Solution Explanation

This problem asks to generate random points within a circle. The key is to understand how to generate uniformly distributed points within a circle. A naive approach might be to generate random x and y coordinates within a bounding square and reject points that fall outside the circle, but this is inefficient (rejection sampling). A more efficient approach uses polar coordinates.

Approach using Polar Coordinates

This approach leverages the properties of polar coordinates to generate uniformly distributed points. Here's the breakdown:

1. Generate a random radius: The radius r should be chosen such that the probability of selecting any given radius is proportional to its length. This ensures that points are uniformly distributed across all possible areas within the circle. We achieve this by generating a random number between 0 and the square of the radius (r²), then taking the square root. This gives us a radius with the correct probability distribution.

2. Generate a random angle: The angle θ (theta) is uniformly chosen between 0 and 2π (360 degrees). This ensures an even distribution around the circle's center.

3. Convert to Cartesian coordinates: We convert the polar coordinates (r, θ) to Cartesian coordinates (x, y) using the following formulas:

   • x = x_center + r * cos(θ)
   • y = y_center + r * sin(θ)

   where x_center and y_center are the coordinates of the circle's center.

Time and Space Complexity

• Time Complexity: The __init__ method has a time complexity of O(1) as it performs constant-time operations. The randPoint method also has a time complexity of O(1), as it involves a fixed number of arithmetic operations regardless of the circle's size.

• Space Complexity: The space complexity is O(1) because the solution uses a fixed amount of memory to store the circle's radius, center coordinates, and generated points. No additional memory scales with input size.

Code Implementation (Python)

import math
import random
 
class Solution:
    def __init__(self, radius: float, x_center: float, y_center: float):
        self.radius = radius
        self.x_center = x_center
        self.y_center = y_center
 
    def randPoint(self) -> List[float]:
        length = math.sqrt(random.uniform(0, self.radius**2))  #Generate random radius
        degree = random.uniform(0, 1) * 2 * math.pi         #Generate random angle
        x = self.x_center + length * math.cos(degree)       #Convert to Cartesian coordinates
        y = self.y_center + length * math.sin(degree)
        return [x, y]
 

This Python code directly implements the polar coordinate approach described above. The __init__ method initializes the circle's parameters, and randPoint generates and returns a random point within the circle. The use of random.uniform ensures uniformly distributed random numbers. The math module provides trigonometric functions.

Note: Other languages (Java, C++, JavaScript, etc.) would follow a very similar structure, using their respective random number generators and mathematical libraries. The core algorithm remains the same.