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.\" ========================================================================
.\"
.IX Title "TriD 3pm"
.TH TriD 3pm 2024-02-05 "perl v5.38.2" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
.\" way too many mistakes in technical documents.
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.nh
.SH NAME
PDL::Graphics::TriD \- PDL 3D interface
.SH SYNOPSIS
.IX Header "SYNOPSIS"
.Vb 1
\& use PDL::Graphics::TriD;
\&
\& # Generate a somewhat interesting sequence of points:
\& $t = sequence(100)/10;
\& $x = sin($t); $y = cos($t), $z = $t;
\& $coords = cat($x, $y, $z)\->transpose;
\& my $red = cos(2*$t); my $green = sin($t); my $blue = $t;
\& $colors = cat($red, $green, $blue)\->transpose;
\&
\& # After each graph, let the user rotate and
\& # wait for them to press \*(Aqq\*(Aq, then make new graph
\& line3d($coords);       # $coords = (3,n,...)
\& line3d($coords,$colors);  # $colors = (3,n,...)
\& line3d([$x,$y,$z]);
\&
\& # Generate a somewhat interesting sequence of surfaces
\& $surf1 = (rvals(100, 100) / 50)**2 + sin(xvals(100, 100) / 10);
\& $surf2 = sqrt(rvals(zeroes(50,50))/2);
\& $x = sin($surface); $y = cos($surface), $z = $surface;
\& $coords = cat($x, $y, $z)\->transpose;
\& $red = cos(2*$surface); $green = sin($surface); $blue = $surface;
\& $colors = cat($red, $green, $blue)\->transpose;
\& imagrgb([$red,$green,$blue]);     # 2\-d ndarrays
\& lattice3d([$surf1]);
\& points3d([$x,$y,$z]);
\& spheres3d([$x,$y,$z]);  # preliminary implementation
\&
\& hold3d(); # the following graphs are on top of each other and the previous
\& line3d([$x,$y,$z]);
\& line3d([$x,$y,$z+1]);
\& $pic = grabpic3d(); # Returns the picture in a (3,$x,$y) float ndarray (0..1).
\&
\& release3d(); # the next graph will again wipe out things.
.Ve
.SH WARNING
.IX Header "WARNING"
These modules are still in a somewhat unfocused state: don't use them yet
if you don't know how to make them work if they happen to do something
strange.
.SH DESCRIPTION
.IX Header "DESCRIPTION"
This module implements a generic 3D plotting interface for PDL.
Points, lines and surfaces (among other objects) are supported.
.PP
With OpenGL, it is easy to manipulate the resulting 3D objects
with the mouse in real time \- this helps data visualization a lot.
.SH "SELECTING A DEVICE"
.IX Header "SELECTING A DEVICE"
The default device for TriD is currently OpenGL.
You can specify a different device either in your program
or in the environment variable \f(CW\*(C`PDL_3D_DEVICE\*(C'\fR.
The one specified in the program takes priority.
.PP
The currently available devices are
.IP GL 8
.IX Item "GL"
OpenGL
.IP GLpic 8
.IX Item "GLpic"
OpenGL but off-line (pixmap) rendering and writing to
a graphics file.
.IP "VRML (\fI Not available this release \fR)" 8
.IX Item "VRML ( Not available this release )"
VRML objects rendering. This writes a VRML file describing the
scene. This VRML file can then be read with  a browser.
.SH "ONLINE AND OFFLINE VISUALIZATION"
.IX Header "ONLINE AND OFFLINE VISUALIZATION"
TriD  offers both on\- and off-line visualization.
Currently the interface  w.r.t. this division is still much
in motion.
.PP
For OpenGL you can select either on\- or off-line rendering.
VRML is currently always offline (this may change  later,
if someone bothers to write  the  java(script)  code to  contact
PDL and wait for the next PDL image over the network.
.SH "COORDINATE SPECIFICATIONS"
.IX Header "COORDINATE SPECIFICATIONS"
Specifying a set of coordinates is generally a context-dependent operation.
For a traditional 3D surface plot, you'll want two of the coordinates
to have just the xvals and yvals of the ndarray, respectively.
For a line, you would generally want to have one coordinate held
at zero and the other advancing.
.PP
This module tries to make a reasonable way of specifying the context
while letting you do whatever you want by overriding the default
interpretation.
.PP
The alternative syntaxes for specifying a set of coordinates (or colors) are
.PP
.Vb 7
\&   $ndarray                             # MUST have 3 as first dim.
\&  [$ndarray]
\&  [$ndarray1,$ndarray2]
\&  [$ndarray1,$ndarray2,$ndarray3]
\&  [CONTEXT,$ndarray]
\&  [CONTEXT,$ndarray1,$ndarray2]
\&  [CONTEXT,$ndarray1,$ndarray2,$ndarray3]
.Ve
.PP
where \f(CW\*(C`CONTEXT\*(C'\fR is a string describing in which context you wish these
ndarrays to be interpreted. Each routine specifies a default context
which is explained in the routines documentation.
Context is usually used only to understand what the user wants
when they specify less than 3 ndarrays.
.PP
The following contexts are currently supported:
.IP SURF2D 8
.IX Item "SURF2D"
A 2\-D lattice. \f(CW\*(C` [$ndarray] \*(C'\fR is interpreted as the Z coordinate over
a lattice over the first dimension. Equivalent to
\&\f(CW\*(C`[$ndarray\->xvals, $ndarray\->yvals, $ndarray]\*(C'\fR.
.IP POLAR2D 8
.IX Item "POLAR2D"
A 2\-D polar coordinate system. \f(CW\*(C` [$ndarray] \*(C'\fR is interpreted as the
z coordinate over theta and r (theta = the first dimension of the ndarray).
.IP COLOR 8
.IX Item "COLOR"
A set of colors. \f(CW\*(C` [$ndarray] \*(C'\fR is interpreted as grayscale color
(equivalent to \f(CW\*(C` [$ndarray,$ndarray,$ndarray] \*(C'\fR).
.IP LINE 8
.IX Item "LINE"
A line made of 1 or 2 coordinates. \f(CW\*(C` [$ndarray] \*(C'\fR is interpreted as
\&\f(CW\*(C`[$ndarray\->xvals,$ndarray,0]\*(C'\fR. \f(CW\*(C` [$ndarray1,$ndarray2] \*(C'\fR is interpreted as
\&\f(CW\*(C`[$ndarray1,$ndarray2,$ndarray1\->xvals]\*(C'\fR.
.PP
What makes contexts useful is that if you want to plot points
instead of the full surface you plotted with
.PP
.Vb 1
\&  imag3d([$zcoords]);
.Ve
.PP
you don't need to start thinking about where to plot the points:
.PP
.Vb 1
\&  points3d([SURF2D,$zcoords]);
.Ve
.PP
will do exactly the same.
.SS "Wrapping your head around 3d surface specifications"
.IX Subsection "Wrapping your head around 3d surface specifications"
Let's begin by thinking about how you might make a 2d data plot.
If you sampled your data at regular intervals, you would have
a time serires y(t) = (y0, y1, y2, ...).  You could plot y vs t
by computing t0 = 0, t1 = dt, t2 = 2 * dt, and then plotting
(t0, y0), (t1, y1), etc.
.PP
Next suppose that you measured x(t) and y(t).  You can still
plot y vs t, but you can also plot y vs x by plotting (x0, y0),
(x1, y1), etc.  The x\-values don't have to increase monotonically:
they could back-track on each other, for example, like the
latitude and longitude of a boat on a lake.  If you use plplot,
you would plot this data using
\&\f(CW\*(C`$pl\->xyplot($x, $y, PLOTTYPE => \*(AqPOINTS\*(Aq)\*(C'\fR.
.PP
Good.  Now let's add a third coordinate, z(t).  If you actually
sampled x and y at regular intervals, so that x and y lie on a
grid, then you can construct a grid for z(x, y), and you would
get a surface.  This is the situation in which you would use
\&\f(CWmesh3d([$surface])\fR.
.PP
Of course, your data is not required to be regularly gridded.
You could, for example, be measuring the flight path of a bat
flying after mosquitos, which could be wheeling and arching
all over the space.  This is what you might plot using
\&\f(CW\*(C`line3d([$x, $y, $z])\*(C'\fR.  You could plot the trajectories of
multiple bats, in which case \f(CW$x\fR, \f(CW$y\fR, and \f(CW$z\fR would have
multiple columns, but in general you wouldn't expect them to be
coordinated.
.PP
More generally, each coordinate is expected to be arranged in a 3D
fashion, similar to \f(CW\*(C`3,x,y\*(C'\fR. The "3" is the actual 3D coordinates of
each point. The "x,y" help with gridding, because each point at \f(CW\*(C`x,y\*(C'\fR
is expected to have as geographical neighbours \f(CW\*(C`x+1,y\*(C'\fR, \f(CW\*(C`x\-1,y\*(C'\fR,
\&\f(CW\*(C`x,y+1\*(C'\fR, \f(CW\*(C`x,y\-1\*(C'\fR, and the grid polygon-building relies on that.
This is how, and why, the 3D earth in \f(CW\*(C`demo 3d\*(C'\fR arranges its data.
.PP
.Vb 1
\& #!/usr/bin/perl
\&
\& use PDL;
\& use PDL::Graphics::TriD;
\&
\& # Draw out a trajectory in three\-space
\& $t = sequence(100)/10;
\& $x = sin($t); $y = cos($t); $z = $t;
\&
\& # Plot the trajectory as (x(t), y(t), z(t))
\& print "using line3d to plot a trajectory (press q when you\*(Aqre done twiddling)\en";
\& line3d [$x,$y,$z];
\&
\& # If you give it a single ndarray, it expects
\& # the data to look like
\& # ((x1, y1, z1), (x2, y2, z2), ...)
\& # which is why we have to do the exchange:
\& $coords = cat($x, $y, $z)\->transpose;
\& print "again, with a different coordinate syntax (press q when you\*(Aqre done twiddling)\en";
\& line3d $coords;
\&
\& # Draw a regularly\-gridded surface:
\& $surface = sqrt(rvals(zeroes(50,50))/2);
\& print "draw a mesh of a regularly\-gridded surface using mesh3d\en";
\& mesh3d [$surface];
\& print "draw a regularly\-gridded surface using imag3d\en";
\& imag3d [$surface], {Lines=>0};
\&
\& # Draw a mobius strip:
\& $two_pi = 8 * atan2(1,1);
\& $t = sequence(51) / 50 * $two_pi;
\& # We want three paths:
\& $mobius1_x = cos($t) + 0.5 * sin($t/2);
\& $mobius2_x = cos($t);
\& $mobius3_x = cos($t) \- 0.5 * sin($t/2);
\& $mobius1_y = sin($t) + 0.5 * sin($t/2);
\& $mobius2_y = sin($t);
\& $mobius3_y = sin($t) \- 0.5 * sin($t/2);
\& $mobius1_z = $t \- $two_pi/2;
\& $mobius2_z = zeroes($t);
\& $mobius3_z = $two_pi/2 \- $t;
\&
\& $mobius_x = cat($mobius1_x, $mobius2_x, $mobius3_x);
\& $mobius_y = cat($mobius1_y, $mobius2_y, $mobius3_y);
\& $mobius_z = cat($mobius1_z, $mobius2_z, $mobius3_z);
\&
\& $mobius_surface = cat($mobius_x, $mobius_y, $mobius_z)\->mv(2,0);
\&
\& print "A mobius strip using line3d one way\en";
\& line3d $mobius_surface;
\& print "A mobius strip using line3d the other way\en";
\& line3d $mobius_surface\->xchg(1,2);
\& print "A mobius strip using mesh3d\en";
\& mesh3d $mobius_surface;
\& print "The same mobius strip using imag3d\en";
\& imag3d $mobius_surface, {Lines => 0};
.Ve
.SH "SIMPLE ROUTINES"
.IX Header "SIMPLE ROUTINES"
Because using the whole object-oriented interface for doing
all your work might be cumbersome, the following shortcut
routines are supported:
.SH FUNCTIONS
.IX Header "FUNCTIONS"
.SS line3d
.IX Subsection "line3d"
3D line plot, defined by a variety of contexts.
.PP
Implemented by \f(CW\*(C`PDL::Graphics::TriD::LineStrip\*(C'\fR.
.PP
.Vb 2
\& line3d ndarray(3,x), {OPTIONS}
\& line3d [CONTEXT], {OPTIONS}
.Ve
.PP
Example:
.PP
.Vb 6
\& pdl> line3d [sqrt(rvals(zeroes(50,50))/2)]
\& \- Lines on surface
\& pdl> line3d [$x,$y,$z]
\& \- Lines over X, Y, Z
\& pdl> line3d $coords
\& \- Lines over the 3D coordinates in $coords.
.Ve
.PP
Note: line plots differ from mesh plots in that lines
only go in one direction. If this is unclear try both!
.PP
See module documentation for more information on
contexts and options
.SS imag3d
.IX Subsection "imag3d"
3D rendered image plot, defined by a variety of contexts
.PP
Implemented by \f(CW\*(C`PDL::Graphics::TriD::SLattice_S\*(C'\fR.
.PP
The variant, \f(CW\*(C`imag3d_ns\*(C'\fR, is implemented by \f(CW\*(C`PDL::Graphics::TriD::SLattice\*(C'\fR.
.PP
.Vb 2
\& imag3d ndarray(3,x,y), {OPTIONS}
\& imag3d [ndarray,...], {OPTIONS}
.Ve
.PP
Example:
.PP
.Vb 1
\& pdl> imag3d [sqrt(rvals(zeroes(50,50))/2)], {Lines=>0};
\&
\& \- Rendered image of surface
.Ve
.PP
See module documentation for more information on
contexts and options
.SS mesh3d
.IX Subsection "mesh3d"
3D mesh plot, defined by a variety of contexts
.PP
Implemented by \f(CW\*(C`PDL::Graphics::TriD::Lattice\*(C'\fR.
.PP
.Vb 2
\& mesh3d ndarray(3,x,y), {OPTIONS}
\& mesh3d [ndarray,...], {OPTIONS}
.Ve
.PP
Example:
.PP
.Vb 1
\& pdl> mesh3d [sqrt(rvals(zeroes(50,50))/2)]
\&
\& \- mesh of surface
.Ve
.PP
Note: a mesh is defined by two sets of lines at
right-angles (i.e. this is how is differs from
line3d).
.PP
See module documentation for more information on
contexts and options
.SS lattice3d
.IX Subsection "lattice3d"
alias for mesh3d
.SS trigrid3d
.IX Subsection "trigrid3d"
Show a triangular mesh, giving \f(CW$vertices\fR and \f(CW$faceidx\fR which is
a series of triplets of indices into the vertices, each describing
one triangle. The order of points matters for the shading \- the normal
vector points towards the clockface if the points go clockwise.
.PP
Options: \f(CW\*(C`Smooth\*(C'\fR (on by default), \f(CW\*(C`Lines\*(C'\fR (off by default),
\&\f(CW\*(C`ShowNormals\*(C'\fR (off by default, useful for debugging).
.PP
Implemented by \f(CW\*(C`PDL::Graphics::TriD::STrigrid_S\*(C'\fR.
.SS trigrid3d_ns
.IX Subsection "trigrid3d_ns"
Like "trigrid3d", but without shading or normals.
.PP
Implemented by \f(CW\*(C`PDL::Graphics::TriD::STrigrid\*(C'\fR.
.SS points3d
.IX Subsection "points3d"
3D points plot, defined by a variety of contexts
.PP
Implemented by \f(CW\*(C`PDL::Graphics::TriD::Points\*(C'\fR.
.PP
.Vb 2
\& points3d ndarray(3), {OPTIONS}
\& points3d [ndarray,...], {OPTIONS}
.Ve
.PP
Example:
.PP
.Vb 2
\& pdl> points3d [sqrt(rvals(zeroes(50,50))/2)];
\& \- points on surface
.Ve
.PP
See module documentation for more information on
contexts and options
.SS spheres3d
.IX Subsection "spheres3d"
3D spheres plot (preliminary implementation)
.PP
This is a preliminary implementation as a proof of
concept.  It has fixed radii for the spheres being
drawn and no control of color or transparency.
.PP
Implemented by \f(CW\*(C`PDL::Graphics::TriD::Spheres\*(C'\fR.
.PP
.Vb 2
\& spheres3d ndarray(3), {OPTIONS}
\& spheres3d [ndarray,...], {OPTIONS}
.Ve
.PP
Example:
.PP
.Vb 1
\& pdl> spheres3d ndcoords(10,10,10)\->clump(1,2,3)
\&
\& \- lattice of spheres at coordinates on 10x10x10 grid
.Ve
.SS imagrgb
.IX Subsection "imagrgb"
2D RGB image plot (see also imag2d)
.PP
Implemented by \f(CW\*(C`PDL::Graphics::TriD::Image\*(C'\fR.
.PP
.Vb 2
\& imagrgb ndarray(3,x,y), {OPTIONS}
\& imagrgb [ndarray,...], {OPTIONS}
.Ve
.PP
This would be used to plot an image, specifying
red, green and blue values at each point. Note:
contexts are very useful here as there are many
ways one might want to do this.
.PP
e.g.
.PP
.Vb 2
\& pdl> $x=sqrt(rvals(zeroes(50,50))/2)
\& pdl> imagrgb [0.5*sin(8*$x)+0.5,0.5*cos(8*$x)+0.5,0.5*cos(4*$x)+0.5]
.Ve
.SS imagrgb3d
.IX Subsection "imagrgb3d"
2D RGB image plot as an object inside a 3D space
.PP
Implemented by \f(CW\*(C`PDL::Graphics::TriD::Image\*(C'\fR.
.PP
.Vb 2
\& imagrdb3d ndarray(3,x,y), {OPTIONS}
\& imagrdb3d [ndarray,...], {OPTIONS}
.Ve
.PP
The ndarray gives the colors. The option allowed is Points,
which should give 4 3D coordinates for the corners of the polygon,
either as an ndarray or as array ref.
The default is [[0,0,0],[1,0,0],[1,1,0],[0,1,0]].
.PP
e.g.
.PP
.Vb 2
\& pdl> imagrgb3d $colors, {Points => [[0,0,0],[1,0,0],[1,0,1],[0,0,1]]};
\& \- plot on XZ plane instead of XY.
.Ve
.SS grabpic3d
.IX Subsection "grabpic3d"
Grab a 3D image from the screen.
.PP
.Vb 1
\& $pic = grabpic3d();
.Ve
.PP
The returned ndarray has dimensions (3,$x,$y) and is of type float
(currently). XXX This should be altered later.
.SS "hold3d, release3d"
.IX Subsection "hold3d, release3d"
Keep / don't keep the previous objects when plotting new 3D objects
.PP
.Vb 2
\& hold3d();
\& release3d();
.Ve
.PP
or
.PP
.Vb 2
\& hold3d(1);
\& hold3d(0);
.Ve
.SS "keeptwiddling3d, nokeeptwiddling3d"
.IX Subsection "keeptwiddling3d, nokeeptwiddling3d"
Wait / don't wait for 'q' after displaying a 3D image.
.PP
Usually, when showing 3D images, the user is given a chance
to rotate it and then press 'q' for the next image. However,
sometimes (for e.g. animation) this is undesirable and it is
more desirable to just run one step of the event loop at
a time.
.PP
.Vb 2
\& keeptwiddling3d();
\& nokeeptwiddling3d();
.Ve
.PP
or
.PP
.Vb 2
\& keeptwiddling3d(1);
\& keeptwiddling3d(0);
.Ve
.PP
When an image is added to the screen, keep twiddling it until
user explicitly presses 'q'.
.PP
.Vb 10
\& keeptwiddling3d();
\& imag3d(..);
\& nokeeptwiddling3d();
\& $o = imag3d($c);
\& do {
\&        $c .= nextfunc($c);
\&        $o\->data_changed;
\& } while(!twiddle3d()); # animate one step, then iterate
\& keeptwiddling3d();
\& twiddle3d(); # wait one last time
.Ve
.SS twiddle3d
.IX Subsection "twiddle3d"
Wait for the user to rotate the image in 3D space.
.PP
Let the user rotate the image in 3D space, either for one step
or until they press 'q', depending on the 'keeptwiddling3d'
setting. If 'keeptwiddling3d' is not set the routine returns
immediately and indicates that a 'q' event was received by
returning 1. If the only events received were mouse events,
returns 0.
.SS close3d
.IX Subsection "close3d"
Close the currently-open 3D window.
.SH "NOT EXPORTED"
.IX Header "NOT EXPORTED"
These functions are not exported, partly because they are not fully
implemented.
.IP contour3d 4
.IX Item "contour3d"
Implemented by \f(CW\*(C`PDL::Graphics::TriD::Contours\*(C'\fR.
.SH CONCEPTS
.IX Header "CONCEPTS"
The key concepts (object types) of TriD are explained in the following:
.SS Object
.IX Subsection "Object"
In this 3D abstraction, everything that you can "draw"
without using indices is an Object. That is, if you have a surface,
each vertex is not an object and neither is each segment of a long
curve. The whole curve (or a set of curves) is the lowest level Object.
.PP
Transformations and groups of Objects are also Objects.
.PP
A Window is simply an Object that has subobjects.
.SS Twiddling
.IX Subsection "Twiddling"
Because there is no eventloop in Perl yet and because it would
be hassleful to do otherwise, it is currently not possible to
e.g. rotate objects with your mouse when the console is expecting
input or the program is doing other things. Therefore, you need
to explicitly say "$window\->\fBtwiddle()\fR" in order to display anything.
.SH OBJECTS
.IX Header "OBJECTS"
The following types of objects are currently supported.
Those that do not have a calling sequence described here should
have their own manual pages.
.PP
There are objects that are not mentioned here; they are either internal
to PDL3D or in rapidly changing states. If you use them, you do so at
your own risk.
.PP
The syntax \f(CW\*(C`PDL::Graphics::TriD::Scale(x,y,z)\*(C'\fR here means that you create
an object like
.PP
.Vb 1
\&        $c = PDL::Graphics::TriD::Scale\->new($x,$y,$z);
.Ve
.SS PDL::Graphics::TriD::LineStrip
.IX Subsection "PDL::Graphics::TriD::LineStrip"
This is just a line or a set of lines. The arguments are 3 1\-or\-more\-D
ndarrays which describe the vertices of a continuous line and an
optional color ndarray (which is 1\-D also and simply
defines the color between red and blue. This will probably change).
.SS PDL::Graphics::TriD::Lines
.IX Subsection "PDL::Graphics::TriD::Lines"
This is just a line or a set of lines. The arguments are 3 1\-or\-more\-D
ndarrays where each contiguous pair of vertices describe a line segment
and an optional color ndarray (which is 1\-D also and simply
defines the color between red and blue. This will probably change).
.SS PDL::Graphics::TriD::Image
.IX Subsection "PDL::Graphics::TriD::Image"
This is a 2\-dimensional RGB image consisting of colored
rectangles. With OpenGL, this is implemented by texturing so this should
be relatively memory and execution-time-friendly.
.SS PDL::Graphics::TriD::Lattice
.IX Subsection "PDL::Graphics::TriD::Lattice"
This is a 2\-D set of points connected by lines in 3\-space.
The constructor takes as arguments 3 2\-dimensional ndarrays.
.SS PDL::Graphics::TriD::Points
.IX Subsection "PDL::Graphics::TriD::Points"
This is simply a set of points in 3\-space. Takes as arguments
the x, y and z coordinates of the points as ndarrays.
.SS PDL::Graphics::TriD::Scale(x,y,z)
.IX Subsection "PDL::Graphics::TriD::Scale(x,y,z)"
Self-explanatory
.SS PDL::Graphics::TriD::Translation(x,y,z)
.IX Subsection "PDL::Graphics::TriD::Translation(x,y,z)"
Ditto
.SS PDL::Graphics::TriD::Quaternion(c,x,y,z)
.IX Subsection "PDL::Graphics::TriD::Quaternion(c,x,y,z)"
One way of representing rotations is with quaternions. See the appropriate
man page.
.SS PDL::Graphics::TriD::ViewPort
.IX Subsection "PDL::Graphics::TriD::ViewPort"
This is a special class: in order to obtain a new viewport, you
need to have an earlier viewport on hand. The usage is:
.PP
.Vb 1
\&  $new_vp = $old_vp\->new_viewport($x0,$y0,$x1,$y1);
.Ve
.PP
where \f(CW$x0\fR etc are the coordinates of the upper left and lower right
corners of the new viewport inside the previous (relative
to the previous viewport in the (0,1) range.
.PP
Every implementation-level window object should implement the new_viewport
method.
.SH AUTHOR
.IX Header "AUTHOR"
Copyright (C) 1997 Tuomas J. Lukka (lukka@husc.harvard.edu). Documentation
contributions from Karl Glazebrook (kgb@aaoepp.aao.gov.au).
All rights reserved. There is no warranty. You are allowed
to redistribute this software / documentation under certain
conditions. For details, see the file COPYING in the PDL
distribution. If this file is separated from the PDL distribution,
the copyright notice should be included in the file.