# Some code.

Decided to publish the code from yesterday’s example, though wordpress only allows for certain somewhat odd file types to be shared, so I am just copy/pasting below.  There are 5 functions, and as usual, you save each to the same folder in MATLAB, open that folder, and then typing

>>billiardgame3(25,10,15,20)

will generate a 3-dimensional billard game with 25 balls of masses between 0 and 10, speeds between 0 and 15 bouncing for 20 seconds, then play this.  It also returns the position vectors [X,Y,Z] associated with this.  I’d be glad to also publish the 2-dimensional version of this, though it should not be too hard to wade through my code and cut it down to two dimensions.

function [v1n,v2n] = collide3(m1,m2,v1,v2)
%solves the conservation of velocity and momentum for two balls of mass m1
%and m2 colliding at velocities v1 and v2, and returns their new
%velocities.
C1 = (m1-m2)/(m1+m2);
C2 = (2*m2)/(m1+m2);
C3 = (2*m1)/(m1+m2);
v1n = C1*v1+C2*v2;
v2n = -C1*v2+C3*v1;
end

%takes vectors X, Y, and Z where each entry contains the x- or y-position of
%a billiard ball and returns an n x 2 matrix A containing the indices where
%and a matrix W containing the indices of which balls collided with walls
%in a w x l x h rectangle

C = X>=w; %oddly, I can’t find a way to make this code nicer. Checks collisions with walls
C = +C;
D = Y>=l;
D = +D;
E = X<=0;
E = +E;
F = Y<=0;
F = +F;
G = Z>=h;
G = +G;
H = Z<=0;
H = +H;
C = C+D+E+F+G+H;
for j = 1:length(C)
if C(j) > 0
W = [W;j];
end
end

X = ones(length(X),1)*X;
Y = ones(length(Y),1)*Y;
Z = ones(length(Z),1)*Z;

B = sqrt((X-X.’).^2+(Y-Y.’).^2+(Z-Z.’).^2); %a matrix whose (i,j)th entry is the distance between particle i and particle j
A = [];
for j = 1:size(B,2)
for k = 1:j-1
if B(k,j)==1
A = [A;k,j];
end
end
end
end

%simulates billiards with elastic collisions on a w x l billiards table.  M
%should be a vector recording the (positive) masses of the billiard balls
%(the function will create as many balls as the length of M).  vx, vy, X0,
%Y0 will similarly be vectors giving the initial x and y velocities of each
%billiard ball, and then initial positions.  The program runs for T seconds
%with time step tstep.  A reasonable setup is
%billiards(9,4.5,randi(10,1,9),3*rand(1,9),3*rand(1,9),.5+randi(8,1,9),.4+ran
%di(4,1,9),5,.2,.01)

X = zeros(floor(T/tstep),length(M)); %initialize the three position arrays, one column per particle
Y = zeros(floor(T/tstep),length(M));
Z = zeros(floor(T/tstep),length(M));

X(1,:) = X0; %set initial position
Y(1,:) = Y0;
Z(1,:) = Z0;

for k = 2:floor(T/tstep)
tryxpos = X(k-1,:)+tstep*vx; %here we check if any collisions will happen in the next step
tryypos = Y(k-1,:)+tstep*vy;
tryzpos = Z(k-1,:)+tstep*vz;

for j = 1:size(A,1)
[vx(A(j,1)),vx(A(j,2))] = collide3(M(A(j,1)),M(A(j,2)),vx(A(j,1)),vx(A(j,2))); %avoiding collisions with particles
[vy(A(j,1)),vy(A(j,2))] = collide3(M(A(j,1)),M(A(j,2)),vy(A(j,1)),vy(A(j,2)));
[vz(A(j,1)),vz(A(j,2))] = collide3(M(A(j,1)),M(A(j,2)),vz(A(j,1)),vz(A(j,2)));
end
for j = 1:length(W)
if tryxpos(W(j)) >= w || tryxpos(W(j))<=0 %avoiding collisions with walls
vx(W(j)) = -vx(W(j));
elseif tryypos(W(j))>=l || tryypos(W(j))<=0
vy(W(j)) = -vy(W(j));
elseif tryzpos(W(j)) >=h || tryzpos(W(j))<=0
vz(W(j)) = -vz(W(j));
end
end
X(k,:) = X(k-1,:)+tstep*vx;%updating the position with the “fixed” velocity vectors
Y(k,:) = Y(k-1,:)+tstep*vy;
Z(k,:) = Z(k-1,:)+tstep*vz;
end
end

function billiardplayer3(X,Y,Z,ptime,w,l,h)
%a program for visualizing billiard movement in 3 dimensions.

figure(1)
axis equal
%maxwindow
for j = 2:size(X,1)
%color the first particle red
plot3(X(j,1),Y(j,1),Z(j,1),’ro’,’MarkerFaceColor’,’r’)
hold on
%keep the rest of the particles blue circles
plot3(X(j,2:size(X,2)),Y(j,2:size(Y,2)),Z(j,2:size(Z,2)),’o’)
hold off
axis([0 w 0 l 0 h])
grid on
pause(ptime)
end
hold on
%draw the path of the first particle
plot3(X(:,1),Y(:,1),Z(:,1),’k’,’LineSmoothing’,’on’)
end

function [X,Y,Z] = billiardgame3(N,massmax,vmax,T)
%generates and plays a game of billiards with N randomly placed billiard
%balls on a table.  The balls have random mass between 0 and massmax,
%velocity between 0 and vmax, and plays for T seconds, with timestep .01.
w = 9;
l = 4.5;
h = 4.5;

M = massmax*rand(1,N);

vx = 2*vmax*rand(1,N)-vmax;
vy = 2*vmax*rand(1,N)-vmax;
vz = 2*vmax*rand(1,N)-vmax;

X0 = w*rand(1,N);
Y0 = l*rand(1,N);
Z0 = h*rand(1,N);

tstep = 0.01;

billiardplayer3(X,Y,Z,0.01,w,l,h)
end

# Bouncing balls

Inspired, as usual, by Leonid’s recent post, I decided to first write a script that would mimic his.  After that, since I had all the numbers worked out, I wrote two more MATLAB programs: one that mimicked elastic collisions in 2-dimensions, and one that mimics them in 3.

In theory, you can specify the number of particles and their radius, as well as the mass, position, and initial velocity for each (I didn’t vectorize radius for some reason, so I cannot model balls of different sizes bouncing around).  However, in practice I just generate random vectors for each of these numbers.  The final aspect is that the domain I put the balls in was a pool table of 9 x 4.5 units, or 9 x 4.5 x 4.5 for the 3D version.  This was just to make calculating the reflecting angle easier when a ball hit the wall.

As with Leonid’s code, mine works by checking whether the next step will cause any collisions, then adjusting the velocity vector so that the collision didn’t happen (using conservation of momentum and kinetic energy).  This algorithm is not “smart” in the sense that by avoiding one collision, it might get pushed into a second collision which it does not detect, and if a particle gets going fast enough, it can reflect off a wall from a large distance (my time step is just 0.01).  You can spot this in some of the figures below.

Anyways, here are some of the outputs.  I did not go through the trouble of turning these into .gifs, but they play fairly smoothly.  What happens is I simulate N particles of varying masses and velocities bouncing around in a 2- or 3- dimensional box for T seconds, then plot the path of one of the particles.  The end position of all the particles, plus this path, is in each picture below (with the “tracked” particle colored in red).

4 particles for 20 seconds. Not many collisions... I can spot 3, I think.

40 particles bouncing for 20 seconds. This particle is involved in a few more collisions than last time (and looks to have been moving faster, too.

Now 40 particles for 20 seconds in 3d. A few more collisions, and it looks like it was going pretty fast in the middle there for a while.

400 particles for 20seconds. Starting to look more like the "random walk" of Robert Brown's pollen, though I would certainly have to mess with how heavy the particles were to more accurately model that.