Waterman library installation and configuration
Installation of the library requires two files waterman.mb and qconvex.exe. The QConvex program is part of the Qhull library developed at the National Science and Technology Research Center for Computation and Visualization of Geometric Structures, The Geometry Center, University of Minnesota. The complete Qhull library can be downloaded from http://www.qhull.org. With the waterman.mb library there is enclosed the qconvex.exe only.
Both files should be copied into the folder where in user configuration of MuPAD the variable READPATH points up. For example, in my configuration of MuPAD, all my works are in the folder
d:\documents\mupad
Therefore in MuPAD I have to execute command:
READPATH := "D:\\Documents\\MuPAD\\":
Finally, in order to have access to qconvex.exe
from MuPAD, we need also additional variableQCONVEX_PATH := "D:\\Documents\\MuPAD\\":
If both files were copied to the folder mentioned above, we can start a new MuPAD notebook with commands:
READPATH := "D:\\Documents\\MuPAD\\":
read("waterman.mb")
QCONVEX_PATH := "D:\\Documents\\MuPAD\\":
Now we can execute any procedure that is part of the library. For example,
plot(WatermanPoly(3,4), Scaling=Constrained)
The Waterman library contains the following procedures:
CCPoints(xrange, yrange, zrange)-procedure to calculate CCP points, obtained output is a plot object that can be displayed on the screen using the plot command. Here is an example,
plot(CCPoints(-2..2,-2..2,-2..2), Scaling=Constrained)
Result was shown in the figure 3.
CCSpheres(xrange, yrange, zrange) - procedure to calculate CCP spheres arrangement for a parallelepiped defined by xrange, yrange, zrange. Note, the output will contain all CCP spheres that have centers inside of the parallelepiped. Here is an example,
plot(CCSpheres(-3..3,-3..3,-3..3), Scaling=Constrained)
WatermanData([x,y,z],r) - procedure to calculate data for all procedures displaying Waterman polyhedra.
WatermanSpheres([x,y,z],r) - procedure to calculate all CCP spheres, with centers on, or inside, of a sphere with the given radius r and center [x,y,z]
. Obtained result is a plot object that can be displayed using the plot command, for example like here,plot(WatermanSpheres([1,0,0],3), Axes=Boxed)
WatermanPoints([x,y,z],r) - procedure to calculate centers (CCP points) of all spheres produced by the WatermanSpheres procedure. Obtained result is a plot object that can be displayed using the plot
command, for example like here,plot(WatermanPoints([0,0,0],2), PointSize=2)
WatermanPoly(sequence,number) - procedure to obtain Waterman polyhedra from Mark Newbold's sequences. Obtained result is a plot object that can be displayed using the plot command, for example like here,
plot(WatermanPoly(2,13), Scaling=Constrained)
Comment
There is one significant difference between our and Newbold's approach. It may happen that for two, or more radii, we obtain exactly the same Waterman polyhedron, for example w2,7=w2,8. In Mark Newbold's work such multiple polyhedron is counted only once, and it occupies only one position of the sequence. In our implementation identical consecutive polyhedra are treated as separate terms of the sequence.
WatermanExplore([x,y,z],Radius) - is a procedure that can be used to produce a Waterman polyhedron for any point of the 3D space as a center of the polyhedron, and for any positive radius. If such polyhedron does not exist, the output is the point P(x,y,z). In each case obtained result is a plot object that can be displayed with the plot command, for example like here,
plot(WatermanExplore([0.6,0.6,0.6],15), Scaling=Constrained)
Most of the procedures in the Waterman library produce objects with completely random colors. The advantage of this approach is that each face of a polyhedron gets an individual character. Therefore, while discussing properties of a polyhedron we are able to refer to the individual faces of it by using their colors. For example, we can say-- the blue pentagon on the top of the polyhedron is ... etc. There is also a serious disadvantage to such approach. For instance, animation of a polyhedron W((0,0,a),3), where a is the animation parameter, will produce a series of blinking images.
I am sure that coloring polyhedra in MuPAD this is another interesting topic that requires more attention.
