Sample Points Uniformly from a Unit Disk
WaymoData Scientist · Technical Screen
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Derive uniform random sampling inside a unit disk, explain the radial distribution, and compare a polar sampler with rejection sampling.
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.
Hints
Hint 1
Compare regions with equal area: Consider how much of the disk lies within a chosen distance of the center.
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?
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