Generate Random Point in a Circle

Generate Random Point in a Circle - Problem

Given the radius and the position of the center of a circle, implement the function randPoint which generates a uniform random point inside the circle.

Implement the Solution class:

Solution(double radius, double x_center, double y_center) initializes the object with the radius of the circle radius and the position of the center (x_center, y_center).
randPoint() returns a random point inside the circle. A point on the circumference of the circle is considered to be in the circle. The answer is returned as an array [x, y].

Input & Output

Example 1 — Unit Circle at Origin

$ Input: radius = 1.0, x_center = 0.0, y_center = 0.0

› Output: [0.52, -0.85] (example output)

💡 Note: Generate a random point inside unit circle centered at origin. The actual output will vary each time due to randomness, but will always satisfy x² + y² ≤ 1.

Example 2 — Larger Circle with Offset Center

$ Input: radius = 5.0, x_center = 3.0, y_center = 4.0

› Output: [6.2, 1.8] (example output)

💡 Note: Generate random point in circle of radius 5 centered at (3,4). Point satisfies (x-3)² + (y-4)² ≤ 25.

Example 3 — Small Circle

$ Input: radius = 0.1, x_center = -1.0, y_center = 2.0

› Output: [-0.95, 2.03] (example output)

💡 Note: Very small circle shows the algorithm works for any radius size. Point is within 0.1 units of center (-1, 2).

Constraints

0 < radius ≤ 10⁸
-10⁷ ≤ x_center, y_center ≤ 10⁷
At most 3 × 10⁴ calls will be made to randPoint

Visualization

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Asked in

G Google 12 f Facebook 8 a Amazon 6 M Microsoft 4

The key insight is that uniform random sampling in a circle requires careful handling of the radius distribution. The polar coordinate approach with √(random) radius scaling ensures uniform area distribution. Best approach uses mathematical transformation: Time O(1), Space O(1).

Common Approaches

✓ Polar Coordinate Transform

⏱️ Time: O(1) Space: O(1)

Generate random angle and radius using polar coordinates. Apply square root to radius to ensure uniform area distribution, then convert to cartesian coordinates.

Rejection Sampling

⏱️ Time: O(1) Space: O(1)

Generate random x and y coordinates within the bounding square of the circle. Check if the point is within the circle using the distance formula. If not, repeat until a valid point is found.

Polar Coordinate Transform — Algorithm Steps

Generate random angle θ ∈ [0, 2π]
Generate random radius using √(uniform[0,1]) × radius
Convert to cartesian: x = r×cos(θ), y = r×sin(θ)
Translate to actual center position

Visualization

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Step-by-Step Walkthrough

Random Angle

Generate θ uniformly in [0, 2π]

Square Root Radius

Use √(random) × radius for uniform area

Convert to Cartesian

x = r×cos(θ), y = r×sin(θ)

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <time.h>

#ifndef M_PI
#define M_PI 3.14159265358979323846
#endif

typedef struct {
    double radius;
    double x_center;
    double y_center;
} Solution;

Solution* createSolution(double radius, double x_center, double y_center) {
    Solution* sol = (Solution*)malloc(sizeof(Solution));
    sol->radius = radius;
    sol->x_center = x_center;
    sol->y_center = y_center;
    srand(time(NULL));
    return sol;
}

void randPoint(Solution* sol, double* result) {
    double angle = ((double)rand() / RAND_MAX) * 2 * M_PI;
    double r = sqrt((double)rand() / RAND_MAX) * sol->radius;
    double x = r * cos(angle);
    double y = r * sin(angle);
    result[0] = sol->x_center + x;
    result[1] = sol->y_center + y;
}

int main() {
    double radius, x_center, y_center;
    scanf("%lf", &radius);
    scanf("%lf", &x_center);
    scanf("%lf", &y_center);
    
    Solution* sol = createSolution(radius, x_center, y_center);
    double result[2];
    randPoint(sol, result);
    printf("[%f,%f]\n", result[0], result[1]);
    
    free(sol);
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(1)

Fixed number of mathematical operations per call

✓ Linear Growth

Space Complexity

O(1)

Only uses a few variables for calculations

✓ Linear Space

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Medium Frequency

~25 min Avg. Time

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Input & Output

Constraints

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Related Problems

Common Approaches

Polar Coordinate Transform — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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