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<h1><a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/half.html"><img
src="qh--half.gif" alt="[halfspace]" align="middle" width="100"
height="100"></a> Examples of Qhull</h1>

<p>This section of the Qhull manual will introduce you to Qhull
and its options. Each example is a file for viewing with <a
href="index.htm#geomview">Geomview</a>.  You will need to
use a Unix computer with a copy of Geomview.
<p>
If you are not running Unix, you can view <a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/welcome.html">pictures</a> 
for some of the examples.  To understand Qhull without Geomview, try the
examples in <a href="qh-quick.htm#programs">Programs</a> and
<a href="qh-quick.htm#programs">Programs/input</a>.  You can also try small
examples that you compute by hand.  Use <a href="rbox.htm">rbox</a>
to generate examples.
<p>
To generate the Geomview examples, execute the shell script <tt>eg/q_eg</tt>.
It uses <tt>rbox</tt>.  The shell script <tt>eg/q_egtest</tt> generates
test examples, and <tt>eg/q_test</tt> exercises the code. If you
find yourself viewing the inside of a 3-d example, use Geomview's
normalization option on the 'obscure' menu.</p>

<p><b>Copyright &copy; 1995-2003 The Geometry Center, Minneapolis MN</b></p>

<hr>

<h2><a href="#TOP">»</a><a name="TOC">Qhull examples: Table of
Contents </a></h2>

<ul>
    <li><a href="#2d">2-d and 3-d examples</a></li>
    <li><a href="#how">How Qhull adds a point</a></li>
    <li><a href="#joggle">Triangulated output or joggled input</a></li>
    <li><a href="#delaunay">Delaunay and Voronoi diagrams</a></li>
    <li><a href="#merge">Facet merging for imprecision</a></li>
    <li><a href="#4d">4-d objects</a></li>
    <li><a href="#half">Halfspace intersections</a></li>
</ul>

<hr>
<ul>
    <li><a href="#TOC">»</a><a name="2d">2-d and 3-d examples</a><ul>
            <li><a href="#01">eg.01.cube</a></li>
            <li><a href="#02">eg.02.diamond.cube</a></li>
            <li><a href="#03">eg.03.sphere</a></li>
            <li><a href="#04">eg.04.circle</a></li>
            <li><a href="#05">eg.05.spiral</a></li>
            <li><a href="#06">eg.06.merge.square</a></li>
            <li><a href="#07">eg.07.box</a></li>
            <li><a href="#08a">eg.08a.cube.sphere</a></li>
            <li><a href="#08b">eg.08b.diamond.sphere</a></li>
            <li><a href="#09">eg.09.lens</a></li>
        </ul>
    </li>
    <li><a href="#TOC">»</a><a name="how">How Qhull adds a point</a><ul>
            <li><a href="#10a">eg.10a.sphere.visible</a></li>
            <li><a href="#10b">eg.10b.sphere.beyond</a></li>
            <li><a href="#10c">eg.10c.sphere.horizon</a></li>
            <li><a href="#10d">eg.10d.sphere.cone</a></li>
            <li><a href="#10e">eg.10e.sphere.new</a></li>
            <li><a href="#14">eg.14.sphere.corner</a></li>
        </ul>
    </li>
    <li><a href="#TOC">»</a> <a name="joggle">Triangulated output or joggled input</a>
    	<ul>
            <li><a href="#15a">eg.15a.surface</a></li>
            <li><a href="#15b">eg.15b.triangle</a></li>
            <li><a href="#15c">eg.15c.joggle</a></li>
        </ul>
    <li><a href="#TOC">»</a><a name="delaunay"> Delaunay and
        Voronoi diagrams</a><ul>
            <li><a href="#17a">eg.17a.delaunay.2</a></li>
            <li><a href="#17b">eg.17b.delaunay.2i</a></li>
            <li><a href="#17c">eg.17c.delaunay.2-3</a></li>
            <li><a href="#17d">eg.17d.voronoi.2</a></li>
            <li><a href="#17e">eg.17e.voronoi.2i</a></li>
            <li><a href="#17f">eg.17f.delaunay.3</a></li>
            <li><a href="#18a">eg.18a.furthest.2-3</a></li>
            <li><a href="#18b">eg.18b.furthest-up.2-3</a></li>
            <li><a href="#18c">eg.18c.furthest.2</a></li>
            <li><a href="#19">eg.19.voronoi.region.3</a></li>
        </ul>
    </li>
    <li><a href="#TOC">»</a><a name="merge">Facet merging for
        imprecision </a><ul>
            <li><a href="#20">eg.20.cone</a></li>
            <li><a href="#21a">eg.21a.roundoff.errors</a></li>
            <li><a href="#21b">eg.21b.roundoff.fixed</a></li>
            <li><a href="#22a">eg.22a.merge.sphere.01</a></li>
            <li><a href="#22b">eg.22b.merge.sphere.-01</a></li>
            <li><a href="#22c">eg.22c.merge.sphere.05</a></li>
            <li><a href="#22d">eg.22d.merge.sphere.-05</a></li>
            <li><a href="#23">eg.23.merge.cube</a></li>
        </ul>
    </li>
    <li><a href="#TOC">»</a><a name="4d">4-d objects</a><ul>
            <li><a href="#24">eg.24.merge.cube.4d-in-3d</a></li>
            <li><a href="#30">eg.30.4d.merge.cube</a></li>
            <li><a href="#31">eg.31.4d.delaunay</a></li>
            <li><a href="#32">eg.32.4d.octant</a></li>
        </ul>
    </li>
    <li><a href="#TOC">»</a><a name="half">Halfspace
        intersections</a><ul>
            <li><a href="#33a">eg.33a.cone</a></li>
            <li><a href="#33b">eg.33b.cone.dual</a></li>
            <li><a href="#33c">eg.33c.cone.halfspace</a></li>
        </ul>
    </li>
</ul>

<hr>

<h2><a href="#TOC">»</a>2-d and 3-d examples</h2>

<h3><a href="#2d">»</a><a name="01">rbox c D3 | qconvex G
&gt;eg.01.cube </a></h3>

<p>The first example is a cube in 3-d. The color of each facet
indicates its normal. For example, normal [0,0,1] along the Z
axis is (r=0.5, g=0.5, b=1.0). With the 'Dn' option in <tt>rbox</tt>,
you can generate hypercubes in any dimension. Above 7-d the
number of intermediate facets grows rapidly. Use '<a
href="qh-optt.htm#TFn">TFn</a>' to track qconvex's progress. Note
that each facet is a square that qconvex merged from coplanar
triangles.</p>

<h3><a href="#2d">»</a><a name="02">rbox c d G3.0 | qconvex G
&gt;eg.02.diamond.cube </a></h3>

<p>The second example is a cube plus a diamond ('d') scaled by <tt>rbox</tt>'s
'G' option. In higher dimensions, diamonds are much simpler than
hypercubes. </p>

<h3><a href="#2d">»</a><a name="03">rbox s 100 D3 | qconvex G
&gt;eg.03.sphere </a></h3>

<p>The <tt>rbox s</tt> option generates random points and
projects them to the d-sphere. All points should be on the convex
hull. Notice that random points look more clustered than you
might expect. You can get a smoother distribution by merging
facets and printing the vertices, e.g.,<i> rbox 1000 s | qconvex
A-0.95 p | qconvex G &gt;eg.99</i>.</p>

<h3><a href="#2d">»</a><a name="04">rbox s 100 D2 | qconvex G
&gt;eg.04.circle </a></h3>

<p>In 2-d, there are many ways to generate a convex hull. One of
the earliest algorithms, and one of the fastest, is the 2-d
Quickhull algorithm [c.f., Preparata &amp; Shamos <a
href="index.htm#pre-sha85">'85</a>]. It was the model for
Qhull.</p>

<h3><a href="#2d">»</a><a name="05">rbox 10 l | qconvex G
&gt;eg.05.spiral </a></h3>

<p>One rotation of a spiral.</p>

<h3><a href="#2d">»</a><a name="06">rbox 1000 D2 | qconvex C-0.03
Qc Gapcv &gt;eg.06.merge.square</a></h3>

<p>This demonstrates how Qhull handles precision errors. Option '<a
href="qh-optc.htm#Cn">C-0.03</a>' requires a clearly convex angle
between adjacent facets. Otherwise, Qhull merges the facets. </p>

<p>This is the convex hull of random points in a square. The
facets have thickness because they must be outside all points and
must include their vertices. The colored lines represent the
original points and the spheres represent the vertices. Floating
in the middle of each facet is the centrum. Each centrum is at
least 0.03 below the planes of its neighbors. This guarantees
that the facets are convex.</p>

<h3><a href="#2d">»</a><a name="07">rbox 1000 D3 | qconvex G
&gt;eg.07.box </a></h3>

<p>Here's the same distribution but in 3-d with Qhull handling
machine roundoff errors. Note the large number of facets. </p>

<h3><a href="#2d">»</a><a name="08a">rbox c G0.4 s 500 | qconvex G
&gt;eg.08a.cube.sphere </a></h3>

<p>The sphere is just barely poking out of the cube. Try the same
distribution with randomization turned on ('<a
href="qh-optq.htm#Qr">Qr</a>'). This turns Qhull into a
randomized incremental algorithm. To compare Qhull and
randomization, look at the number of hyperplanes created and the
number of points partitioned. Don't compare CPU times since Qhull's
implementation of randomization is inefficient. The number of
hyperplanes and partitionings indicate the dominant costs for
Qhull. With randomization, you'll notice that the number of
facets created is larger than before. This is especially true as
you increase the number of points. It is because the randomized
algorithm builds most of the sphere before it adds the cube's
vertices.</p>

<h3><a href="#2d">»</a><a name="08b">rbox d G0.6 s 500 | qconvex G
&gt;eg.08b.diamond.sphere </a></h3>

<p>This is a combination of the diamond distribution and the
sphere.</p>

<h3><a href="#2d">»</a><a name="09">rbox 100 L3 G0.5 s | qconvex 
G &gt;eg.09.lens </a></h3>

<p>Each half of the lens distribution lies on a sphere of radius
three. A directed search for the furthest facet below a point
(e.g., qh_findbest in <tt>geom.c</tt>) may fail if started from
an arbitrary facet. For example, if the first facet is on the
opposite side of the lens, a directed search will report that the
point is inside the convex hull even though it is outside. This
problem occurs whenever the curvature of the convex hull is less
than a sphere centered at the test point. </p>

<p>To prevent this problem, Qhull does not use directed search
all the time. When Qhull processes a point on the edge of the
lens, it partitions the remaining points with an exhaustive
search instead of a directed search (see qh_findbestnew in <tt>geom2.c</tt>).
</p>

<h2><a href="#TOC">»</a>How Qhull adds a point</h2>

<h3><a href="#how">»</a><a name="10a">rbox 100 s P0.5,0.5,0.5 |
qconvex Ga QG0 &gt;eg.10a.sphere.visible</a></h3>

<p>The next 4 examples show how Qhull adds a point. The point
[0.5,0.5,0.5] is at one corner of the bounding box. Qhull adds a
point using the beneath-beyond algorithm. First Qhull finds all
of the facets that are visible from the point. Qhull will replace
these facets with new facets.</p>

<h3><a href="#how">»</a><a name="10b">rbox 100 s
P0.5,0.5,0.5|qconvex Ga QG-0 &gt;eg.10b.sphere.beyond </a></h3>

<p>These are the facets that are not visible from the point.
Qhull will keep these facets.</p>

<h3><a href="#how">»</a><a name="10c">rbox 100 s P0.5,0.5,0.5 |
qconvex PG Ga QG0 &gt;eg.10c.sphere.horizon </a></h3>

<p>These facets are the horizon facets; they border the visible
facets. The inside edges are the horizon ridges. Each horizon
ridge will form the base for a new facet.</p>

<h3><a href="#how">»</a><a name="10d">rbox 100 s P0.5,0.5,0.5 |
qconvex Ga QV0 PgG &gt;eg.10d.sphere.cone </a></h3>

<p>This is the cone of points from the new point to the horizon
facets.  Try combining this image with <tt>eg.10c.sphere.horizon</tt>
and <tt>eg.10a.sphere.visible</tt>.
</p>

<h3><a href="#how">»</a><a name="10e">rbox 100 s P0.5,0.5,0.5 |
qconvex Ga &gt;eg.10e.sphere.new</a></h3>

<p>This is the convex hull after [0.5,0.5,0.5] has been added.
Note that in actual practice, the above sequence would never
happen. Unlike the randomized algorithms, Qhull always processes
a point that is furthest in an outside set. A point like
[0.5,0.5,0.5] would be one of the first points processed.</p>

<h3><a href="#how">»</a><a name="14">rbox 100 s P0.5,0.5,0.5 |
qhull Ga QV0g Q0 &gt;eg.14.sphere.corner</a></h3>

<p>The '<a href="qh-optq.htm#QVn">QVn</a>', '<a
href="qh-optq.htm#QGn">QGn </a>' and '<a href="qh-optp.htm#Pdk">Pdk</a>'
options define good facets for Qhull. In this case '<a
href="qh-optq.htm#QVn">QV0</a>' defines the 0'th point
[0.5,0.5,0.5] as the good vertex, and '<a href="qh-optq.htm#Qg">Qg</a>'
tells Qhull to only build facets that might be part of a good
facet. This technique reduces output size in low dimensions. It
does not work with facet merging.</p>

<h2><a href="#TOC">»</a>Triangulated output or joggled input</h2>

<h3><a href="#joggle">»</a><a name="15a">rbox 500 W0 | qconvex QR0 Qc Gvp &gt;eg.15a.surface</a></h3>

<p>This is the convex hull of 500 points on the surface of
a cube.  Note the large, non-simplicial facet for each face. 
Qhull merges non-convex facets.

<p>If the facets were not merged, Qhull
would report precision problems.  For example, turn off facet merging
with option '<a href="qh-optq.htm#Q0">Q0</a>'.  Qhull may report concave
facets, flipped facets, or other precision errors:
<blockquote>
rbox 500 W0 | qhull QR0 Q0
</blockquote>

<p>
<h3><a href="#joggle">»</a><a name="15b">rbox 500 W0 | qconvex QR0 Qt Qc Gvp &gt;eg.15b.triangle</a></h3>

<p>Like the previous examples, this is the convex hull of 500 points on the 
surface of a cube.  Option '<a href="qh-optq.htm#Qt">Qt</a>' triangulates the
non-simplicial facets.  Triangulated output is  
particularly helpful for Delaunay triangulations.

<p>
<h3><a href="#joggle">»</a><a name="15c">rbox 500 W0 | qconvex QR0 QJ5e-2 Qc Gvp &gt;eg.15c.joggle</a></h3>

<p>This is the convex hull of 500 joggled points on the surface of
a cube.  The option '<a href="qh-optq.htm#QJn">QJ5e-2</a>'
sets a very large joggle to make the effect visible.  Notice
that all of the facets are triangles.  If you rotate the cube,
you'll see red-yellow lines for coplanar points.
<p>
With option '<a href="qh-optq.htm#QJn">QJ</a>', Qhull joggles the
input to avoid precision problems.  It adds a small random number
to each input coordinate.  If a precision
error occurs, it increases the joggle and tries again.  It repeats
this process until no precision problems occur.
<p>
Joggled input is a simple solution to precision problems in
computational geometry.  Qhull can also merge facets to handle
precision problems.  See <a href="qh-impre.htm#joggle">Merged facets or joggled input</a>.  

<h2><a href="#TOC">»</a>Delaunay and Voronoi diagrams</h2>

<h3><a href="#delaunay">»</a><a name="17a">qdelaunay Qt
&lt;eg.data.17 GnraD2 &gt;eg.17a.delaunay.2</a></h3>

<p>
The input file, <tt>eg.data.17</tt>, consists of a square, 15 random
points within the outside half of the square, and 6 co-circular
points centered on the square.

<p>The Delaunay triangulation is the triangulation with empty
circumcircles.  The input for this example is unusual because it
includes six co-circular points.  Every triangular subset of these 
points has the same circumcircle.  Option '<a href="qh-optq.htm#Qt">Qt</a>'
triangulates the co-circular facet.</p>

<h3><a href="#delaunay">»</a><a name="17b">qdelaunay &lt;eg.data.17
GnraD2 &gt;eg.17b.delaunay.2i</a></h3>

<p>This is the same example without triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>').  qdelaunay
merges the non-unique Delaunay triangles into a hexagon.</p>

<h3><a href="#delaunay">»</a><a name="17c">qdelaunay &lt;eg.data.17
Ga &gt;eg.17c.delaunay.2-3 </a></h3>

<p>This is how Qhull generated both diagrams. Use Geomview's
'obscure' menu to turn off normalization, and Geomview's
'cameras' menu to turn off perspective. Then load this <a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/delaunay.html">object</a>
with one of the previous diagrams.</p>

<p>The points are lifted to a paraboloid by summing the squares
of each coordinate. These are the light blue points. Then the
convex hull is taken. That's what you see here. If you look up
the Z-axis, you'll see that points and edges coincide.</p>

<h3><a href="#delaunay">»</a><a name="17d">qvoronoi QJ
&lt;eg.data.17 Gna &gt;eg.17d.voronoi.2</a></h3>

<p>The Voronoi diagram is the dual of the Delaunay triangulation.
Here you see the original sites and the Voronoi vertices.   
Notice the each
vertex is equidistant from three sites. The edges indicate the
Voronoi region for a site. Qhull does not draw the unbounded
edges. Instead, it draws extra edges to close the unbounded
Voronoi regions. You may find it helpful to enclose the input
points in a square.  You can compute the unbounded
rays from option '<a href="qh-optf.htm#Fo2">Fo</a>'.
</p>

<p>Instead
of triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>'), this 
example uses joggled input ('<a href="qh-optq.htm#QJn">QJ</a>').  
Normally, you should use neither 'QJ' nor 'Qt' for Voronoi diagrams.

<h3><a href="#delaunay">»</a><a name="17e">qvoronoi &lt;eg.data.17
Gna &gt;eg.17e.voronoi.2i </a></h3>

<p>This looks the same as the previous diagrams, but take a look
at the data.  Run 'qvoronoi p &lt;eg/eg.data.17'.  This prints
the Voronoi vertices.  

<p>With 'QJ', there are four nearly identical Voronoi vertices
within 10^-11 of the origin.  Option 'QJ' joggled the input.  After the joggle,
the cocircular 
input sites are no longer cocircular.  The corresponding Voronoi vertices are 
similar but not identical.

<p>This example does not use options 'Qt' or 'QJ'.  The cocircular
input sites define one Voronoi vertex near the origin. </p>

<p>Option 'Qt' would triangulate the corresponding Delaunay region into
four triangles.  Each triangle is assigned the same Voronoi vertex.</p>

<h3><a href="#delaunay">»</a><a name="17f"> rbox c G0.1 d |
qdelaunay Gt Qz &lt;eg.17f.delaunay.3 </a></h3>

<p>This is the 3-d Delaunay triangulation of a small cube inside
a prism. Since the outside ridges are transparent, it shows the
interior of the outermost facets. If you slice open the
triangulation with Geomview's ginsu, you will see that the innermost
facet is a cube. Note the use of '<a href="qh-optq.htm#Qz">Qz</a>'
to add a point &quot;at infinity&quot;. This avoids a degenerate
input due to cospherical points.</p>

<h3><a href="#delaunay">»</a><a name="18a">rbox 10 D2 d | qdelaunay
Qu G &gt;eg.18a.furthest.2-3 </a></h3>

<p>The furthest-site Voronoi diagram contains Voronoi regions for
points that are <i>furthest </i>from an input site. It is the
dual of the furthest-site Delaunay triangulation. You can
determine the furthest-site Delaunay triangulation from the
convex hull of the lifted points (<a href="#17c">eg.17c.delaunay.2-3</a>).
The upper convex hull (blue) generates the furthest-site Delaunay
triangulation. </p>

<h3><a href="#delaunay">»</a><a name="18b">rbox 10 D2 d | qdelaunay
Qu Pd2 G &gt;eg.18b.furthest-up.2-3</a></h3>

<p>This is the upper convex hull of the preceding example. The
furthest-site Delaunay triangulation is the projection of the
upper convex hull back to the input points. The furthest-site
Voronoi vertices are the circumcenters of the furthest-site
Delaunay triangles. </p>

<h3><a href="#delaunay">»</a><a name="18c">rbox 10 D2 d | qvoronoi
Qu Gv &gt;eg.18c.furthest.2</a></h3>

<p>This shows an incomplete furthest-site Voronoi diagram. It
only shows regions with more than two vertices. The regions are
artificially truncated. The actual regions are unbounded. You can
print the regions' vertices with 'qvoronoi Qu <a
href="qh-opto.htm#o">o</a>'. </p>

<p>Use Geomview's 'obscure' menu to turn off normalization, and
Geomview's 'cameras' menu to turn off perspective. Then load this
with the upper convex hull.</p>

<h3><a href="#delaunay">»</a><a name="19">rbox 10 D3 | qvoronoi QV5
p | qconvex G &gt;eg.19.voronoi.region.3 </a></h3>

<p>This shows the Voronoi region for input site 5 of a 3-d
Voronoi diagram.</p>

<h2><a href="#TOC">»</a>Facet merging for imprecision</h2>

<h3><a href="#merge">»</a><a name="20">rbox r s 20 Z1 G0.2 |
qconvex G &gt;eg.20.cone </a></h3>

<p>There are two things unusual about this <a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/cone.html">cone</a>.
One is the large flat disk at one end and the other is the
rectangles about the middle. That's how the points were
generated, and if those points were exact, this is the correct
hull. But <tt>rbox</tt> used floating point arithmetic to
generate the data. So the precise convex hull should have been
triangles instead of rectangles. By requiring convexity, Qhull
has recovered the original design.</p>

<h3><a href="#merge">»</a><a name="21a">rbox 200 s | qhull Q0
R0.01 Gav Po &gt;eg.21a.roundoff.errors </a></h3>

<p>This is the convex hull of 200 cospherical points with
precision errors ignored ('<a href="qh-optq.htm#Q0">Q0</a>'). To
demonstrate the effect of roundoff error, we've added a random
perturbation ('<a href="qh-optc.htm#Rn">R0.01</a>') to every
distance and hyperplane calculation. Qhull, like all other convex
hull algorithms with floating point arithmetic, makes
inconsistent decisions and generates wildly wrong results. In
this case, one or more facets are flipped over. These facets have
the wrong color. You can also turn on 'normals' in Geomview's
appearances menu and turn off 'facing normals'. There should be
some white lines pointing in the wrong direction. These
correspond to flipped facets. </p>

<p>Different machines may not produce this picture. If your
machine generated a long error message, decrease the number of
points or the random perturbation ('<a href="qh-optc.htm#Rn">R0.01</a>').
If it did not report flipped facets, increase the number of
points or perturbation.</p>

<h3><a href="#merge">»</a><a name="21b">rbox 200 s | qconvex Qc
R0.01 Gpav &gt;eg.21b.roundoff.fixed </a></h3>

<p>Qhull handles the random perturbations and returns an
imprecise <a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/fixed.html">sphere</a>.
In this case, the output is a weak approximation to the points.
This is because a random perturbation of '<a
href="qh-optc.htm#Rn">R0.01 </a>' is equivalent to losing all but
1.8 digits of precision. The outer planes float above the points
because Qhull needs to allow for the maximum roundoff error. </p>
<p> 
If you start with a smaller random perturbation, you 
can use joggle ('<a href="qh-optq.htm#QJn">QJn</a>') to avoid
precision problems.  You need to set <i>n</i> significantly
larger than the random perturbation.  For example, try 
'rbox 200 s | qconvex Qc R1e-4 QJ1e-1'.

<h3><a href="#merge">»</a><a name="22a">rbox 1000 s| qconvex C0.01
Qc Gcrp &gt;eg.22a.merge.sphere.01</a></h3>

<h3><a href="#merge">»</a><a name="22b">rbox 1000 s| qconvex
C-0.01 Qc Gcrp &gt;eg.22b.merge.sphere.-01</a></h3>

<h3><a href="#merge">»</a><a name="22c">rbox 1000 s| qconvex C0.05
Qc Gcrpv &gt;eg.22c.merge.sphere.05</a></h3>

<h3><a href="#merge">»</a><a name="22d">rbox 1000 s| qconvex
C-0.05 Qc Gcrpv &gt;eg.22d.merge.sphere.-05</a></h3>

<p>The next four examples compare post-merging and pre-merging ('<a
href="qh-optc.htm#Cn2">Cn</a>' vs. '<a href="qh-optc.htm#Cn">C-n</a>').
Qhull uses '-' as a flag to indicate pre-merging.</p>

<p>Post-merging happens after the convex hull is built. During
post-merging, Qhull repeatedly merges an independent set of
non-convex facets. For a given set of parameters, the result is
about as good as one can hope for.</p>

<p>Pre-merging does the same thing as post-merging, except that
it happens after adding each point to the convex hull. With
pre-merging, Qhull guarantees a convex hull, but the facets are
wider than those from post-merging. If a pre-merge option is not
specified, Qhull handles machine round-off errors.</p>

<p>You may see coplanar points appearing slightly outside
the facets of the last example.  This is becomes Geomview moves
line segments forward toward the viewer.  You can avoid the
effect by setting 'lines closer' to '0' in Geomview's camera menu.

<h3><a href="#merge">»</a><a name="23">rbox 1000 | qconvex s
Gcprvah C0.1 Qc &gt;eg.23.merge.cube</a></h3>

<p>Here's the 3-d imprecise cube with all of the Geomview
options. There's spheres for the vertices, radii for the coplanar
points, dots for the interior points, hyperplane intersections,
centrums, and inner and outer planes.  The radii are shorter than
the spheres because this uses post-merging ('<a href="qh-optc.htm#Cn2">C0.1</a>')
instead of pre-merging.

<h2><a href="#TOC">»</a>4-d objects</h2>

<p>With Qhull and Geomview you can develop an intuitive sense of
4-d surfaces. When you get into trouble, think of viewing the
surface of a 3-d sphere in a 2-d plane.</p>

<h3><a href="#4d">»</a><a name="24">rbox 5000 D4 | qconvex s GD0v
Pd0:0.5 C-0.02 C0.1 &gt;eg.24.merge.cube.4d-in-3d</a></h3>

<p>Here's one facet of the imprecise cube in 4-d. It's projected
into 3-d (the '<a href="qh-optg.htm#GDn">GDn</a>' option drops
dimension n). Each ridge consists of two triangles between this
facet and the neighboring facet. In this case, Geomview displays
the topological ridges, i.e., as triangles between three
vertices. That is why the cube looks lopsided.</p>

<h3><a href="#4d">»</a><a name="30">rbox 5000 D4 | qconvex s
C-0.02 C0.1 Gh &gt;eg.30.4d.merge.cube </a></h3>

<p><a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/4dcube.html">Here</a>
is the equivalent in 4-d of the imprecise <a href="#06">square</a>
and imprecise <a href="#23">cube</a>. It's the imprecise convex
hull of 5000 random points in a hypercube. It's a full 4-d object
so Geomview's <tt>ginsu </tt>does not work. If you view it in
Geomview, you'll be inside the hypercube. To view 4-d objects
directly, either load the <tt>4dview</tt> module or the <tt>ndview
</tt>module. For <tt>4dview</tt>, you must have started Geomview
in the same directory as the object. For <tt>ndview</tt>,
initialize a set of windows with the prefab menu, and load the
object through Geomview. The <tt>4dview</tt> module includes an
option for slicing along any hyperplane. If you do this in the x,
y, or z plane, you'll see the inside of a hypercube.</p>

<p>The '<a href="qh-optg.htm#Gh">Gh</a>' option prints the
geometric intersections between adjacent facets. Note the strong
convexity constraint for post-merging ('<a href="qh-optc.htm#Cn2">C0.1</a>').
It deletes the small facets.</p>

<h3><a href="#4d">»</a><a name="31">rbox 20 D3 | qdelaunay G
&gt;eg.31.4d.delaunay </a></h3>

<p>The Delaunay triangulation of 3-d sites corresponds to a 4-d
convex hull. You can't see 4-d directly but each facet is a 3-d
object that you can project to 3-d. This is exactly the same as
projecting a 2-d facet of a soccer ball onto a plane.</p>

<p>Here we see all of the facets together. You can use Geomview's
<tt>ndview</tt> to look at the object from several directions.
Try turning on edges in the appearance menu. You'll notice that
some edges seem to disappear. That's because the object is
actually two sets of overlapping facets.</p>

<p>You can slice the object apart using Geomview's <tt>4dview</tt>.
With <tt>4dview</tt>, try slicing along the w axis to get a
single set of facets and then slice along the x axis to look
inside. Another interesting picture is to slice away the equator
in the w dimension.</p>

<h3><a href="#4d">»</a><a name="32">rbox 30 s D4 | qconvex s G
Pd0d1d2D3</a></h3>

<p>This is the positive octant of the convex hull of 30 4-d
points. When looking at 4-d, it's easier to look at just a few
facets at once. If you picked a facet that was directly above
you, then that facet looks exactly the same in 3-d as it looks in
4-d. If you pick several facets, then you need to imagine that
the space of a facet is rotated relative to its neighbors. Try
Geomview's <tt>ndview</tt> on this example.</p>

<h2><a href="#TOC">»</a>Halfspace intersections</h2>

<h3><a href="#half">»</a><a name="33a">rbox 10 r s Z1 G0.3 |
qconvex G &gt;eg.33a.cone </a></h3>

<h3><a href="#half">»</a><a name="33b">rbox 10 r s Z1 G0.3 |
qconvex FV n | qhalf G &gt;eg.33b.cone.dual</a></h3>

<h3><a href="#half">»</a><a name="33c">rbox 10 r s Z1 G0.3 |
qconvex FV n | qhalf Fp | qconvex G &gt;eg.33c.cone.halfspace</a></h3>

<p>These examples illustrate halfspace intersection. The first
picture is the convex hull of two 20-gons plus an apex. The
second picture is the dual of the first. Try loading <a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/half.html">both</a>
at once. Each vertex of the second picture corresponds to a facet
or halfspace of the first. The vertices with four edges
correspond to a facet with four neighbors. Similarly the facets
correspond to vertices. A facet's normal coefficients divided by
its negative offset is the vertice's coordinates. The coordinates
are the intersection of the original halfspaces. </p>

<p>The third picture shows how Qhull can go back and forth
between equivalent representations. It starts with a cone,
generates the halfspaces that define each facet of the cone, and
then intersects these halfspaces. Except for roundoff error, the
third picture is a duplicate of the first. </p>
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