n维空间点分布调整：如何修改Matlab代码实现图A所示分布

Hey there! Let's break down how to tweak your existing code to generate the point distribution shown in Figure A. First, let's clarify what your original code produces: it generates uniformly random points inside an n-dimensional solid sphere (the rand(N,1).^(1/d) term ensures volume-based uniformity across the sphere's interior).

Assuming Figure A shows uniformly random points on the surface of an n-dimensional sphere (the most common contrast to your current code's output), here are the key changes you need to make:

Key Modifications #

1. Fix the Radius to the Sphere's Surface #

The original code uses radius = r * (rand(N,1).^(1/d)); to spread points throughout the sphere's volume. To lock all points onto the sphere's surface, replace this line with a fixed radius equal to r:

% Replace the original radius calculation with this
radius = r * ones(N, 1); % All points sit exactly at radius r from the center

2. Simplify (Optional) the Vector Normalization #

Your original code normalizes the random points to unit length before scaling. You can make this cleaner using MATLAB's built-in vecnorm function (available in R2017b and later):

% Replace the original X scaling line with this
X = X .* (radius ./ vecnorm(X, 2, 2)) + C_rep;
% For older MATLAB versions, keep your original logic but remove the rand-based radius

If Figure A Shows a Hypercube Distribution #

If instead Figure A displays points uniformly distributed inside an n-dimensional hypercube (square in 2D, cube in 3D), you can completely replace the point-generation logic:

1. Delete all lines from X = randn(N,d); up to X = X.*repmat(...) + C_rep;
2. Add this line instead:

% Generate uniform points inside a hypercube centered at C_point, with side length 2r
X = C_rep + (rand(N,d) - 0.5) * 2*r;

Verify the Change #

After making these adjustments, run the code. The 2D plot will show all points lying exactly on the circle (for the sphere surface case) or filling a square (for the hypercube case), matching Figure A's distribution.

内容的提问来源于stack exchange，提问作者Shdotcom