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<title>rbox -- generate point distributions</title>
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<h1><a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/cone.html"><img
src="qh--cone.gif" alt="[CONE]" align="middle" width="100"
height="100"></a>rbox -- generate point distributions</h1>

<blockquote>
       rbox  generates  random or regular points according to the
       options given, and  outputs  the  points  to  stdout.  The
       points  are  generated in a cube, unless 's', 'x', or 'y'
           are given.

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="synopsis">rbox synopsis</a></h3>
<pre>
rbox- generate various point distributions.  Default is random in cube.

args (any order, space separated):
  3000    number of random points in cube, lens, spiral, sphere or grid
  D3      dimension 3-d
  c       add a unit cube to the output ('c G2.0' sets size)
  d       add a unit diamond to the output ('d G2.0' sets size)
  l       generate a regular 3-d spiral
  r       generate a regular polygon, ('r s Z1 G0.1' makes a cone)
  s       generate cospherical points
  x       generate random points in simplex, may use 'r' or 'Wn'
  y       same as 'x', plus simplex
  Cn,r,m  add n nearly coincident points within radius r of m points
  Pn,m,r  add point [n,m,r] first, pads with 0

  Ln      lens distribution of radius n.  Also 's', 'r', 'G', 'W'.
  Mn,m,r  lattice (Mesh) rotated by [n,-m,0], [m,n,0], [0,0,r], ...
          '27 M1,0,1' is {0,1,2} x {0,1,2} x {0,1,2}.  Try 'M3,4 z'.
  W0.1    random distribution within 0.1 of the cube's or sphere's surface
  Z0.5 s  random points in a 0.5 disk projected to a sphere
  Z0.5 s G0.6 same as Z0.5 within a 0.6 gap

  Bn      bounding box coordinates, default 0.5
  h       output as homogeneous coordinates for cdd
  n       remove command line from the first line of output
  On      offset coordinates by n
  t       use time as the random number seed (default is command line)
  tn      use n as the random number seed
  z       print integer coordinates, default 'Bn' is 1e+06
</pre>

<h3><a href="#TOP">&#187;</a><a name="outputs">rbox outputs</a></h3>
<blockquote>

The format of the output is the following: first line contains
       the dimension and a comment, second line contains the
           number of points, and the following lines contain the points,
       one  point per line. Points are represented by their coordinate values.

<p>For example, <tt>rbox c 10 D2</tt> generates
<blockquote>
<pre>
2 RBOX c 10 D2
14
-0.4999921736307369 -0.3684622117955817
0.2556053225468894 -0.0413498678629751
0.0327672376602583 -0.2810408135699488
-0.452955383763607 0.17886471718444
0.1792964061529342 0.4346928963760779
-0.1164979223315585 0.01941637230982666
0.3309653464993139 -0.4654278894564396
-0.4465383649305798 0.02970019358182344
0.1711493843897706 -0.4923018137852678
-0.1165843490665633 -0.433157762450313
  -0.5   -0.5
  -0.5    0.5
   0.5   -0.5
   0.5    0.5
</pre>

</blockquote>

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="examples">rbox examples</a></h3>

<pre>
       rbox 10
              10 random points in the unit cube centered  at  the
              origin.

       rbox 10 s D2
              10 random points on a 2-d circle.

       rbox 100 W0
              100 random points on the surface of a cube.

       rbox 1000 s D4
              1000 random points on a 4-d sphere.

       rbox c D5 O0.5
              a 5-d hypercube with one corner at the origin.

       rbox d D10
              a 10-d diamond.

       rbox x 1000 r W0
              100 random points on the surface of a fixed simplex

       rbox y D12
              a 12-d simplex.

       rbox l 10
              10 random points along a spiral

       rbox l 10 r
              10 regular points  along  a  spiral  plus  two  end
              points

       rbox 1000 L10000 D4 s
              1000 random points on the surface of a narrow lens.

           rbox 1000 L100000 s G1e-6
                  1000 random points near the edge of a narrow lens

       rbox c G2 d G3
              a cube with coordinates +2/-2 and  a  diamond  with
              coordinates +3/-3.

       rbox 64 M3,4 z
              a  rotated,  {0,1,2,3} x {0,1,2,3} x {0,1,2,3} lat-
              tice (Mesh) of integer points.

       rbox P0 P0 P0 P0 P0
              5 copies of the origin in 3-d.  Try 'rbox P0 P0  P0
              P0 P0 | qhull QJ'.

       r 100 s Z1 G0.1
              two  cospherical  100-gons plus another cospherical
              point.

       100 s Z1
              a cone of points.

       100 s Z1e-7
              a narrow cone of points with many precision errors.
</pre>

<h3><a href="#TOP">&#187;</a><a name="notes">rbox notes</a></h3>
<blockquote>
Some combinations of arguments generate odd results.

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="options">rbox options</a></h3>

<pre>
       n      number of points

       Dn     dimension n-d (default 3-d)

       Bn     bounding box coordinates (default 0.5)

       l      spiral distribution, available only in 3-d

       Ln     lens  distribution  of  radius n.  May be used with
              's', 'r', 'G', and 'W'.

       Mn,m,r lattice  (Mesh)  rotated  by  {[n,-m,0],   [m,n,0],
              [0,0,r],  ...}.   Use  'Mm,n'  for a rigid rotation
              with r = sqrt(n^2+m^2).  'M1,0'  is  an  orthogonal
              lattice.   For  example,  '27  M1,0'  is  {0,1,2} x
              {0,1,2} x {0,1,2}.

       s      cospherical points randomly generated in a cube and
              projected to the unit sphere

       x      simplicial  distribution.   It  is fixed for option
              'r'.  May be used with 'W'.

       y      simplicial distribution plus a simplex.   Both  'x'
              and 'y' generate the same points.

       Wn     restrict  points  to distance n of the surface of a
              sphere or a cube

       c      add a unit cube to the output

       c Gm   add a cube with all combinations of +m  and  -m  to
              the output

       d      add a unit diamond to the output.

       d Gm   add a diamond made of 0, +m and -m to the output

       Cn,r,m add n nearly coincident points within radius r of m points

       Pn,m,r add point [n,m,r] to the output first.  Pad coordi-
              nates with 0.0.

       n      Remove the command line from the first line of out-
              put.

       On     offset the data by adding n to each coordinate.

       t      use  time  in  seconds  as  the  random number seed
              (default is command line).

       tn     set the random number seed to n.

       z      generate integer coordinates.  Use 'Bn'  to  change
              the  range.   The  default  is 'B1e6' for six-digit
              coordinates.  In R^4, seven-digit coordinates  will
              overflow hyperplane normalization.

       Zn s   restrict points to a disk about the z+ axis and the
              sphere (default Z1.0).  Includes the opposite pole.
              'Z1e-6'  generates  degenerate  points under single
              precision.

       Zn Gm s
              same as Zn with an empty center (default G0.5).

       r s D2 generate a regular polygon

       r s Z1 G0.1
              generate a regular cone
</pre>

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