Check if any point overlaps the given Circle and Rectangle

Given two opposite diagonal points of a rectangle (X1, Y1), (X2, Y2) and the center, radius of the circle R, (Xc, Yc), the task is to check if there exists any point P that belongs to both the circle as well as the rectangle. 
Examples:

Input: R = 2, Xc = 0, Yc = 0, X1 = 1, Y1 = 0, X2 = 3, Y2 = 3 
Output: true 
Explanation:
Clearly, from the below illustration, the circle and the rectangle intersect. 

Input: R = 1, Xc = 1, Yc = 1, X1 = 3, Y1 = 3, X2 = 5, Y2 = 6 
Output: false 

Approach: The idea is to simply check if the circle and the rectangle intersect or not. There are essentially 2 possible cases when the intersection occurs. 

1. Case 1: The side of the rectangle touches or intersects the circle. In order to check whether the shapes intersect, we need to find a point on or inside the rectangle that is closest to the center of the circle. If this point lies on or inside the circle, it is guaranteed that both the shapes intersect. Let the closest point be denoted by (Xn, Yn). Then the distance between the closest point and the center of the circle can be found using sqrt((Xc- Xn)² + (Yc- Yn)²). If this distance ? the radius of the circle, the two shapes intersect.
2. Case 2: The center of the circle lies inside the rectangle. Since the center of the circle lies inside the rectangle, the closest point will be (Xc, Yc).

On close observation, it can be observed that the point of interest only depends on the locations of (X1, Y1) and (X2, Y2) relative to (Xc, Yc). Therefore, the closest point in both the above cases can be calculated as:

1. Xn= max(X1, min(Xc, X2))
2. Yn= max(Y1, min(Yc, Y2))

Below is the implementation of the above approach: 

C++

// C++ implementation to check if any
// point overlaps the given circle
// and rectangle
#include<bits/stdc++.h>
using namespace std;

// Function to check if any point
// overlaps the given circle
// and rectangle
bool checkOverlap(int R, int Xc, int Yc,
                         int X1, int Y1,
                         int X2, int Y2)
{

    // Find the nearest point on the
    // rectangle to the center of
    // the circle
    int Xn = max(X1, min(Xc, X2));
    int Yn = max(Y1, min(Yc, Y2));

    // Find the distance between the
    // nearest point and the center
    // of the circle
    // Distance between 2 points,
    // (x1, y1) & (x2, y2) in
    // 2D Euclidean space is
    // ((x1-x2)**2 + (y1-y2)**2)**0.5
    int Dx = Xn - Xc;
    int Dy = Yn - Yc;
    return (Dx * Dx + Dy * Dy) <= R * R;
}

// Driver code
int main()
{
    int R = 1;
    int Xc = 0, Yc = 0;
    int X1 = 1, Y1 = -1;
    int X2 = 3, Y2 = 1;

    if(checkOverlap(R, Xc, Yc,
                       X1, Y1,
                       X2, Y2))
    {
        cout << "True" << endl;
    }
    else
    {
        cout << "False";
    }
}

// This code is contributed by BhupendraSingh

Java

// Java implementation to check if any
// point overlaps the given circle
// and rectangle
class GFG{

// Function to check if any point
// overlaps the given circle
// and rectangle
static boolean checkOverlap(int R, int Xc, int Yc,
                                   int X1, int Y1,
                                   int X2, int Y2)
{

    // Find the nearest point on the
    // rectangle to the center of
    // the circle
    int Xn = Math.max(X1, Math.min(Xc, X2));
    int Yn = Math.max(Y1, Math.min(Yc, Y2));

    // Find the distance between the
    // nearest point and the center
    // of the circle
    // Distance between 2 points,
    // (x1, y1) & (x2, y2) in
    // 2D Euclidean space is
    // ((x1-x2)**2 + (y1-y2)**2)**0.5
    int Dx = Xn - Xc;
    int Dy = Yn - Yc;
    return (Dx * Dx + Dy * Dy) <= R * R;
}

// Driver code
public static void main(String[] args)
{
    int R = 1;
    int Xc = 0, Yc = 0;
    int X1 = 1, Y1 = -1;
    int X2 = 3, Y2 = 1;

    if(checkOverlap(R, Xc, Yc,
                       X1, Y1,
                       X2, Y2))
    {
        System.out.print("True" + "\n");
    }
    else
    {
        System.out.print("False");
    }
}
}

// This code is contributed by 29AjayKumar

Python3

# Python3 implementation to check if any
# point overlaps the given Circle
# and Rectangle

# Function to check if any point
# overlaps the given Circle
# and Rectangle
def checkOverlap(R, Xc, Yc, X1, Y1, X2, Y2):

    # Find the nearest point on the
    # rectangle to the center of
    # the circle
    Xn = max(X1, min(Xc, X2))
    Yn = max(Y1, min(Yc, Y2))

    # Find the distance between the
    # nearest point and the center
    # of the circle
    # Distance between 2 points,
    # (x1, y1) & (x2, y2) in
    # 2D Euclidean space is
    # ((x1-x2)**2 + (y1-y2)**2)**0.5
    Dx = Xn - Xc
    Dy = Yn - Yc
    return (Dx**2 + Dy**2) <= R**2

# Driver code
if(__name__ == "__main__"):
    R = 1
    Xc, Yc = 0, 0
    X1, Y1 = 1, -1
    X2, Y2 = 3, 1

    print(checkOverlap(R, Xc, Yc, X1, Y1, X2, Y2))

C#

// C# implementation to check if any
// point overlaps the given circle
// and rectangle
using System;
class GFG{

// Function to check if any point
// overlaps the given circle
// and rectangle
static bool checkOverlap(int R, int Xc, int Yc,
                                int X1, int Y1,
                                int X2, int Y2)
{

    // Find the nearest point on the
    // rectangle to the center of
    // the circle
    int Xn = Math.Max(X1,
             Math.Min(Xc, X2));
    int Yn = Math.Max(Y1,
             Math.Min(Yc, Y2));

    // Find the distance between the
    // nearest point and the center
    // of the circle
    // Distance between 2 points,
    // (x1, y1) & (x2, y2) in
    // 2D Euclidean space is
    // ((x1-x2)**2 + (y1-y2)**2)**0.5
    int Dx = Xn - Xc;
    int Dy = Yn - Yc;
    return (Dx * Dx + Dy * Dy) <= R * R;
}

// Driver code
public static void Main()
{
    int R = 1;
    int Xc = 0, Yc = 0;
    int X1 = 1, Y1 = -1;
    int X2 = 3, Y2 = 1;

    if(checkOverlap(R, Xc, Yc,
                       X1, Y1,
                       X2, Y2))
    {
        Console.Write("True" + "\n");
    }
    else
    {
        Console.Write("False");
    }
}
}

// This code is contributed by Nidhi_biet

Javascript

<script>

// Javascript implementation to check if any
// point overlaps the given circle
// and rectangle

// Function to check if any point
// overlaps the given circle
// and rectangle
function checkOverlap(R, Xc, Yc, X1, Y1, X2, Y2)
{

    // Find the nearest point on the
    // rectangle to the center of
    // the circle
    let Xn = Math.max(X1, Math.min(Xc, X2));
    let Yn = Math.max(Y1, Math.min(Yc, Y2));

    // Find the distance between the
    // nearest point and the center
    // of the circle
    // Distance between 2 points,
    // (x1, y1) & (x2, y2) in
    // 2D Euclidean space is
    // ((x1-x2)**2 + (y1-y2)**2)**0.5
    let Dx = Xn - Xc;
    let Dy = Yn - Yc;
    return (Dx * Dx + Dy * Dy) <= R * R;
}


// Driver Code

    let R = 1;
    let Xc = 0, Yc = 0;
    let X1 = 1, Y1 = -1;
    let X2 = 3, Y2 = 1;

    if(checkOverlap(R, Xc, Yc,
                       X1, Y1,
                       X2, Y2))
    {
        document.write("True" + "\n");
    }
    else
    {
        document.write("False");
    }

</script>

Output:

True

Time Complexity: O(1)

Auxiliary Space: O(1)