Generate and Plot N Points Picking Random Points on Triangle

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Sophie Culhane
Sophie Culhane 2020-10-22
评论： Torsten 2023-2-15
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

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Alan Stevens
Alan Stevens 2020-10-22
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.
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Alan Stevens
Alan Stevens 2023-2-15
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!

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