An example of scattering

A common visualization problem is to draw curves in three dimensions. We may want to draw streamlines for a fluid or gas, for example. It may be difficult to interpret such plots, since it is hard to see if one curve lies behind another. So, it is common to draw the curves as ribbon or tubes instead. In the images below the leftmost curve is drawn using the plot3-command and in the other two the curve has been drawn as a tube. The leftmost curve is almost impossible to interpret (at least if one has not seen the tubes).

Curve 1
 Curve 2
 Curve 3

Using ribbons or tubes it is also possible to add information to a curve. A ribbon can twist showing the rotation, curl, of the flow and a tube can have a varying diameter, showing the divergence, for example (see page 16-33 - 16-38 in the Matlab manual "3-D Visualization").

The Matlab commands streamline, streamribbon and streamtube can draw streamlines provided the input is presented in a mesh-form (as if produced by meshgrid). Update 2010: one of the course participants told me that streamline, streamribbon and streamtube can handle vector data in the latest Matlab versions, but you are not allowed to use these functions for the lab (since I want you to practice on using polygons). But do try to use the functions as well (e.g. collect the xyz-coordinates, of the j:th trajectory, in a three-column matrix and store it in T{j}. Then type streamtube(T)).

In the following exercise we will not be able to use the existing commands, since a curve is given as: (x(k), y(k), z(k)), k = 1, ..., n. This means that you have to draw the ribbon yourself. Make a simple-minded implementation. If you want a harder exercise, you can draw tubes instead. Now to the visualization problem.

You have a group of stationary, point-charges situated around the origin. We shoot several particles (all with the same charge), at the group, and your job is to draw the trajectories of the moving particles. This problem is slightly harder than drawing streamlines, since the trajectories may intersect. This is what a typical image may look like (I have drawn tubes using a tube-drawing command of my own construction):

Scatter
When looking at the image I would use the whole screen (the above image is too small) and I would rotate it. In the following image I have zoomed in:


Zoom

The gray spheres are the stationary charged particles (they should be points, but then we cannot see them clearly).

Now some mathematics. Let r(t) = (x(t), y(t), z(t)), be the position of one moving particle. Assuming Coulomb forces, no gravity etc., the differential equation for the motion is given by (dropping (t) in r(t)):

r'' = c [ (r - p1) / || r - p1 ||3 + (r - p2) / || r - p2 ||3 + ... + (r - pn) / || r - pn ||3 ]

where p1, ..., pn are the positions for the n fixed particles. c is a constant having to do with the mass of the particle and the force between particles.  || || denotes the ordinary two-norm (distance). Use ode45 two solve the system. Test at least, the following configuration, defined by the code below:

global P c  % P and c are used in f
% P contains the coordinates for the fixed particles. One particle per column.
P = [
 -0.0466  0.0368  0.1156 -0.1553 -0.1079  0.1564  0.1155  0.1430  0.0826 0.1333
 -0.0516 -0.0825 -0.0364 -0.1542  0.0528 -0.1747  0.0316  0.0713  0.1680 0.0453
  0.1293  0.0281 -0.1349 -0.1852  0.1727 -0.0044 -0.1318 -0.1272 -0.1713 0.0056];

% The constant c is 0.001.
c = 0.001;

tspan = linspace(0, 15, 50); 
phi_s = linspace(0, 2 * pi, 8);

% Find the trajectories for a few particles...
for R = [0.1 0.2 0.3]  % R is the radius, and not r
  for phi = phi_s(1:end-1)
    [t, y] = ode45(@f, tspan, [2 R*cos(phi) R*sin(phi) -0.3 0 0]');
%   more stuff ...
  end
end
If you do not know how to write f, you can find it here, f.m .

When giving the initial conditions I have assumed that the positions come first and then the velocities. So the initial position, for one particle, is  [2 R*cos(phi) R*sin(phi)] and the initial velocity is [-0.3 0 0].

Question So try to visualize the trajectories using ribbons or tubes, where you have written your own ribbon- or tube function (ribbons are easier). Here is an old tube assignment (in Swedish only) with hints of how to make a tube. You can get a reasonable, but not perfect translation, of the old assignment by using Google's language tools.


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