Knots & Tubes

Generate tubes from 3D curves and plot knots from closed curves.

The scripts
- `generate_closed_tube`
- `generate_open_tube`

given a 3d curve as a sequence of points build a mesh of triangles or polygons realizing a tube around the curve.

A closed tube is realized by the commands:

~~~~~~~~~~~~~~~~~~~
[pnts,conn,line1,line2] = generate_closed_tube(typ,R,nr,xyz) ;
[pnts,conn,line1,line2] = generate_closed_tube(typ,R,nr,x,y,z) ;
~~~~~~~~~~~~~~~~~~~

where R is the ray of the tube, nr is the number of edge used to approximate the circle of ray R moving along the curve.
The points on the guiding curve are specified by using xyz a matrix 3 by N whose columns are the points, or by using the vectors x, y and z. `typ` is the kind of polygons are used to describe the tube

- typ = 0 the mesh is composed by rectangles
- typ = 1 to 4 the mesh is composed by triangles

return values are

- pnts the points of the mesh os the tube as a matrix 3 by npts
- conn cell array where each element is a vector of integer with represents the connection of the faces with the points
- line1 and line2 are vectors of integer which represents the equator lines of the tube-torus.

This command assume that the curve is closed, otherwise use

~~~~~~~~~~~~~~~~~~~
[pnts,conn] = generate_open_tube(typ,R,nr,xyz) ;
[pnts,conn] = generate_open_tube(typ,R,nr,x,y,z) ;
~~~~~~~~~~~~~~~~~~~

which generate an open tube. For typ = 0 there are two polygonal faces at the begin and end of the tube, For typ > 0 this polygon is split in triangles.

**Plotting**
to visualize a tube the utility function

~~~~~~~~~~~~~~~~~~~~
plot_tube( pnts, conn, line1, line2, ... ) ;
~~~~~~~~~~~~~~~~~~~~

can be used. As optional parameter you can use

- 'FaceColor', [r,g,b] = color of the faces
- 'EdgeColor', [r,g,b] = color of the edges
- 'Line1Color', [r,g,b] = color of line 1
- 'Line2Color', [r,g,b] = color of line 2

**Input Output**
A tube can be saved in a text file by using

~~~~~~~~~~~~~~~~~~~~
bbox = save_tube(name,pnts,conn) ;
~~~~~~~~~~~~~~~~~~~~

The file name+'.txt' is generated with a simple description of the tube as a list of points followed by face connections.

To read an already saved tube use

~~~~~~~~~~~~~~~~~~~~
[pnts,conn] = read_tube( fname ) ;
~~~~~~~~~~~~~~~~~~~~

with the obvious meaning.

It is possibile to save a tube or a list o tubes to files which can be used by a mesh generator like NETGEN or TETGEN.

For netgen use:

~~~~~~~~~~~~~~~~~~~~
bbox = save_netgen(fileName,name,pnts,conn) ;
~~~~~~~~~~~~~~~~~~~~

For tetgen use:

~~~~~~~~~~~~~~~~~~~~
bbox = save_tetgen(fileName,names,pin,pnts,conn);
~~~~~~~~~~~~~~~~~~~~

tetgen needs the knowledge of an internal point of the tube stored in `pin`.

**Examples**
In the directory `example` many examples shows as to use the library to build knots and plot it.