Sampling Points from a Circle Using Polar Coordinates

Solution & Explanation

To generate a uniform random sample of points from a circle using polar coordinates, it's crucial to understand the distribution of points within the circle. A naive approach might lead to an uneven distribution, with more points clustering near the center. The solution involves carefully selecting the radius and angle to ensure a uniform distribution across the entire circle.

Steps to Generate Uniform Random Points:

1. Random Angle Generation:

   • The angle $\theta$ should be uniformly distributed between $0$ and $2\pi$. This ensures that every direction around the circle is equally likely.
   • Formula: $\theta = 2\pi \cdot U_\theta$, where $U_\theta \sim U(0, 1)$.

2. Random Radius Generation:

   • The naive method would suggest picking the radius $r$ uniformly between $0$ and the circle's radius $R$. However, this results in a higher density of points near the center.
   • To achieve uniform distribution, the radius $r$ should be derived from a transformation of a uniform random variable. Specifically, use the square root transformation:
     • Formula: $r = R \cdot \sqrt{U_r}$, where $U_r \sim U(0, 1)$.
   • The square root transformation compensates for the increasing area available as $r$ increases, ensuring that points are evenly distributed throughout the circle.

3. Conversion to Cartesian Coordinates:

   • Once you have $r$ and $\theta$, convert these polar coordinates to Cartesian coordinates for practical use:
     • $x = r \cdot \cos(\theta)$
     • $y = r \cdot \sin(\theta)$
   • This conversion is essential if you need to work with the points in a Cartesian coordinate system.

Explanation:

• Why $r = R \cdot \sqrt{U_r}$?

  • The area of a circle segment between $r$ and $r + \Delta r$ is proportional to $r \cdot \Delta r$. Thus, to maintain uniform density, the probability density function (PDF) of $r$ should be proportional to $r$. The transformation $r = R \cdot \sqrt{U_r}$ achieves this by ensuring that the cumulative distribution function (CDF) of $r$ is linear.

• Ensuring Uniform Distribution:

  • The square root transformation of $r$ balances the distribution of points, ensuring that each small area within the circle has an equal probability of containing a point.

By following these steps, you ensure that the generated points are uniformly distributed across the circle, avoiding clustering near the center and achieving the desired even spread.