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2 or 3 from the dimension of the face.]

<li> geom := (geoentity<sub>1</sub> ...
geoentity<sub><em>m</em></sub>)
<li> <a name = "geocorresp">geoentity := (<strong>vertex</strong> cpnum) |
(<strong>bezier_curve</strong> deg cpnum<sub>1</sub> ... 
cpnum<sub><em>p</em></sub>) |
(<strong>bezier_triangle</strong> deg cpnum<sub>1</sub> ... 
cpnum<sub><em>p</em></sub>) |
(<strong>bezier_quad</strong> deg<sub>1</sub> deg<sub>2</sub> 
cpnum<sub>1</sub> ... cpnum<sub><em>p</em></sub>).
[The <strong>vertex</strong> entity is allowed only for topological
vertices, and a topological vertex must have exactly one such
entity.  The <strong>bezier_curve</strong> 
entity is allowed only for edges.  The
<strong>bezier_triangle</strong> and 
<strong>bezier_quad</strong> entities are allowed only for surfaces.
The degree entries are positive integers.  The number of
control points <em>p</em> is deg+1 for a curve, (deg+1)*(deg+2)/2
for a triangle, and (deg<sub>1</sub>+1)*(deg<sub>2</sub>+1) 
for a quad.  Each
control point 
cpnum<sub>1</sub>, ..., cpnum<sub><em>p</em></sub> is a nonnegative integer
that is an index into the control point list.]</a>
</ul>

The Ascii format for a mesh is:
<ul>
<li>
asciiMesh := code intrinsicDim embeddedDim globalPropVal vertexList
brepVertexList
brepEdgeList {brepSurfaceList {brepChamberList}} [The number
of brep entity lists must be the intrinsic dimension plus 1. 
In other words, if the mesh's intrinsic dimension is 1, then
it must possess a brepVertexList and a brepEdgeList but may
not possess a brepSurfaceList nor a brepChamberList.]
<li>
code := <strong>mesh_v2.01</strong>
<li>
intrinsicDim := <strong> 1 </strong> |
<strong> 2 </strong> | <strong> 3 </strong>
[Intrinsic dim must be less than or equal to embedded dim.]
<li>
embeddedDim := <strong>2</strong> | <strong> 3 </strong>
<li>
globalPropVal := ( prop<sub>1</sub> val<sub>1</sub> ... 
prop<sub><em>m</em></sub> val<sub><em>m</em></sub> )
[The property/value pairs are arbitrary strings.  If
a property or value string contains spaces, then it
should be enclosed in an outer pair of parentheses.
Currently, the only global property value with significance to
QMG 2.0 is <strong>geo_global_id</strong>.]
<li>
vertexList := (
globalId<sub>1</sub> xc<sub>1</sub> yc<sub>1</sub>
{zc<sub>1</sub>} ... 
globalId<sub><em>n</em></sub> 
xc<sub><em>n</em></sub> yc<sub><em>n</em></sub>
{zc<sub><em>n</em></sub>})
[This is a list of the <em>n</em> nodes in the mesh.  Each
entry consists of a global ID, which must be a nonnegative
integer, followed by either 2 or 3 coordinates, which are
real numbers.  The number of coordinates is equal to the
embedded dimension (2 or 3).  The global ID's do not have
to be in increasing order, but they must be distinct.]
<li>
brepVertexList := ((globalId<sub>1</sub>) ( ) (globalId<sub>2</sub>) ( ) ...
(globalId<sub><em>m</em></sub>) ( )) [This is the list of mesh entities
associated with the brep's vertices.  There is one globalId per
brep vertex.]
<li>
brepEdgeList := (edgeVlist<sub>1</sub> edgeSlist<sub>1</sub> ...
edgeVlist<sub><em>m</em></sub> edgeSlist<sub><em>m</em></sub>)
[This is the list of mesh entities associated with the
brep's edges.  In particular, there is an edgeVlist and edgeSlist
for each brep edge, so <em>m</em> here denotes the number of
brep edges.]
<li> edgeVlist := 
(globalId<sub>1</sub> curveindex<sub>1</sub> paramcoord<sub>1</sub> ...
globalId<sub><em>p</em></sub> curveindex<sub><em>p</em></sub> 
paramcoord<sub><em>p</em></sub>)
[This is the list of mesh nodes lying on a particular brep edge.
Each triple in this list specifies which mesh node, which geometric
curve within the brep edge, and the parametric coordinate 
(a real number in [0,1]) of the mesh node.]
<li> edgeSlist :=
(globalId<sub>1</sub> ... globalId<sub>2<em>r</em></sub>)
[This is a list of global node ID's.  They are arranged in
pairs, for a total of <em>r</em> pairs.  Each pair indicates
the endpoints of a mesh edge that lies on the brep edge.
The union of these mesh edges should equal the brep edge.
Each global vertex ID lying in this list must also occur
in edgeVlist.]
<li>
brepSurfaceList := (surfaceVlist<sub>1</sub> surfaceSlist<sub>1</sub> ...
surfaceVlist<sub><em>m</em></sub> surfaceSlist<sub><em>m</em></sub>)
[This is the list of mesh entities associated with the
brep's surfaces.  In particular, there is a surfaceVlist and surfaceSlist
for each brep surface, so <em>m</em> here denotes the number of
brep surfaces.]
<li>
surfaceVlist := ( ) | 
(globalId<sub>1</sub> patchindex<sub>1</sub> paramcoordU<sub>1</sub> 
paramcoordV<sub>1</sub> ...
globalId<sub><em>p</em></sub> patchindex<sub><em>p</em></sub> 
paramcoordU<sub><em>p</em></sub>
paramcoordV<sub><em>p</em></sub>)
[This is the list of mesh nodes lying on a particular brep surface.
If the embedded dimension is 2, then this list is required to be
empty, that is, ( ).  If the embedded dimension is 3, then 
this list is divided into four-tuples.
Each four-tuple includes the global ID of the vertex, the index
of the geometric patch containing the mesh node, and the parametric
coordinates within the patch of the mesh node.]
<li> surfaceSlist :=
(globalId<sub>1</sub> ... globalId<sub>3<em>r</em></sub>)
[This is a list of global node ID's.  They are arranged in
triples, for a total of <em>r</em> triples.  Each triple indicates
the endpoints of a mesh triangle that lies on the brep surface.
The union of these mesh triangles should equal the brep surface.
Each global vertex ID lying in this list must also occur
in surfaceVlist.]

<li>
brepChamberList :=( ( ) chamberSlist<sub>1</sub> ... ( ) 
chamberSlist<sub><em>m</em></sub>)
[This is the list of mesh entities associated with the
brep's chambers (assuming the intrinsic and embedded dimension are both 3).  
In particular, there is a chamberSlist
for each chamber, so <em>m</em> here denotes the number of
brep chambers.]
<li>
chamberSlist := (globalId<sub>1</sub> .....
globalId<sub>4<em>r</em></sub>)
[This is a list of global node ID's.  They are arranged in
four-tuples, for a total of <em>r</em> four-tuples.  Each four-tuple indicates
the endpoints of a mesh tetrahedron that lies in the brep chamber.
The union of these mesh tetrahedra should equal the brep chamber.]

</ul>


<h2><a name="matrep">Matlab internal representation of objects</a></h2>

<a name = "zba">
The Matlab internal representation of objects uses zero-based array
addressing.  This implemented via a Matlab class called "zba".  </a>
In a zero-based array, rows and column numbering starts with 0
instead of 1.  Many array operations with which you are familiar
have been reimplemented for zba's.  You can extend the operations
available for zba's by adding new routines
to the @zba subdirectory.  
To convert an array to a zba, use the <code>zba</code> function.  To convert
back, use the <code>double</code> function.
<pre>
<code>
&gt;&gt; z = zba([3,5,7]);
&gt;&gt; z(0)

ans =

     3

&gt;&gt; y = double(z);
&gt;&gt; y(1)

ans =

     3
</code>
</pre>


The user can directly access entries of these objects to
examine and modify a brep or mesh.  Breps or meshes can also be
created directly by the user or with m-files.  (This is in contrast
to QMG 1.0 and 1.1, in which the object was stored in 'chunk' format
and was accessible and modifiable only by using certain
accessor functions.)

<p>
The zba's in QMG are usually nested, with the outer level being
a cell array.  Every level of indexing in a nested zba is zero-based.

<p>
A brep is represented by a zero-based cell column vector with 6+<em>i</em> 
entries,
where <em>i</em> is the intrinsic dimension of the brep.  (This 
documentation assumes that you are already familiar with
subscripting of vectors, matrices, and cell arrays in Matlab.)
Say the brep is b.  The entries of b are as follows.
<ul>
<li>
b{0} is a string which is always 'brep_v2.0'. 
<li>
b{1}
is the intrinsic dimension.
<li> 
b{2} is the embedded dimension.
<li>
b{3} is the property-value list.  This is a 2-by-k cell
array, where each cell entry is a string.  In other words,
b{3}{0,0} is the first property, b{3}{1,0} is its value, and so on.

<li>
Entry b{4} is the control-point table.  This is an <em>e</em>-by-<em>m</em> 
matrix (with
zero-based numbering), where <em>m</em> 
is the number of control points and <em>e</em>
is the embedded dimension of the brep.

<li>
Entry b{5} holds vertices, b{6} holds edges, b{7} holds surfaces,
and b{8} holds chambers.  Not all of these entries are present; the
number present depends on the intrinsic dimension of the brep.
The format of these entries is as follows.
Each of b{d+5}, for d=0,...,3, is a 5-by-<em>t</em> cell
array,
where <em>t</em> denotes the number of faces of the particular dimension.
The cells are as follows.
<ul>
<li>
b{d+5}{0,i} (for i between 0 and <em>t</em>-1) is the name of the
ith face (a string).
<li>
b{d+5}{1,i} is the property-value pair list of this face.
It is a 2-by-<em>k</em> cell array of strings, where <em>k</em> is the number
of property values.

<li>
b{d+5}{2,i} is the list of subfaces of the ith face.  This is a
1-by-<em>r</em> cell array,
where <em>r</em> is the number of subfaces. 
Each entry is a string with the name of the subface.  This name
may optionally be preceded by + or - to indicate orientation
(current not used).

<li>
b{d+5}{3,i}
is the list of lower-dimensional
internal boundaries.  This is a 1-by-<em>s</em> cell array, 
where <em>s</em> is the number
of lower dimensional internal boundaries.  Each entry is a string
with the name of the subface.

<li>
bd{d+5}{4,i}
is the cell array geometric entities making up the
face.  This cell array is 3-by-p, where p is the number
of entities.  Entry {0,j} of this cell array is
a string, either 'vertex', 'bezier_curve', 'bezier_triangle',
or 'bezier_quad'.  As mentioned <a href="#geocorresp">above</a>,
the choice of entity type is restricted according to the face dimension.
Entry {1,j} of the cell array is the degree, either empty (for a vertex),
one integer (for a curve or triangle) or two integers (for a patch).
Entry {2,j} of the cell array is a vector of control point indices.
The length of this vector follows the rule mentioned 
<a href="#geocorresp">above</a>.

</ul>
</ul>

The matlab format for a mesh is as follows.  A mesh m is a zba
cell-vector with 6+<em>i</em> entries, where <em>i</em>
is the embedded dimension.  The entries as follows.
<ul>
<li>
m{0} is the string 'mesh_v2.01'.
<li>
m{1} is the mesh's intrinsic dimension.
<li>
m{2} is the mesh's embedded dimension.
<li>
m{3} is a 2-by-<em>k</em> cell array of strings, which hold
the property-value pairs of the mesh.
<li>
m{4} is the mesh's node list.  This is a <em>k</em>-by-<em>r</em> matrix, 
where
<em>k</em>=3 if the embedded dim is 2, else <em>k</em>=4 
if the embedded dim is 3.
The number of columns <em>r</em> is equal to the number of vertices in
the complex.  The first entry of each column is the vertex's global
ID, which must be a nonnegative integer (stored as a double though).
The remaining entries in each column are the vertex's real-space
coordinates.

<li>
Entry m{5} holds mesh entities
associated with brep vertices, m{6} 
holds mesh entities associated with brep edges, 
m{7} holds mesh entities associated with brep surfaces, and
and m{8} holds 
mesh entities associated with brep chambers.  
Not all of these entries are present; the
number present depends on the intrinsic dimension of the brep.
The format of these entries is as follows.
Each of m{d+5}, for d=0,...,3, is a 2-by-<em>t</em> cell
array,
where <em>t</em> denotes the number of brep
faces of the particular dimension.
The cells are as follows.
<ul>
<li>
m{d+5}{0,i} is the list of vertices associated with the
ith topological entity of dimension d.  This list of vertices
is a <em>k</em>-by-<em>r</em> matrix defined as follows.
If d=0, then <em>k</em>=<em>r</em>=1, and the single entry
is the global id of the mesh vertex lying on topological vertex.
If d=1, then <em>k</em>=3, and each column of this array corresponds
to a mesh node lying on the topological edge.  The first entry in the
column is the mesh's global ID (a nonnegative integer
stored as a double), the second entry is the curve index of the
curve of the edge that contains the vertex (a nonnegative
integer stored as a double), and the third entry is the parametric
coordinate (a double in [0,1]) of the point within the curve.
<p>
If d=2, then  <em>k</em>=<em>r</em>=0 when the embedded dimension is 2.
<p>
If d=2 and the embedded dimension is 3, then <em>k</em>=4.  Each column
of m{d+5}{0,i} is a vertex lying on the ith topological surface.
The first entry is the vertex global ID, the next entry is
the index of the patch within the topological surface, and the
last two entries are the parametric coordinates of the mesh node
within that patch.
<p>
If d=3, then   <em>k</em>=<em>r</em>=0.
<li>
m{d+5}{1,i} is a matrix of mesh nodes (nonnegative global ID's, which
are integers but are stored as doubles) making up the mesh entities
lying on the face.  If d=0, this matrix is empty.
If d=1, then m{d+5}{1,i} is a 2-by-<em>r</em>
matrix of mesh nodes; each column in this matrix is a mesh edge
lying on the brep edge.  If d=2, then m{d+5}{1,i} is a 3-by-<em>r</em>
matrix; each column is a mesh triangle lying on the brep surface.
If d=3, then m{d+5}{1,i} is a 4-by-<em>r</em> matrix; each
column is a tetrahedron.
</ul>
</ul>

<p>
Two file-formats for meshes and breps can be used in
Matlab/QMG.  The
first is Ascii format: to read a brep or mesh from
a file in Ascii format, use <code>gm_read</code>.  To write it
out to a file, use <code>gm_write</code>.
Breps and meshes can also be saved in Matlab's native mat-file
format using the <code>save</code> and <code>load</code>
commands.

<h2><a name = "tclrep">Tcl representation of objects</a></h2>

In Tcl/Tk, breps and meshes are represented internally using 
C data structures.  This data structure is not documented
here; refer to the source-code files GeoFmt.h
and MshFmt2.h in the QMGROOT/src/tcl directory and the comments
in those files.

<p>
Breps and meshes are Tcl dual-ported objects, which means that
they stay in the C data structures until a command is executed
on the object that is unable to directly handle breps or meshes,
for instance, the concat function.  Then the object is
converted to its string representation, which is the
Ascii format described <a href="#asciirep">above</a>.
Converting a large object to Ascii can be very
expensive, so you should avoid Tcl operations that cause
a conversion unless you need them.

<p>
To make it easier to manipulate geometric objects in Tcl, 
there are also the two functions <code>gm_obj2list</code>
and its inverse <code>gm_list2obj</code>.  The former
is roughly equivalent to converting the object to Ascii format
and then replacing all parentheses that occur at outer levels
with curly braces so that entries can be indexed and modified using the
<code>lindex</code> and <code>lreplace</code> 
functions of Tcl.  (The <code>gm_obj2list</code> function does not
actually convert to a string on an intermediate step.  Instead,
it converts directly to Tcl's internal list representation, which
is more efficient.)
The <code>gm_list2obj</code> operation inverts this procedure.

<p>
Two file-formats for meshes and breps are available in
Tcl/QMG.  The
first is Ascii format.  To read a brep or mesh from
a file in Ascii format, use <code>gm_read</code>.  To write it
out to a file, use <code>gm_write</code>.
Breps and meshes can also be saved in a binary file format
based on the XDR standards.  The functions for this purpose are
<code>gmxdr_read</code> and <code>gmxdr_write</code>.
This XDR format is platform independent because the XDR routines
handle byte-order issues.

<hr>
<p>
This documentation is written by 
<a href="http://www.cs.cornell.edu/home/vavasis/vavasis.html">Stephen A. 
Vavasis</a> and is
copyright &#169;1999 by <a href="http://www.info.cornell.edu/CUHomePage.html">Cornell 
University</a>.
Permission to reproduce this documentation is granted provided this
notice remains attached.  There is no warranty of any kind on
this software or its documentation.  See the accompanying file
<a href="copyright.html">'copyright'</a>
for a full statement of the copyright.
<p>
<address>
Stephen A. Vavasis, Computer Science Department, Cornell University,
Ithaca, NY 14853, [e-mail]
</address>
</body>
</html>