of S, are closer to P than any other point in 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.
-- : voronoi (X, Y)
-- : voronoi (X, Y, OPTIONS)
-- : voronoi (..., "linespec")
-- : voronoi (HAX, ...)
-- : H = voronoi (...)
-- : [VX, VY] = voronoi (...)
Plot the Voronoi diagram of points ‘(X, Y)’.
The Voronoi facets with points at infinity are not drawn.
The OPTIONS argument, which must be a string or cell array of
strings, contains options passed to the underlying qhull command.
See the documentation for the Qhull library for details
<http://www.qhull.org/html/qh-quick.htm#options>.
If "linespec" is given it is used to set the color and line style
of the plot.
If an axes graphics handle HAX is supplied then the Voronoi diagram
is drawn on the specified axes rather than in a new figure.
If a single output argument is requested then the Voronoi diagram
will be plotted and a graphics handle H to the plot is returned.
[VX, VY] = voronoi (...) returns the Voronoi vertices instead of
plotting the diagram.
x = rand (10, 1);
y = rand (size (x));
h = convhull (x, y);
[vx, vy] = voronoi (x, y);
plot (vx, vy, "-b", x, y, "o", x(h), y(h), "-g");
legend ("", "points", "hull");
See also: Note: voronoin, *note delaunay:
XREFdelaunay, Note: convhull.
-- : [C, F] = voronoin (PTS)
-- : [C, F] = voronoin (PTS, OPTIONS)
Compute N-dimensional Voronoi facets.
The input matrix PTS of size [n, dim] contains n points in a space
of dimension dim.
C contains the points of the Voronoi facets. The list F contains,
for each facet, the indices of the Voronoi points.
An optional second argument, which must be a string or cell array
of strings, contains options passed to the underlying qhull
command. See the documentation for the Qhull library for details
<http://www.qhull.org/html/qh-quick.htm#options>.
The default options depend on the dimension of the input:
• 2-D and 3-D: OPTIONS = ‘{"Qbb"}’
• 4-D and higher: OPTIONS = ‘{"Qbb", "Qx"}’
If OPTIONS is not present or ‘[]’ then the default arguments are
used. Otherwise, OPTIONS replaces the default argument list. To
append user options to the defaults it is necessary to repeat the
default arguments in OPTIONS. Use a null string to pass no
arguments.
See also: Note: voronoi, *note convhulln:
XREFconvhulln, Note: delaunayn.
An example of the use of ‘voronoi’ is
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");
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 ‘polyarea’ and ‘inpolygon’ functions respectively.
-- : polyarea (X, Y)
-- : polyarea (X, Y, DIM)
Determine area of a polygon by triangle method.
The variables X and Y define the vertex pairs, and must therefore
have the same shape. They can be either vectors or arrays. If
they are arrays then the columns of X and Y are treated separately
and an area returned for each.
If the optional DIM argument is given, then ‘polyarea’ works along
this dimension of the arrays X and Y.
An example of the use of ‘polyarea’ might be
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
Facets of the Voronoi diagram with a vertex at infinity have infinity
area. A simplified version of ‘polyarea’ for rectangles is available
with ‘rectint’
-- : AREA = rectint (A, B)
Compute area or volume of intersection of rectangles or N-D boxes.
Compute the area of intersection of rectangles in A and rectangles
in B. N-dimensional boxes are supported in which case the volume,
or hypervolume is computed according to the number of dimensions.
2-dimensional rectangles are defined as ‘[xpos ypos width height]’
where xpos and ypos are the position of the bottom left corner.
Higher dimensions are supported where the coordinates for the
minimum value of each dimension follow the length of the box in
that dimension, e.g., ‘[xpos ypos zpos kpos ... width height depth
k_length ...]’.
Each row of A and B define a rectangle, and if both define multiple
rectangles, then the output, AREA, is a matrix where the i-th row
corresponds to the i-th row of a and the j-th column corresponds to
the j-th row of b.
See also: Note: polyarea.
-- : IN = inpolygon (X, Y, XV, YV)
-- : [IN, ON] = inpolygon (X, Y, XV, YV)
For a polygon defined by vertex points ‘(XV, YV)’, return true if
the points ‘(X, Y)’ are inside (or on the boundary) of the polygon;
Otherwise, return false.
The input variables X and Y, must have the same dimension.
The optional output ON returns true if the points are exactly on
the polygon edge, and false otherwise.
See also: Note: delaunay.
An example of the use of ‘inpolygon’ might be
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]);
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