Generate and Plot N Points Picking Random Points on Triangle

My task is to write a function that takes as input a positive integer n. We are asked to consider the traingle whose corner points are (0,0), (2,0), and (1,2). Let p0 = (1,1). The function's purpose is to generate and plot n points p1,p2, ... pn where pk is the point determined by randomly picking one of the three corners of the triangle, and letting pk be the midpoint between the chosen corner point and pk-1 for 1 <= k <= n. I have a small program written, but I have no clue where to go from there. Please help me to get started on finding a program that works. I am aware that this program does not do what I want, it is just a starting point. Here is what I have so far:

function ex3(n)

%

%

rng('shuffle')

x = zeros(1,n);

y = x;

k = 1;

while k <=n

x(k) = 2*rand;

y(k) = 1*rand;

if y(k) > 1/2*x(k)

k = k + 1;

end

end

Border_x = [0,2,2,0];

Border_y = [0,0,1,0];

plot(Border_x,Border_y,'r',x,y,'b.','markersize',1)

axis tight

axis equal

[Sophie Culhane]

I would appreciate a starting point and I should be able to figure out a program from there. Thank you.

Answer 1

[Alan Stevens]

More like the following;

rng('shuffle');
n = 1000;  % Set number of points as desired
% Let (0,0) be vertex 1, (0,2) be vertex 2, (1,2) be vertex 3
X = [0 2 1];
Y = [0 0 2];
x = zeros(1,n);
y = x;
x(1) = 0.5; y(1) = 0.1;  % starting point (change as desired)
for i = 2:n
    p = randi(3,1); % Choose 1 2 or 3 at random
    x(i) = (X(p) + x(i-1))/2;
    y(i) = (Y(p) + y(i-1))/2;  
end
Border_x = [0,2,1,0];
Border_y = [0,0,2,0];
plot(Border_x,Border_y,'r',x,y,'b.','markersize',1)
axis tight   
axis equal

This generates a triangular Sierpinski gasket (better with a larger value of n). See what effect changing the starting point has.

[Sophie Culhane]

This program did not work for me. Is there an alternative to this program that would also work?

[Alan Stevens]

Strange! What was the error message?

[Sophie Culhane]

I’m sorry I noticed a typo in one of my lines. This program works, thank you for the help!

[Ana Clemente Sánchez]

Hi @Alan Stevens!

Thank you for the program, but it only covers part of the triangle's area (see attached). How could one make it cover the whole area?

Many thanks and kind regards,

Ana

openfig('untitled.fig');

[Torsten]

https://itecnote.com/tecnote/matlab-generate-and-plot-a-point-cloud-distributed-within-a-triangle/

But I cannot guarantee that the solution is correct.

[Alan Stevens]

The object you've been asked to generate is known as a Sierpinski triangle. It is a fractal structure. The gaps are an intrinsic part of it!

[Torsten]

@Alan Stevens

The object you've been asked to generate is known as a Sierpinski triangle.

@Ana Clemente Sánchez is not @Sophie Culhane , I guess.