Info: (octave) Delaunay Triangulation

octave: Delaunay Triangulation

 
 30.1 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.
 
  -- : TRI = delaunay (X, Y)
  -- : TETR = delaunay (X, Y, Z)
  -- : TRI = delaunay (X)
  -- : TRI = delaunay (..., OPTIONS)
      Compute the Delaunay triangulation for a 2-D or 3-D set of points.
 
      For 2-D sets, the return value TRI is a set of triangles which
      satisfies the Delaunay circum-circle criterion, i.e., no data point
      from [X, Y] is within the circum-circle of the defining triangle.
      The set of triangles TRI is a matrix of size [n, 3].  Each row
      defines a triangle and the three columns are the three vertices of
      the triangle.  The value of ‘TRI(i,j)’ is an index into X and Y for
      the location of the j-th vertex of the i-th triangle.
 
      For 3-D sets, the return value TETR is a set of tetrahedrons which
      satisfies the Delaunay circum-circle criterion, i.e., no data point
      from [X, Y, Z] is within the circum-circle of the defining
      tetrahedron.  The set of tetrahedrons is a matrix of size [n, 4].
      Each row defines a tetrahedron and the four columns are the four
      vertices of the tetrahedron.  The value of ‘TETR(i,j)’ is an index
      into X, Y, Z for the location of the j-th vertex of the i-th
      tetrahedron.
 
      The input X may also be a matrix with two or three columns where
      the first column contains x-data, the second y-data, and the
      optional third column contains z-data.
 
      The optional last 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 are ‘{"Qt", "Qbb", "Qc", "Qz"}’.
 
      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.
 
           x = rand (1, 10);
           y = rand (1, 10);
           tri = delaunay (x, y);
           triplot (tri, x, y);
           hold on;
           plot (x, y, "r*");
           axis ([0,1,0,1]);
 
DONTPRINTYET       See also: delaunayn XREFdelaunayn, *noteconvhull:
DONTPRINTYET DONTPRINTYET       See also: delaunayn XREFdelaunayn, convhull

      XREFconvhull, voronoi XREFvoronoi, *notetriplot:
DONTPRINTYET DONTPRINTYET DONTPRINTYET       See also: delaunayn XREFdelaunayn, convhull

      XREFconvhull, voronoi XREFvoronoi, triplot

      XREFtriplot, trimesh XREFtrimesh, *notetetramesh:
DONTPRINTYET DONTPRINTYET DONTPRINTYET       See also: delaunayn XREFdelaunayn, convhull

      XREFconvhull, voronoi XREFvoronoi, triplot

      XREFtriplot, trimesh XREFtrimesh, tetramesh

      XREFtetramesh, trisurf XREFtrisurf.
 
    For 3-D inputs ‘delaunay’ returns a set of tetrahedra that satisfy
 the Delaunay circum-circle criteria.  Similarly, ‘delaunayn’ returns the
 N-dimensional simplex satisfying the Delaunay circum-circle criteria.
 The N-dimensional extension of a triangulation is called a tessellation.
 
  -- : T = delaunayn (PTS)
  -- : T = delaunayn (PTS, OPTIONS)
      Compute the Delaunay triangulation for an N-dimensional set of
      points.
 
      The Delaunay triangulation is a tessellation of the convex hull of
      a set of points such that no N-sphere defined by the N-triangles
      contains any other points from the set.
 
      The input matrix PTS of size [n, dim] contains n points in a space
      of dimension dim.  The return matrix T has size [m, dim+1].  Each
      row of T contains a set of indices back into the original set of
      points PTS which describes a simplex of dimension dim.  For
      example, a 2-D simplex is a triangle and 3-D simplex is a
      tetrahedron.
 
      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 = ‘{"Qt", "Qbb", "Qc", "Qz"}’
 
         • 4-D and higher: OPTIONS = ‘{"Qt", "Qbb", "Qc", "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.
 
DONTPRINTYET       See also: delaunay XREFdelaunay, *noteconvhulln:
DONTPRINTYET DONTPRINTYET       See also: delaunay XREFdelaunay, convhulln

      XREFconvhulln, voronoin XREFvoronoin, *notetrimesh:
DONTPRINTYET DONTPRINTYET       See also: delaunay XREFdelaunay, convhulln

      XREFconvhulln, voronoin XREFvoronoin, trimesh

      XREFtrimesh, tetramesh XREFtetramesh.
 
    An example of a Delaunay triangulation of a set of points is
 
      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*");
octave: Delaunay Triangulation

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