Circle and Rectangle Overlapping

Circle and Rectangle Overlapping - Problem

Math Geometry Hard

You are given a circle represented as (radius, xCenter, yCenter) and an axis-aligned rectangle represented as (x1, y1, x2, y2), where (x1, y1) are the coordinates of the bottom-left corner, and (x2, y2) are the coordinates of the top-right corner of the rectangle.

Return true if the circle and rectangle are overlapped otherwise return false. In other words, check if there is any point (xi, yi) that belongs to the circle and the rectangle at the same time.

Input & Output

Example 1 — Circle Overlapping Rectangle

$ Input: radius = 1, xCenter = 0, yCenter = 0, x1 = 1, y1 = -1, x2 = 3, y2 = 1

› Output: true

💡 Note: Circle at (0,0) with radius 1 overlaps the rectangle. The closest point on rectangle to circle center is (1,0), and distance = 1 which equals the radius, so they overlap.

Example 2 — Circle Not Overlapping Rectangle

$ Input: radius = 1, xCenter = 1, yCenter = 1, x1 = 1, y1 = -3, x2 = 2, y2 = -1

› Output: false

💡 Note: Circle at (1,1) with radius 1 does not reach the rectangle. The closest point on rectangle is (1,-1), distance = 2 > radius 1, so no overlap.

Example 3 — Circle Contains Rectangle

$ Input: radius = 5, xCenter = 0, yCenter = 0, x1 = -1, y1 = -1, x2 = 1, y2 = 1

› Output: true

💡 Note: Large circle with radius 5 completely contains the small rectangle, so they definitely overlap.

Constraints

1 ≤ radius ≤ 2000
-10⁴ ≤ xCenter, yCenter ≤ 10⁴
-10⁴ ≤ x1 < x2 ≤ 10⁴
-10⁴ ≤ y1 < y2 ≤ 10⁴

Visualization

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Asked in

G Google 15 🍎 Apple 12 M Microsoft 8

The key insight is to find the closest point on the rectangle to the circle's center using coordinate clamping, then check if this distance is within the circle's radius. Best approach is Mathematical Closest Point with Time: O(1), Space: O(1)

Common Approaches

✓ Binary Search

⏱️ Time: N/A Space: N/A

Brute Force Point Sampling

⏱️ Time: O(n²) Space: O(1)

Generate a dense grid of points covering both shapes and check each point to see if it lies within both the circle and rectangle. This approach is conceptually simple but computationally expensive.

Mathematical Closest Point

⏱️ Time: O(1) Space: O(1)

Calculate the closest point on the rectangle to the circle's center, then check if this distance is less than or equal to the circle's radius. This is the optimal geometric approach.

Algorithm Steps — Algorithm Steps

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

int checkOverlap(int radius, int xCenter, int yCenter, int x1, int y1, int x2, int y2) {
    // Find the closest point on the rectangle to the circle center
    int closestX, closestY;
    
    // Clamp circle center to rectangle bounds
    if (xCenter < x1) {
        closestX = x1;
    } else if (xCenter > x2) {
        closestX = x2;
    } else {
        closestX = xCenter;
    }
    
    if (yCenter < y1) {
        closestY = y1;
    } else if (yCenter > y2) {
        closestY = y2;
    } else {
        closestY = yCenter;
    }
    
    // Calculate squared distance from circle center to closest point
    int dx = xCenter - closestX;
    int dy = yCenter - closestY;
    int distanceSquared = dx * dx + dy * dy;
    
    // Check if distance is less than or equal to radius
    return distanceSquared <= radius * radius;
}

int main() {
    int radius, xCenter, yCenter, x1, y1, x2, y2;
    
    scanf("%d", &radius);
    scanf("%d", &xCenter);
    scanf("%d", &yCenter);
    scanf("%d", &x1);
    scanf("%d", &y1);
    scanf("%d", &x2);
    scanf("%d", &y2);
    
    int result = checkOverlap(radius, xCenter, yCenter, x1, y1, x2, y2);
    printf("%s\n", result ? "true" : "false");
    
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

✓ Linear Growth

Space Complexity

✓ Linear Space

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