# How to give trail to particles/ How to fade out plots with time (complex example)

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Niklas Kurz on 6 Apr 2022
Commented: DGM on 19 Apr 2022
Inspired from a webproject I have created this n-body-simulation:
First of a function to acces on:
% calculating acceleration of particles
function [a] = getAcc(pos,mass,softening)
% gravitation constant
G = 1;
% Number of particles
N = size(pos,1);
% acceleration matrix
a = zeros(N,3);
% loop to fill in acceleration by interaction with each particle
for i = 1:N
for j = 1:N
if i ~= j
dx = pos(j,1)-pos(i,1);
dy = pos(j,2)-pos(i,2);
dz = pos(j,3)-pos(i,3);
dr = (dx.^2 + dy.^2 + dz.^2+ softening.^2).^(-3/2);
a(i,1) = a(i,1) + G * mass(j,1) * (dr * dx);
a(i,2) = a(i,2) + G * mass(j,1) * (dr * dy);
a(i,3) = a(i,3) + G * mass(j,1) * (dr * dz);
end
end
end
followed by the Main script:
clear
%% setting initial conditions
% Number of particles
N = 10;
% max distance between partices at start
rmax = 4;
pos = rmax * rand(N,3);
% max velocity for particles
vmax = 1;
vel = vmax * rand(N,3);
% max mass of particles
mmax = 5;
mass = mmax * ones(N,1);
% minimal distance parameter
softening = 0.5;
% calculating acceleration of particles
acc = getAcc(pos,mass,softening);
% selecting start time
t_start = 0;
% selecting timestep
dt = 0.04;
% selecting end time
t_end = 250;
% initializing figure
figure
grid on
view(3)
%% starting main loop
for i = 1:t_end
vel = vel + acc * dt/2;
[pos] = pos + vel * dt;
acc = getAcc(pos, mass, softening);
vel = vel + acc * dt/2;
t_start = t_start + dt;
hold all
for j = 1:N
plot3(pos(j,1),pos(j,2),pos(j,3),'b.','MarkerSize',4)
end
pause(0.005)
end
However I'm running into a problem here: the particles path will not disapear gradually since all positions are hold. I saw some clever ways implementing a fade but I'm not able adapting it here. Maybe someone brave enough working through all the code may give some advises.

DGM on 7 Apr 2022
Edited: DGM on 7 Apr 2022
Here's one way using scatter3()
% setting initial conditions
% Number of particles
N = 3; % i'm using fewer points for ease of viewing
% max distance between partices at start
rmax = 4;
pos = rmax * rand(N,3);
% max velocity for particles
vmax = 1;
vel = vmax * rand(N,3);
% max mass of particles
mmax = 5;
mass = mmax * ones(N,1);
% minimal distance parameter
softening = 0.5;
% calculating acceleration of particles
acc = getAcc(pos,mass,softening);
% selecting start time
t_start = 0;
% selecting timestep
dt = 0.04;
% selecting end time
t_end = 250;
% initializing figure
grid on
view(3)
% preallocate outputs
xpos = zeros(N,t_end);
ypos = zeros(N,t_end);
zpos = zeros(N,t_end);
% starting main loop
for i = 1:t_end
vel = vel + acc * dt/2;
[pos] = pos + vel * dt;
acc = getAcc(pos, mass, softening);
vel = vel + acc * dt/2;
t_start = t_start + dt;
% store outputs
xpos(:,i) = pos(:,1);
ypos(:,i) = pos(:,2);
zpos(:,i) = pos(:,3);
end
% each trajectory has its own solid color
cmap = hsv(N);
% alpha is a linear gradient from 1 to 0
alph = fliplr(linspace(0,1,t_end));
for ks = 1:N
hs = scatter3(xpos(ks,:),ypos(ks,:),zpos(ks,:),10,cmap(ks,:),'filled');
hold on;
hs.MarkerFaceAlpha = 'flat';
end function [a] = getAcc(pos,mass,softening)
% gravitation constant
G = 1;
% Number of particles
N = size(pos,1);
% acceleration matrix
a = zeros(N,3);
% loop to fill in acceleration by interaction with each particle
for i = 1:N
for j = 1:N
if i ~= j
dx = pos(j,1)-pos(i,1);
dy = pos(j,2)-pos(i,2);
dz = pos(j,3)-pos(i,3);
dr = (dx.^2 + dy.^2 + dz.^2+ softening.^2).^(-3/2);
a(i,1) = a(i,1) + G * mass(j,1) * (dr * dx);
a(i,2) = a(i,2) + G * mass(j,1) * (dr * dy);
a(i,3) = a(i,3) + G * mass(j,1) * (dr * dz);
end
end
end
end
I imagine this isn't really what you want though. I imagine you want something that fades points dynamically as new points are plotted. That can be done, but it'll be slow.
DGM on 19 Apr 2022
There may be some way of addressing the individual descendant (marker) objects in a scatter plot that I don't know about (perhaps something undocumented). If that were possible, then you might only need N scatter objects instead of N*t_end objects. I don't know if that is an option, but it might be a big boost to speed if it were.

Image Analyst on 6 Apr 2022
Perhaps if you used scatter() to create the markers. You can create an array with the colors for all the markers so have your older ones be faded and your newer ones be brighter/bolder/more colored. You can then put hold on and call plot() if you want lines between the markers.