**Up:** Home page for Qhull

**Up:** Qhull manual: Table of Contents

**To:** Programs
• Options
• Output
• Formats
• Geomview
• Print
• Qhull
• Precision
• Trace
• Functions

**Copyright © 1995-2015 C.B. Barber**

» Programs • Options • Output • Formats • Geomview • Print • Qhull • Precision • Trace • Functions

Geomview is the graphical viewer for visualizing Qhull output in 2-d, 3-d and 4-d.

Geomview displays each facet of the convex hull. The color of a facet is determined by the coefficients of the facet's normal equation. For imprecise hulls, Geomview displays the inner and outer hull. Geomview can also display points, ridges, vertices, coplanar points, and facet intersections.

For 2-d Delaunay triangulations, Geomview displays the corresponding paraboloid. Geomview displays the 2-d Voronoi diagram. For halfspace intersections, it displays the dual convex hull.

**General**- G
- display Geomview output
- Gt
- display transparent 3-d Delaunay triangulation
- GDn
- drop dimension n in 3-d and 4-d output
**Specific**- Ga
- display all points as dots
- Gc
- display centrums (2-d, 3-d)
- Gp
- display coplanar points and vertices as radii
- Gh
- display hyperplane intersections
- Gi
- display inner planes only (2-d, 3-d)
- Go
- display outer planes only (2-d, 3-d)
- Gr
- display ridges (3-d)
- Gv
- display vertices as spheres
- Gn
- do not display planes

By default, option 'G' displays edges in 2-d, outer planes in 3-d, and ridges in 4-d.

A ridge can be explicit or implicit. An explicit ridge is a *(d-1)*-dimensional
simplex between two facets. In 4-d, the explicit ridges are
triangles. An implicit ridge is the topological intersection of
two neighboring facets. It is the union of explicit ridges.

For non-simplicial 4-d facets, the explicit ridges can be quite complex. When displaying a ridge in 4-d, Qhull projects the ridge's vertices to one of its facets' hyperplanes. Use 'Gh' to project ridges to the intersection of both hyperplanes. This usually results in a cleaner display.

For 2-d Delaunay triangulations, Geomview displays the corresponding paraboloid. Geomview displays the 2-d Voronoi diagram. For halfspace intersections, it displays the dual convex hull.

Each input point is displayed as a green dot.

The centrum is defined by a green radius sitting on a blue plane. The plane corresponds to the facet's hyperplane. If you sight along a facet's hyperplane, you will see that all neighboring centrums are below the facet. The radius is defined by 'C-n' or 'Cn'.

The result is a 2-d or 3-d object. In 4-d, this corresponds to
viewing the 4-d object from the nth axis without perspective.
It's best to view 4-d objects in pieces. Use the 'Pdk' 'Pg'
'PG' 'QGn'
and 'QVn' options to select a few
facets. If one of the facets is perpendicular to an axis, then
projecting along that axis will show the facet exactly as it is
in 4-d. If you generate many facets, use Geomview's `ginsu`
module to view the interior

To view multiple 4-d dimensions at once, output the object
without 'GDn' and read it with Geomview's `ndview`. As you
rotate the object in one set of dimensions, you can see how it
changes in other sets of dimensions.

For additional control over 4-d objects, output the object
without 'GDn' and read it with Geomview's `4dview`. You
can slice the object along any 4-d plane. You can also flip the
halfspace that's deleted when slicing. By combining these
features, you can get some interesting cross sections.

In 3-d, the intersection is a black line. It lies on two neighboring hyperplanes, c.f., the blue squares associated with centrums ('Gc '). In 4-d, the ridges are projected to the intersection of both hyperplanes. If you turn on edges (Geomview's 'appearances' menu), each triangle corresponds to one ridge. The ridges may overlap each other.

The inner plane of a facet is below all of its vertices. It is
parallel to the facet's hyperplane. The inner plane's color is
the opposite of the outer plane's color, i.e., *[1-r,1-g,1-b] *.
Its edges are determined by the vertices.

By default, Geomview displays the precise plane (no merging) or both inner and output planes (if merging). If merging, Geomview does not display the inner plane if the the difference between inner and outer is too small.

The outer plane of a facet is above all input points. It is parallel to the facet's hyperplane. Its color is determined by the facet's normal, and its edges are determined by the vertices.

Coplanar points ('Qc'), interior
points ('Qi'), outside points ('TCn' or 'TVn'),
and vertices are displayed as red and yellow radii. The radii are
perpendicular to the corresponding facet. Vertices are aligned
with an interior point. The radii define a ball which corresponds
to the imprecision of the point. The imprecision is the maximum
of the roundoff error, the centrum radius, and *maxcoord * (1 -
**A-n**)*. It is at
least 1/20'th of the maximum coordinate, and ignores post merging
if pre-merging is done.

If 'Gv' (print vertices as spheres) is also selected, option 'Gp' displays coplanar points as radii. Select options Qc' and/or 'Qi'. Options 'Qc Gpv' displays coplanar points while 'Qci Gpv' displays coplanar and interior points. Option 'Qc' is automatically selected if 'Qi' is not selected with options 'Gpv'.

A ridge connects the two vertices that are shared by neighboring facets. It is displayed in green. A ridge is the topological edge between two facets while the hyperplane intersection is the geometric edge between two facets. Ridges are always displayed in 4-d.

A 3-d Delaunay triangulation looks like a convex hull with interior facets. Option 'Gt' removes the outside ridges to reveal the outermost facets. It automatically sets options 'Gr' and 'GDn'. See example eg.17f.delaunay.3.

The radius of the sphere corresponds to the imprecision of the data. See 'Gp' for determining the radius.

**Up:** Home page for Qhull

**Up:** Qhull manual: Table of Contents

**To:** Programs
• Options
• Output
• Formats
• Geomview
• Print
• Qhull
• Precision
• Trace
• Functions

Comments to: qhull@qhull.org

Created: Sept. 25, 1995 --- Last modified: see top