Sample Points Uniformly from a Unit Disk

## Sample Points Uniformly from a Unit Disk Write a function that takes the requested number of points, `n`, and returns `n` random two-dimensional points distributed uniformly over the area inside the unit circle centered at the origin. The output is a collection of coordinate pairs. Explain why the sampling rule is uniform over the disk's area. The task does not prescribe a random seed or a fixed sequence of coordinates; many sampled outputs are valid. ### Clarifications Uniformity is over area, rather than over the radius alone or only around the circumference. The input represents a number of requested points. Discuss the empty output when no points are requested. ```hint Compare regions with equal area Consider how much of the disk lies within a chosen distance of the center. ``` ### What a Strong Answer Covers - Generates points inside the radius-one disk and returns the requested number of coordinate pairs. - Explains the radial distribution needed for uniform area coverage. - Distinguishes random sampling from returning one deterministic coordinate list. - States the time and storage cost in terms of the requested number of points. ### Follow-up Questions - Why does choosing a radius uniformly between zero and one concentrate points near the center? - How would rejection sampling from the surrounding square produce another valid uniform sampler?