Problem Description

You need to implement a class that generates random points uniformly distributed inside a circle. The circle is defined by its radius and center position coordinates.

The Solution class has two methods:

• Constructor __init__(radius, x_center, y_center): Initializes the circle with:

  • radius: the radius of the circle
  • x_center, y_center: the coordinates of the circle's center

• Random Point Generator randPoint(): Returns a randomly generated point [x, y] that lies inside or on the boundary of the circle

The key challenge is ensuring the points are uniformly distributed within the circle. This means every point inside the circle should have an equal probability of being selected.

The solution uses polar coordinates to generate uniform random points:

1. Random radius: Uses length = math.sqrt(random.uniform(0, radius²)) to ensure uniform distribution. The square root is crucial - without it, points would cluster near the center.
2. Random angle: Uses degree = random.uniform(0, 1) * 2π to get a random angle from 0 to 2π radians.
3. Convert to Cartesian coordinates:
   • x = x_center + length * cos(degree)
   • y = y_center + length * sin(degree)

The mathematical insight: In polar coordinates, the area element is proportional to r * dr * dθ. To achieve uniform distribution in Cartesian space, we need to sample r² uniformly (not r directly), then take the square root. This compensates for the fact that circles at larger radii have more area than those at smaller radii.

Intuition

The first instinct might be to generate random polar coordinates directly: pick a random radius r between 0 and radius, and a random angle θ between 0 and 2π. However, this approach would create a non-uniform distribution with points clustering near the center.

Why does this happen? Imagine dividing the circle into concentric rings. A ring at radius r has circumference 2πr, which means it has more area than rings closer to the center. If we pick radius values uniformly, we're giving equal probability to each radius value, but larger radii represent more area. This causes points to cluster near the center.

To visualize this: if we divide the circle into two regions - inner circle (radius 0 to r/2) and outer ring (radius r/2 to r), the outer ring actually contains 75% of the total area, while the inner circle only has 25%. Yet uniform radius selection would place points in each region with 50% probability.

The key insight is that the area of a circle grows with the square of the radius (A = πr²). Therefore, to achieve uniform distribution across the area, we need the probability of selecting a radius to be proportional to the radius itself (since more area exists at larger radii).

Mathematically, if we want uniform distribution over area, the cumulative distribution function (CDF) for radius should be proportional to r²/R² (where R is the circle's radius). To generate a random variable with this distribution, we:

1. Generate a uniform random value u between 0 and 1
2. Set u = r²/R²
3. Solve for r: r = R * sqrt(u)

This is why the solution uses length = math.sqrt(random.uniform(0, radius²)) - it's equivalent to generating a uniform value between 0 and 1, multiplying by radius², then taking the square root.

Solution Approach

The implementation consists of a class with initialization and random point generation methods. Here's how each component works:

1. Class Initialization (__init__ method):

def __init__(self, radius: float, x_center: float, y_center: float):
    self.radius = radius
    self.x_center = x_center
    self.y_center = y_center

Store the circle's parameters as instance variables for use in point generation.

2. Random Point Generation (randPoint method):

The method generates uniformly distributed points using polar coordinates with proper transformation:

def randPoint(self) -> List[float]:
    length = math.sqrt(random.uniform(0, self.radius**2))
    degree = random.uniform(0, 1) * 2 * math.pi
    x = self.x_center + length * math.cos(degree)
    y = self.y_center + length * math.sin(degree)
    return [x, y]

Step-by-step breakdown:

1. Generate random radius with proper distribution:

   • random.uniform(0, self.radius**2) generates a uniform random value between 0 and radius²
   • math.sqrt() transforms this to ensure uniform area distribution
   • This gives us length, the distance from center to our random point

2. Generate random angle:

   • random.uniform(0, 1) generates a value between 0 and 1
   • Multiply by 2 * math.pi to get an angle in radians from 0 to 2π
   • This gives us degree, the angular position of our point

3. Convert polar to Cartesian coordinates:

   • x = self.x_center + length * math.cos(degree) calculates the x-coordinate
   • y = self.y_center + length * math.sin(degree) calculates the y-coordinate
   • The center coordinates (x_center, y_center) are added to translate the point from origin to the actual circle center

4. Return the result:

   • Return the point as a list [x, y]

Key algorithmic patterns used:

• Inverse Transform Sampling: Using the inverse CDF to generate random variables with a specific distribution
• Coordinate Transformation: Converting between polar and Cartesian coordinate systems
• Translation: Shifting points from origin-centered circle to the actual center position

The time complexity is O(1) for each randPoint() call, and space complexity is O(1) as we only store the circle parameters.

Example Walkthrough

Let's walk through generating random points in a circle with radius 2.0 centered at (1.0, 3.0).

Step 1: Initialize the circle

solution = Solution(radius=2.0, x_center=1.0, y_center=3.0)

Step 2: Generate a random point

Let's trace through one call to randPoint():

1. Generate random radius with uniform distribution:

   • Suppose random.uniform(0, 4.0) returns 3.0 (since radius² = 4.0)
   • Calculate length = sqrt(3.0) ≈ 1.732
   • This point will be ~1.732 units from the center

2. Generate random angle:

   • Suppose random.uniform(0, 1) returns 0.25
   • Calculate degree = 0.25 * 2π ≈ 1.571 radians (90 degrees)
   • This point will be directly above the center

3. Convert to Cartesian coordinates:

   • x = 1.0 + 1.732 * cos(1.571) ≈ 1.0 + 1.732 * 0 = 1.0
   • y = 3.0 + 1.732 * sin(1.571) ≈ 3.0 + 1.732 * 1 = 4.732
   • Final point: [1.0, 4.732]

Step 3: Verify the distribution

To understand why we use sqrt(random.uniform(0, radius²)) instead of just random.uniform(0, radius), consider dividing our circle into two regions:

• Inner circle: radius 0 to 1.0 (area = π)
• Outer ring: radius 1.0 to 2.0 (area = 4π - π = 3π)

The outer ring has 3 times the area of the inner circle, so it should receive 75% of the points.

With our approach:

• Probability of point in inner circle: P(length ≤ 1) = P(sqrt(u) ≤ 1) = P(u ≤ 1) = 1/4 = 25%
• Probability of point in outer ring: P(length > 1) = 75%

This matches the area distribution perfectly! If we had used random.uniform(0, radius) directly, both regions would get 50% of points, causing clustering near the center.

Multiple sample points: Calling randPoint() multiple times might generate:

• [2.414, 3.707] - point in the upper-right quadrant
• [0.123, 2.156] - point in the lower-left quadrant
• [1.000, 3.000] - rare but possible point exactly at center
• [-0.876, 3.543] - point in the upper-left quadrant

Each point satisfies: (x - 1.0)² + (y - 3.0)² ≤ 4.0

Solution Implementation

Time and Space Complexity

Time Complexity: O(1)

• The __init__ method simply stores three values, which takes O(1) time.
• The randPoint method performs a fixed number of operations:
  • random.uniform() is called twice - O(1) each
  • math.sqrt() is called once - O(1)
  • math.cos() and math.sin() are called once each - O(1) each
  • Basic arithmetic operations (multiplication, addition, exponentiation) - O(1) each
• All operations are constant time regardless of the radius or center values.

Space Complexity: O(1)

• The __init__ method stores three floating-point values (radius, x_center, y_center) - O(1) space.
• The randPoint method uses a fixed number of local variables (length, degree, x, y) - O(1) space.
• No additional data structures that scale with input size are used.
• The returned list contains exactly 2 elements regardless of input - O(1) space.

Common Pitfalls

1. Incorrect Radius Sampling (Most Critical)

Pitfall: The most common mistake is sampling the radius directly using random.uniform(0, radius) instead of using the square root transformation.

Why it's wrong:

• Direct radius sampling causes points to cluster near the center
• The area of a ring at distance r from center is proportional to r (circumference = 2πr)
• Larger radii have more area, so they should have proportionally more points
• Without the square root transformation, the distribution becomes biased toward the center

Incorrect implementation:

# WRONG - causes center clustering
random_radius = random.uniform(0, self.radius)  

Correct implementation:

# CORRECT - uniform distribution
random_radius = math.sqrt(random.uniform(0, self.radius ** 2))

2. Rejection Sampling Inefficiency

Pitfall: Using rejection sampling (generating points in a square and rejecting those outside the circle).

Why it's problematic:

• Wastes approximately 21.5% of generated points (1 - π/4)
• Non-deterministic runtime - could theoretically loop forever
• Less efficient than the polar coordinate method

Inefficient approach:

# INEFFICIENT - rejection sampling
while True:
    x = random.uniform(-self.radius, self.radius)
    y = random.uniform(-self.radius, self.radius)
    if x**2 + y**2 <= self.radius**2:
        return [self.x_center + x, self.y_center + y]

3. Angle Unit Confusion

Pitfall: Mixing degrees and radians when generating the angle.

Wrong implementations:

# WRONG - using degrees with math.cos/sin (which expect radians)
random_angle = random.uniform(0, 360)
x = self.x_center + random_radius * math.cos(random_angle)  # Expects radians!

# WRONG - incorrect radian range
random_angle = random.uniform(0, 2)  # Should be 0 to 2π, not 0 to 2

Correct implementation:

random_angle = random.uniform(0, 2 * math.pi)  # Or random.uniform(0, 1) * 2 * math.pi

4. Forgetting Center Translation

Pitfall: Generating points around the origin (0, 0) instead of the actual circle center.

Wrong implementation:

# WRONG - forgets to add center coordinates
x = random_radius * math.cos(random_angle)  # Missing self.x_center
y = random_radius * math.sin(random_angle)  # Missing self.y_center

Correct implementation:

x = self.x_center + random_radius * math.cos(random_angle)
y = self.y_center + random_radius * math.sin(random_angle)

5. Edge Case: Zero Radius

Pitfall: Not handling the case when radius is 0 or very small.

Potential issue: While mathematically valid, a circle with radius 0 should always return the center point. The current implementation handles this correctly, but some alternative approaches might have numerical stability issues.

Complete Correct Solution:

def randPoint(self) -> List[float]:
    # Correct square root transformation for uniform distribution
    length = math.sqrt(random.uniform(0, self.radius**2))
    # Correct radian range: 0 to 2π
    degree = random.uniform(0, 2 * math.pi)
    # Correct translation to circle center
    x = self.x_center + length * math.cos(degree)
    y = self.y_center + length * math.sin(degree)
    return [x, y]