Minimum Time Taken for N Points to coincide

Given an array arr[][] of size N, consisting of pairs of coordinates such that arr[i][0] and arr[i][1] represents the X and Y coordinates in a 2D plane. Given another array V[] that denotes the maximum velocity of each point in any direction, the task is to find the minimum time such that all the given points meet at any arbitrary points in the 2D plane.

Examples:

Input: N = 4, arr[][] = {{1, 2}, {-3, 34}, {-1, -2}, {2, -2}}, V[] = {3, 2, 4, 5} 
Output: 1.1157499537009508

Input: N = 2, arr[][] = {{1. 2}, {-1, -2}}, V[] = {2, 3} 
Output: 0.8944271909999159

Approach: The given problem can be solved based on the following observations:

• The given N points in the coordinates axis can move in any direction and if these points are allowed to move for T time then they can reach any point within the circle (by reducing V[i] accordingly) having a radius equal to V[i] * T and centre equal to initial position of the point, where V[i] represents the speed of the i^th point.
• Therefore, to minimize the common area of these N circles radius needs to be minimized and if there exists a common meeting point in time T then there must exist a meeting point for T’ > T as there will be a more common area for T’.
• Hence, the idea is to check for the existence of a common meeting point i.e., check for the existence of a common area of N circles.
• Below is the illustration for all possible combinations of 3 circles and each has a non-zero area of intersection then all the 3 circles have a common area of intersection as:
  • In Fig 1, 2, 3, 4, 5 the sum of the radius of two of the three circles is less than the distance between them which means there is no common area among the three circles.
  • In Fig 6, 7, 8, 9 one or two circle(s) lie within another circle.
  • In Fig 10, 11, 12, 13 find the intersection between two circles and check whether those intersection points lie within the other circle or not.

From the above observations, the idea is to use Binary Search for finding the minimum time with a unique point of intersection for all the given N points. Follow the steps below to solve the given problem:

• Declare a function, say intersection(X1, Y1, R1, X2, Y2, R2, X3, Y3, R3) that takes the coordinates and radius of the circles as a parameter and perform the following steps:
  • If the sum of radiuses of any two circles is less than the sum of distances between their centers, then return false as there doesn’t exist any such common area between them.
  • Now, check if any two circles have a common area or not.  If found to be true, then return true. Otherwise, return false.
• Declare a function, say isGood(mid, N, X, Y, V) that takes the current possible time, coordinates of the N points, and velocity of each point as a parameter and perform the following steps:
  • If the value of N is at least 3, then check for all possible combinations of three circles to get the common area by calling the function intersection(X1, Y1, R1, X2, Y2, R2, X3, Y3, R3). If there exists any such combination that doesn’t have any common area then return false. Otherwise, return true.
  • If the value of N is 2, then check if the two circles have a common area or not.  If found to be true, then return true. Otherwise, return false.
• Initialize two variables, say tl as 0.0 and tu as 100000.0 as the minimum and maximum time required to get the unique point of intersection respectively.
• Now, iterate over the range [0, 1000] to get the better precision of the resultant time and perform a binary search by following the below steps:
  • Find the middle value, say mid as (tl  +  tu)/2.0.
  • Now check if the above time mid has at least one common area of intersections or not by calling the function isGood(mid, N, X, Y, V). If found to be true, then update tu as mid. Otherwise, update tl as t.
• After completing the above steps, print the value of tu as the resultant minimum time for all the given N points intersecting at one point.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include<bits/stdc++.h>
using namespace std;

// Function to check for common area
// between three circles
bool intersection(int X1, int Y1, double R1,
                  int X2, int Y2, double R2,
                  int X3, int Y3, double R3)
{

  // Find the distance between the
  // centers of circle 1 & circle 2
  double d12 = sqrt((X1 - X2) * (X1 - X2)
                    + (Y1 - Y2) * (Y1 - Y2));

  // Find the distance between the
  // centers of circle 1 & circle 3
  double d13 = sqrt((X1 - X3) * (X1 - X3)
                    + (Y1 - Y3) * (Y1 - Y3));

  // Find the distance between the
  // centers of circle 2 & circle 3
  double d23 = sqrt((X2 - X3) * (X2 - X3)
                    + (Y2 - Y3) * (Y2 - Y3));

  // If sum of radius is less than
  // the distance between their
  // centers then false
  if ((R1 + R2 < d12) || (R1 + R3 < d13)
      || (R2 + R3 < d23)) {

    return false;
  }
  else {

    // If circle 1 lies within
    // circle 2 or if circle
    // 2 lies within circle 1
    if (abs(R1 - R2) >= d12) {

      // If circle 1 lies
      // within circle 2
      if (R1 < R2) {

        // Check whether R1
        // (common area between
        // R1 and R2) and
        // R3 intersect
        return R1 + R3 >= d13;
      }
      else {

        // Check whether R2
        //(common area between
        // R1 and R2) and
        // R3 intersect
        return R2 + R3 >= d23;
      }
    }

    // If circle 1 lies within
    // circle 3 or if circle
    // 3 lies within circle 1
    else if (abs(R1 - R3) >= d13) {

      // If circle 1 lies
      // within circle 3
      if (R1 < R3) {

        // Check whether R1
        // (common area between
        // R1 and R3) and
        // R2 intersect
        return R1 + R2 >= d12;
      }
      else {

        // Check whether R3
        //(common area between
        // R1 and R3) and
        // R2 intersect
        return R2 + R3 >= d23;
      }
    }

    // If circle 2 lies within
    // circle 3 or if circle
    // 3 lies within circle 2
    else if (abs(R2 - R3) >= d23) {

      // If circle 2
      // lies within circle 3
      if (R2 < R3) {

        // Check whether R2
        // (common area between
        // R2 and R3) and
        // R1 intersect
        return R1 + R2 >= d12;
      }
      else {

        // Check whether R3
        // (common area between
        // R2 and R3) and
        // R1 intersect
        return R1 + R3 >= d13;
      }
    }
    else
    {

      double x121, y121, x122,
      y122, x131, y131,
      x132, y132, x231,
      y231, x232, y232,
      a, b;

      // Find the point of
      // intersection for
      // circle 1 & circle 2
      a = (R1 * R1 - R2 * R2)
        / (2 * d12 * d12);

      b = sqrt(2 * (R1 * R1 + R2 * R2)
               / (d12 * d12)
               - (R1 * R1 - R2 * R2)
               * (R1 * R1 - R2 * R2)
               / (pow(d12, 4))
               - 1)
        / 2;

      // First point of
      // intersection (x121, y121)
      x121 = (X1 + X2) / 2.0 + a * (X2 - X1)
        + b * (Y2 - Y1);
      y121 = (Y1 + Y2) / 2.0 + a * (Y2 - Y1)
        + b * (X1 - X2);

      // Check whether the point
      // of intersection lies
      // within circle 3 or not
      if (R3 >= sqrt(
        (x121 - X3) * (x121 - X3)
        + (y121 - Y3) * (y121 - Y3))) {

        return true;
      }

      // Second point of
      // intersection(x122, y122)
      x122 = (X1 + X2) / 2.0 + a * (X2 - X1)
        - b * (Y2 - Y1);

      y122 = (Y1 + Y2) / 2.0 + a * (Y2 - Y1)
        - b * (X1 - X2);

      // Check whether the point
      // of intersection lies
      // within circle 3 or not
      if (R3 >= sqrt(
        (x122 - X3) * (x122 - X3)
        + (y122 - Y3) * (y122 - Y3))) {

        return true;
      }

      // Find the point of
      // intersection for
      // circle 1 & circle 3
      a = (R1 * R1 - R3 * R3) / (2 * d13 * d13);
      b = sqrt(2 * (R1 * R1 + R3 * R3)
               / (d13 * d13)
               - (R1 * R1 - R3 * R3)
               * (R1 * R1 - R3 * R3)
               / (pow(d13, 4))
               - 1)
        / 2;

      // First point of
      // intersection(x131, y131)
      x131 = (X1 + X3) / 2.0 + a * (X3 - X1)
        + b * (Y3 - Y1);
      y131 = (Y1 + Y3) / 2.0 + a * (Y3 - Y1)
        + b * (X1 - X3);

      // Check whether the point
      // of intersection lies
      // within circle 2 or not
      if (R2 >= sqrt(
        (x131 - X2) * (x131 - X2)
        + (y131 - Y2) * (y131 - Y2))) {

        return true;
      }

      // Second point of
      // intersection(x132, y132)
      x132 = (X1 + X3) / 2.0 + a * (X3 - X1)
        - b * (Y3 - Y1);
      y132 = (Y1 + Y3) / 2.0 + a * (Y3 - Y1)
        - b * (X1 - X3);

      // Check whether the point
      // of intersection lies
      // within circle 2 or not
      if (R2 >= sqrt(
        (x132 - X2) * (x132 - X2)
        + (y132 - Y2) * (y132 - Y2))) {

        return true;
      }

      // Find the point of
      // intersection for
      // circle 2 & circle 3
      a = (R2 * R2 - R3 * R3) / (2 * d23 * d23);
      b = sqrt(2 * (R2 * R2 + R3 * R3)
               / (d23 * d23)
               - (R2 * R2 - R3 * R3)
               * (R2 * R2 - R3 * R3)
               / (pow(d23, 4))
               - 1)
        / 2;

      // First point of
      // intersection(x231, y231)
      x231 = (X2 + X3) / 2.0 + a * (X3 - X2)
        + b * (Y3 - Y2);
      y231 = (Y2 + Y3) / 2.0 + a * (Y3 - Y2)
        + b * (X2 - X3);

      // Check whether the point
      // of intersection lies
      // within circle 1 or not
      if (R1 >= sqrt(
        (x231 - X1) * (x231 - X1)
        + (y231 - Y1) * (y231 - Y1))) {
        return true;
      }

      // Second point of
      // intersection(x232, y232)
      x232 = (X2 + X3) / 2.0 + a * (X3 - X2)
        - b * (Y3 - Y2);
      y232 = (Y2 + Y3) / 2.0 + a * (Y3 - Y2)
        - b * (X2 - X3);

      // Check whether the point
      // of intersection lies
      // within circle 1 or not
      return R1 >= sqrt(
        (x232 - X1) * (x232 - X1)
        + (y232 - Y1) * (y232 - Y1));
    }
  }
}

// Function to check if there is
// a common area between N
// circles
bool isGood(double t, int N, int X[],
            int Y[], int V[])
{
  if (N >= 3) {

    // Check for a common area
    // for all combination
    // of 3 circles
    for (int i = 0; i < N; i++) {
      for (int j = i + 1; j < N; j++) {
        for (int k = j + 1; k < N; k++) {

          // t * V give the
          // radius of the circle
          if (intersection(
            X[i], Y[i], t * V[i], X[j],
            Y[j], t * V[j], X[k], Y[k],
            t * V[k]) == false)
            return false;
        }
      }
    }

    return true;
  }

  // For N = 2
  else {

    //(x2 - x1) ^ 2 + (y2 - y1) ^ 2 <=
    //(r1 + r2) ^ 2 for 2 circles
    // to intersect
    return sqrt((X[0] - X[1])
                * (X[0] - X[1])
                + (Y[0] - Y[1])
                * (Y[0] - Y[1]))
      <= t * (V[0] + V[1]);
  }
}

// Function to find minimum time
void binarySearch(int N, int X[], int Y[], int V[])
{

  // Time depends on the area
  // of the 2D plane
  // Area =(Max(X) - Min(X))*
  // (Max(Y) - Min(Y))
  double tl = 0.0, tu = 100000.0, t;

  // Number of iteration
  // depends on the precision
  for (int i = 0; i < 1000; i++) {

    t = (tl + tu) / 2.0;

    // If there is a common area
    // between N circles
    // for time t
    if (isGood(t, N, X, Y, V)) {
      tu = t;
    }

    // If there is no common area
    // between N circles
    // for time t
    else {
      tl = t;
    }
  }

  // Print the minimum time
  cout << fixed << setprecision(16) << tu << endl;
}

// Driver Code
int main()
{
  int N = 4;
  int X[] = { 1, -3, -1, 2 };
  int Y[] = { 2, 4, -2, -2 };
  int V[] = { 3, 2, 4, 5 };

  // Function Call
  binarySearch(N, X, Y, V);
}

// This code is contributed by ipg2016107.

Java

// Java program for the above approach

class GFG {

    // Function to check for common area
    // between three circles
    public static boolean
    intersection(int X1, int Y1, double R1,
                 int X2, int Y2, double R2,
                 int X3, int Y3, double R3)
    {

        // Find the distance between the
        // centers of circle 1 & circle 2
        double d12 = Math.sqrt((X1 - X2) * (X1 - X2)
                               + (Y1 - Y2) * (Y1 - Y2));

        // Find the distance between the
        // centers of circle 1 & circle 3
        double d13 = Math.sqrt((X1 - X3) * (X1 - X3)
                               + (Y1 - Y3) * (Y1 - Y3));

        // Find the distance between the
        // centers of circle 2 & circle 3
        double d23 = Math.sqrt((X2 - X3) * (X2 - X3)
                               + (Y2 - Y3) * (Y2 - Y3));

        // If sum of radius is less than
        // the distance between their
        // centers then false
        if ((R1 + R2 < d12) || (R1 + R3 < d13)
            || (R2 + R3 < d23)) {

            return false;
        }
        else {

            // If circle 1 lies within
            // circle 2 or if circle
            // 2 lies within circle 1
            if (Math.abs(R1 - R2) >= d12) {

                // If circle 1 lies
                // within circle 2
                if (R1 < R2) {

                    // Check whether R1
                    // (common area between
                    // R1 and R2) and
                    // R3 intersect
                    return R1 + R3 >= d13;
                }
                else {

                    // Check whether R2
                    //(common area between
                    // R1 and R2) and
                    // R3 intersect
                    return R2 + R3 >= d23;
                }
            }

            // If circle 1 lies within
            // circle 3 or if circle
            // 3 lies within circle 1
            else if (Math.abs(R1 - R3) >= d13) {

                // If circle 1 lies
                // within circle 3
                if (R1 < R3) {

                    // Check whether R1
                    // (common area between
                    // R1 and R3) and
                    // R2 intersect
                    return R1 + R2 >= d12;
                }
                else {

                    // Check whether R3
                    //(common area between
                    // R1 and R3) and
                    // R2 intersect
                    return R2 + R3 >= d23;
                }
            }

            // If circle 2 lies within
            // circle 3 or if circle
            // 3 lies within circle 2
            else if (Math.abs(R2 - R3) >= d23) {

                // If circle 2
                // lies within circle 3
                if (R2 < R3) {

                    // Check whether R2
                    // (common area between
                    // R2 and R3) and
                    // R1 intersect
                    return R1 + R2 >= d12;
                }
                else {

                    // Check whether R3
                    // (common area between
                    // R2 and R3) and
                    // R1 intersect
                    return R1 + R3 >= d13;
                }
            }
            else {

                double x121, y121, x122,
                    y122, x131, y131,
                    x132, y132, x231,
                    y231, x232, y232,
                    a, b;

                // Find the point of
                // intersection for
                // circle 1 & circle 2
                a = (R1 * R1 - R2 * R2)
                    / (2 * d12 * d12);

                b = Math.sqrt(2 * (R1 * R1 + R2 * R2)
                                  / (d12 * d12)
                              - (R1 * R1 - R2 * R2)
                                    * (R1 * R1 - R2 * R2)
                                    / (Math.pow(d12, 4))
                              - 1)
                    / 2;

                // First point of
                // intersection (x121, y121)
                x121 = (X1 + X2) / 2.0 + a * (X2 - X1)
                       + b * (Y2 - Y1);
                y121 = (Y1 + Y2) / 2.0 + a * (Y2 - Y1)
                       + b * (X1 - X2);

                // Check whether the point
                // of intersection lies
                // within circle 3 or not
                if (R3 >= Math.sqrt(
                              (x121 - X3) * (x121 - X3)
                              + (y121 - Y3) * (y121 - Y3))) {

                    return true;
                }

                // Second point of
                // intersection(x122, y122)
                x122 = (X1 + X2) / 2.0 + a * (X2 - X1)
                       - b * (Y2 - Y1);

                y122 = (Y1 + Y2) / 2.0 + a * (Y2 - Y1)
                       - b * (X1 - X2);

                // Check whether the point
                // of intersection lies
                // within circle 3 or not
                if (R3 >= Math.sqrt(
                              (x122 - X3) * (x122 - X3)
                              + (y122 - Y3) * (y122 - Y3))) {

                    return true;
                }

                // Find the point of
                // intersection for
                // circle 1 & circle 3
                a = (R1 * R1 - R3 * R3) / (2 * d13 * d13);
                b = Math.sqrt(2 * (R1 * R1 + R3 * R3)
                                  / (d13 * d13)
                              - (R1 * R1 - R3 * R3)
                                    * (R1 * R1 - R3 * R3)
                                    / (Math.pow(d13, 4))
                              - 1)
                    / 2;

                // First point of
                // intersection(x131, y131)
                x131 = (X1 + X3) / 2.0 + a * (X3 - X1)
                       + b * (Y3 - Y1);
                y131 = (Y1 + Y3) / 2.0 + a * (Y3 - Y1)
                       + b * (X1 - X3);

                // Check whether the point
                // of intersection lies
                // within circle 2 or not
                if (R2 >= Math.sqrt(
                              (x131 - X2) * (x131 - X2)
                              + (y131 - Y2) * (y131 - Y2))) {

                    return true;
                }

                // Second point of
                // intersection(x132, y132)
                x132 = (X1 + X3) / 2.0 + a * (X3 - X1)
                       - b * (Y3 - Y1);
                y132 = (Y1 + Y3) / 2.0 + a * (Y3 - Y1)
                       - b * (X1 - X3);

                // Check whether the point
                // of intersection lies
                // within circle 2 or not
                if (R2 >= Math.sqrt(
                              (x132 - X2) * (x132 - X2)
                              + (y132 - Y2) * (y132 - Y2))) {

                    return true;
                }

                // Find the point of
                // intersection for
                // circle 2 & circle 3
                a = (R2 * R2 - R3 * R3) / (2 * d23 * d23);
                b = Math.sqrt(2 * (R2 * R2 + R3 * R3)
                                  / (d23 * d23)
                              - (R2 * R2 - R3 * R3)
                                    * (R2 * R2 - R3 * R3)
                                    / (Math.pow(d23, 4))
                              - 1)
                    / 2;

                // First point of
                // intersection(x231, y231)
                x231 = (X2 + X3) / 2.0 + a * (X3 - X2)
                       + b * (Y3 - Y2);
                y231 = (Y2 + Y3) / 2.0 + a * (Y3 - Y2)
                       + b * (X2 - X3);

                // Check whether the point
                // of intersection lies
                // within circle 1 or not
                if (R1 >= Math.sqrt(
                              (x231 - X1) * (x231 - X1)
                              + (y231 - Y1) * (y231 - Y1))) {
                    return true;
                }

                // Second point of
                // intersection(x232, y232)
                x232 = (X2 + X3) / 2.0 + a * (X3 - X2)
                       - b * (Y3 - Y2);
                y232 = (Y2 + Y3) / 2.0 + a * (Y3 - Y2)
                       - b * (X2 - X3);

                // Check whether the point
                // of intersection lies
                // within circle 1 or not
                return R1 >= Math.sqrt(
                                 (x232 - X1) * (x232 - X1)
                                 + (y232 - Y1) * (y232 - Y1));
            }
        }
    }

    // Function to check if there is
    // a common area between N
    // circles
    public static boolean isGood(double t, int N, int[] X,
                                 int[] Y, int[] V)
    {
        if (N >= 3) {

            // Check for a common area
            // for all combination
            // of 3 circles
            for (int i = 0; i < N; i++) {
                for (int j = i + 1; j < N; j++) {
                    for (int k = j + 1; k < N; k++) {

                        // t * V give the
                        // radius of the circle
                        if (!intersection(
                                X[i], Y[i], t * V[i], X[j],
                                Y[j], t * V[j], X[k], Y[k],
                                t * V[k]))
                            return false;
                    }
                }
            }

            return true;
        }

        // For N = 2
        else {

            //(x2 - x1) ^ 2 + (y2 - y1) ^ 2 <=
            //(r1 + r2) ^ 2 for 2 circles
            // to intersect
            return Math.sqrt((X[0] - X[1])
                                 * (X[0] - X[1])
                             + (Y[0] - Y[1])
                                   * (Y[0] - Y[1]))
                <= t * (V[0] + V[1]);
        }
    }

    // Function to find minimum time
    public static void binarySearch(int N, int[] X,
                                    int[] Y, int[] V)
    {
        // Time depends on the area
        // of the 2D plane
        // Area =(Max(X) - Min(X))*
        // (Max(Y) - Min(Y))
        double tl = 0.0, tu = 100000.0, t;

        // Number of iteration
        // depends on the precision
        for (int i = 0; i < 1000; i++) {

            t = (tl + tu) / 2.0;

            // If there is a common area
            // between N circles
            // for time t
            if (isGood(t, N, X, Y, V)) {
                tu = t;
            }

            // If there is no common area
            // between N circles
            // for time t
            else {
                tl = t;
            }
        }

        // Print the minimum time
        System.out.println(tu);
    }

    // Driver Code
    public static void main(String[] args)
    {
        int N = 4;
        int[] X = { 1, -3, -1, 2 };
        int[] Y = { 2, 4, -2, -2 };
        int[] V = { 3, 2, 4, 5 };

        // Function Call
        binarySearch(N, X, Y, V);
    }
}

Python3

# Python program for the above approach
import math

 # Function to check for common area
 # between three circles
def intersection(X1, Y1, R1, X2, Y2, R2, X3, Y3, R3):

    # Find the distance between the
    # centers of circle 1 & circle 2
    d12 = math.sqrt((X1 - X2) * (X1 - X2) + (Y1 - Y2) * (Y1 - Y2))

    # Find the distance between the
    # centers of circle 1 & circle 3
    d13 = math.sqrt((X1 - X3) * (X1 - X3) + (Y1 - Y3) * (Y1 - Y3))

    # Find the distance between the
    # centers of circle 2 & circle 3
    d23 = math.sqrt((X2 - X3) * (X2 - X3) + (Y2 - Y3) * (Y2 - Y3))

    # If sum of radius is less than
    # the distance between their
    # centers then false
    if R1 + R2 < d12 or R1 + R3 < d13 or R2 + R3 < d23:
        return False
    else:

        # If circle 1 lies within
        # circle 2 or if circle
        # 2 lies within circle 1
        if abs(R1 - R2) >= d12:

            # If circle 1 lies
            # within circle 2
            if R1 < R2:

                 # Check whether R1
                 # (common area between
                 # R1 and R2) and
                 # R3 intersect
                return R1 + R3 >= d13
            else:

                # Check whether R2
                # (common area between
                # R1 and R2) and
                # R3 intersect
                return R2 + R3 >= d23

        # If circle 1 lies within
        # circle 3 or if circle
        # 3 lies within circle 1
        elif abs(R1 - R3) >= d13:

            # If circle 1 lies
            # within circle 3
            if (R1 < R3):

                 # Check whether R1
                 # (common area between
                 # R1 and R3) and
                 # R2 intersect
                return R1 + R2 >= d12
            else:

                # Check whether R3
                # (common area between
                # R1 and R3) and
                # R2 intersect
                return R2 + R3 >= d23

        # If circle 2 lies within
        # circle 3 or if circle
        # 3 lies within circle 2
        elif(abs(R2 - R3) >= d23):

            # If circle 2
            # lies within circle 3
            if (R2 < R3):
                # Check whether R2
                # (common area between
                # R2 and R3) and
                # R1 intersect
                return R1 + R2 >= d12
            else:
                # Check whether R3
                # (common area between
                # R2 and R3) and
                # R1 intersect
                return R1 + R3 >= d13
        else:

            # Find the point of
            # intersection for
            # circle 1 & circle 2
            a = (R1 * R1 - R2 * R2) / (2 * d12 * d12)
            b = math.sqrt(2 * (R1 * R1 + R2 * R2) / (d12 * d12) - (R1 * R1 - R2 * R2) * (R1 * R1 - R2 * R2)
/ (pow(d12, 4)) - 1)


            # First point of
            # intersection (x121, y121)
            x121 = (X1 + X2) / 2.0 + a * (X2 - X1) + b * (Y2 - Y1)
            y121 = (Y1 + Y2) / 2.0 + a * (Y2 - Y1)+ b * (X1 - X2)

            # Check whether the point
            # of intersection lies
            # within circle 3 or not
            if (R3 >= math.sqrt((x121 - X3) * (x121 - X3) + (y121 - Y3) * (y121 - Y3))):
                return True

            # Second point of
            # intersection(x122, y122)
            x122 = (X1 + X2) / 2.0 + a * (X2 - X1) - b * (Y2 - Y1)

            y122 = (Y1 + Y2) / 2.0 + a * (Y2 - Y1) - b * (X1 - X2)

            # Check whether the point
            # of intersection lies
            # within circle 3 or not
            if (R3 >= math.sqrt((x122 - X3) * (x122 - X3)+ (y122 - Y3) * (y122 - Y3))):
                return True

            # Find the point of
            # intersection for
            # circle 1 & circle 3
            a = (R1 * R1 - R3 * R3) / (2 * d13 * d13)
            b = math.sqrt(2 * (R1 * R1 + R3 * R3)/ (d13 * d13)- (R1 * R1 - R3 * R3)* (R1 * R1 - R3 * R3)/ (math.pow(d13, 4))- 1)/ 2

            #  First point of
            #  intersection(x131, y131)
            x131 = (X1 + X3) / 2.0 + a * (X3 - X1)
            + b * (Y3 - Y1)
            y131 = (Y1 + Y3) / 2.0 + a * (Y3 - Y1)
            + b * (X1 - X3)

            # Check whether the point
            # of intersection lies
            # within circle 2 or not
            if (R2 >= math.sqrt(
            (x131 - X2) * (x131 - X2)
            + (y131 - Y2) * (y131 - Y2))):
                return True

            # Second point of
            # intersection(x132, y132)
            x132 = (X1 + X3) / 2.0 + a * (X3 - X1)
            - b * (Y3 - Y1)
            y132 = (Y1 + Y3) / 2.0 + a * (Y3 - Y1)
            - b * (X1 - X3)

            # Check whether the point
            # of intersection lies
            # within circle 2 or not
            if (R2 >= math.sqrt(
            (x132 - X2) * (x132 - X2)
            + (y132 - Y2) * (y132 - Y2))):
                return True

            # Find the point of
            # intersection for
            # circle 2 & circle 3
            a = (R2 * R2 - R3 * R3) / (2 * d23 * d23)
            # b = math.sqrt(2 * (R2 * R2 + R3 * R3) / (d23 * d23) - (R2 * R2 - R3 * R3) * (R2 * R2 - R3 * R3) / (pow(d23, 4)) - 1)/2

            # First point of
            # intersection(x231, y231)
            x231 = (X2 + X3) / 2.0 + a * (X3 - X2)
            + b * (Y3 - Y2)
            y231 = (Y2 + Y3) / 2.0 + a * (Y3 - Y2)
            + b * (X2 - X3)

            # Check whether the point
            # of intersection lies
            # within circle 1 or not
            if (R1 >= math.sqrt(
            (x231 - X1) * (x231 - X1)
            + (y231 - Y1) * (y231 - Y1))):
                return True

             # Second point of
            # intersection(x232, y232)
            x232 = (X2 + X3) / 2.0 + a * (X3 - X2)
            - b * (Y3 - Y2)
            y232 = (Y2 + Y3) / 2.0 + a * (Y3 - Y2)
            - b * (X2 - X3)

            # Check whether the point
            # of intersection lies
            # within circle 1 or not
            return R1 >= math.sqrt(
            (x232 - X1) * (x232 - X1)
            + (y232 - Y1) * (y232 - Y1))


# Function to check if there is
# a common area between N
# circles
def isGood(t, N, X, Y, V):
    if N >= 3:

        # Check for a common area
        # for all combination
        # of 3 circles
        for i in range(N):
            for j in range(i+1, N):
                for k in range(j+1, N):

                    # t * V give the
                    # radius of the circle
                    if (intersection(X[i], Y[i], t * V[i], X[j],Y[j], t * V[j], X[k], Y[k],t * V[k])):
                        pass
                    else:
                        return False

        return True
    # For N = 2
    else:

        # (x2 - x1) ^ 2 + (y2 - y1) ^ 2 <=
        # (r1 + r2) ^ 2 for 2 circles
        # to intersect
        return math.sqrt((X[0] - X[1]) * (X[0] - X[1]) + (Y[0] - Y[1]) * (Y[0] - Y[1])) <= t * (V[0] + V[1])

# Function to find minimum time
def binarySearch(N, X, Y, V):
    # Time depends on the area
    # of the 2D plane
    # Area =(Max(X) - Min(X))*
    # (Max(Y) - Min(Y))
    tl = 0.0
    tu = 100000.0

    # Number of iteration
    # depends on the precision
    for i in range(1000):
        t = (tl + tu) / 2.0

        # If there is a common area
        # between N circles
        # for time t
        if (isGood(t, N, X, Y, V)):
            tu = t

        # If there is no common area
        # between N circles
        # for time t
        else:
            tl = t

    # Print the minimum time
    print(tu)

# Driver Code

N = 4
X = [1, -3, -1, 2]
Y = [2, 4, -2, -2]
V = [3, 2, 4, 5]

# Function Call
binarySearch(N, X, Y, V)

# This code is contributed by Gautam goel(gautamgoel962).

C#

// C# program for the above approach
using System;

class GFG{

// Function to check for common area
// between three circles
public static bool intersection(int X1, int Y1, double R1,
                                int X2, int Y2, double R2,
                                int X3, int Y3, double R3)
{

    // Find the distance between the
    // centers of circle 1 & circle 2
    double d12 = Math.Sqrt((X1 - X2) * (X1 - X2) +
                           (Y1 - Y2) * (Y1 - Y2));

    // Find the distance between the
    // centers of circle 1 & circle 3
    double d13 = Math.Sqrt((X1 - X3) * (X1 - X3) +
                           (Y1 - Y3) * (Y1 - Y3));

    // Find the distance between the
    // centers of circle 2 & circle 3
    double d23 = Math.Sqrt((X2 - X3) * (X2 - X3) +
                           (Y2 - Y3) * (Y2 - Y3));

    // If sum of radius is less than
    // the distance between their
    // centers then false
    if ((R1 + R2 < d12) || (R1 + R3 < d13) ||
        (R2 + R3 < d23))
    {
        return false;
    }
    else
    {

        // If circle 1 lies within
        // circle 2 or if circle
        // 2 lies within circle 1
        if (Math.Abs(R1 - R2) >= d12)
        {

            // If circle 1 lies
            // within circle 2
            if (R1 < R2)
            {

                // Check whether R1
                // (common area between
                // R1 and R2) and
                // R3 intersect
                return R1 + R3 >= d13;
            }
            else
            {

                // Check whether R2
                //(common area between
                // R1 and R2) and
                // R3 intersect
                return R2 + R3 >= d23;
            }
        }

        // If circle 1 lies within
        // circle 3 or if circle
        // 3 lies within circle 1
        else if (Math.Abs(R1 - R3) >= d13)
        {

            // If circle 1 lies
            // within circle 3
            if (R1 < R3)
            {

                // Check whether R1
                // (common area between
                // R1 and R3) and
                // R2 intersect
                return R1 + R2 >= d12;
            }
            else
            {

                // Check whether R3
                //(common area between
                // R1 and R3) and
                // R2 intersect
                return R2 + R3 >= d23;
            }
        }

        // If circle 2 lies within
        // circle 3 or if circle
        // 3 lies within circle 2
        else if (Math.Abs(R2 - R3) >= d23)
        {

            // If circle 2
            // lies within circle 3
            if (R2 < R3)
            {

                // Check whether R2
                // (common area between
                // R2 and R3) and
                // R1 intersect
                return R1 + R2 >= d12;
            }
            else
            {

                // Check whether R3
                // (common area between
                // R2 and R3) and
                // R1 intersect
                return R1 + R3 >= d13;
            }
        }
        else
        {

            double x121, y121, x122,
                   y122, x131, y131,
                   x132, y132, x231,
                   y231, x232, y232,
                   a, b;

            // Find the point of
            // intersection for
            // circle 1 & circle 2
            a = (R1 * R1 - R2 * R2) /
                 (2 * d12 * d12);

            b = Math.Sqrt(2 * (R1 * R1 + R2 * R2) /
                (d12 * d12) - (R1 * R1 - R2 * R2) *
                              (R1 * R1 - R2 * R2) /
                         (Math.Pow(d12, 4)) - 1) / 2;

            // First point of
            // intersection (x121, y121)
            x121 = (X1 + X2) / 2.0 + a * (X2 - X1) +
                                     b * (Y2 - Y1);
            y121 = (Y1 + Y2) / 2.0 + a * (Y2 - Y1) +
                                     b * (X1 - X2);

            // Check whether the point
            // of intersection lies
            // within circle 3 or not
            if (R3 >= Math.Sqrt((x121 - X3) * (x121 - X3) +
                                (y121 - Y3) * (y121 - Y3)))
            {
                return true;
            }

            // Second point of
            // intersection(x122, y122)
            x122 = (X1 + X2) / 2.0 + a * (X2 - X1) -
                                     b * (Y2 - Y1);

            y122 = (Y1 + Y2) / 2.0 + a * (Y2 - Y1) -
                                     b * (X1 - X2);

            // Check whether the point
            // of intersection lies
            // within circle 3 or not
            if (R3 >= Math.Sqrt((x122 - X3) * (x122 - X3) +
                                (y122 - Y3) * (y122 - Y3)))
            {
                return true;
            }

            // Find the point of
            // intersection for
            // circle 1 & circle 3
            a = (R1 * R1 - R3 * R3) / (2 * d13 * d13);
            b = Math.Sqrt(2 * (R1 * R1 + R3 * R3) /
                (d13 * d13) - (R1 * R1 - R3 * R3) *
                              (R1 * R1 - R3 * R3) /
                         (Math.Pow(d13, 4)) - 1) / 2;

            // First point of
            // intersection(x131, y131)
            x131 = (X1 + X3) / 2.0 + a * (X3 - X1) +
                                     b * (Y3 - Y1);
            y131 = (Y1 + Y3) / 2.0 + a * (Y3 - Y1) +
                                     b * (X1 - X3);

            // Check whether the point
            // of intersection lies
            // within circle 2 or not
            if (R2 >= Math.Sqrt((x131 - X2) * (x131 - X2) +
                                (y131 - Y2) * (y131 - Y2)))
            {
                return true;
            }

            // Second point of
            // intersection(x132, y132)
            x132 = (X1 + X3) / 2.0 + a * (X3 - X1) -
                                     b * (Y3 - Y1);
            y132 = (Y1 + Y3) / 2.0 + a * (Y3 - Y1) -
                                     b * (X1 - X3);

            // Check whether the point
            // of intersection lies
            // within circle 2 or not
            if (R2 >= Math.Sqrt((x132 - X2) * (x132 - X2) +
                                (y132 - Y2) * (y132 - Y2)))
            {
                return true;
            }

            // Find the point of
            // intersection for
            // circle 2 & circle 3
            a = (R2 * R2 - R3 * R3) / (2 * d23 * d23);
            b = Math.Sqrt(2 * (R2 * R2 + R3 * R3) /
                (d23 * d23) - (R2 * R2 - R3 * R3) *
                              (R2 * R2 - R3 * R3) /
                         (Math.Pow(d23, 4)) - 1) / 2;

            // First point of
            // intersection(x231, y231)
            x231 = (X2 + X3) / 2.0 + a * (X3 - X2) +
                                     b * (Y3 - Y2);
            y231 = (Y2 + Y3) / 2.0 + a * (Y3 - Y2) +
                                     b * (X2 - X3);

            // Check whether the point
            // of intersection lies
            // within circle 1 or not
            if (R1 >= Math.Sqrt((x231 - X1) * (x231 - X1) +
                                (y231 - Y1) * (y231 - Y1)))
            {
                return true;
            }

            // Second point of
            // intersection(x232, y232)
            x232 = (X2 + X3) / 2.0 + a * (X3 - X2) -
                                     b * (Y3 - Y2);
            y232 = (Y2 + Y3) / 2.0 + a * (Y3 - Y2) -
                                     b * (X2 - X3);

            // Check whether the point
            // of intersection lies
            // within circle 1 or not
            return R1 >= Math.Sqrt((x232 - X1) * (x232 - X1) +
                                   (y232 - Y1) * (y232 - Y1));
        }
    }
}

// Function to check if there is
// a common area between N
// circles
public static bool isGood(double t, int N, int[] X,
                                  int[] Y, int[] V)
{
    if (N >= 3)
    {

        // Check for a common area
        // for all combination
        // of 3 circles
        for(int i = 0; i < N; i++)
        {
            for(int j = i + 1; j < N; j++)
            {
                for(int k = j + 1; k < N; k++)
                {

                    // t * V give the
                    // radius of the circle
                    if (!intersection(X[i], Y[i], t * V[i],
                                      X[j], Y[j], t * V[j],
                                      X[k], Y[k], t * V[k]))
                        return false;
                }
            }
        }
        return true;
    }

    // For N = 2
    else
    {

        //(x2 - x1) ^ 2 + (y2 - y1) ^ 2 <=
        //(r1 + r2) ^ 2 for 2 circles
        // to intersect
        return Math.Sqrt((X[0] - X[1]) * (X[0] - X[1]) +
                         (Y[0] - Y[1]) * (Y[0] - Y[1])) <=
                                     t * (V[0] + V[1]);
    }
}

// Function to find minimum time
public static void binarySearch(int N, int[] X,
                                int[] Y, int[] V)
{

    // Time depends on the area
    // of the 2D plane
    // Area =(Max(X) - Min(X))*
    // (Max(Y) - Min(Y))
    double tl = 0.0, tu = 100000.0, t;

    // Number of iteration
    // depends on the precision
    for(int i = 0; i < 1000; i++)
    {
        t = (tl + tu) / 2.0;

        // If there is a common area
        // between N circles
        // for time t
        if (isGood(t, N, X, Y, V))
        {
            tu = t;
        }

        // If there is no common area
        // between N circles
        // for time t
        else
        {
            tl = t;
        }
    }

    // Print the minimum time
    Console.WriteLine(tu);
}

// Driver Code
public static void Main(string[] args)
{
    int N = 4;
    int[] X = { 1, -3, -1, 2 };
    int[] Y = { 2, 4, -2, -2 };
    int[] V = { 3, 2, 4, 5 };

    // Function Call
    binarySearch(N, X, Y, V);
}
}

// This code is contributed by AnkThon

Javascript

<script>
// Javascript program for the above approach


// Function to check for common area
// between three circles
function intersection(X1, Y1, R1, X2, Y2, R2, X3, Y3, R3) {

  // Find the distance between the
  // centers of circle 1 & circle 2
  let d12 = Math.sqrt((X1 - X2) * (X1 - X2)
    + (Y1 - Y2) * (Y1 - Y2));

  // Find the distance between the
  // centers of circle 1 & circle 3
  let d13 = Math.sqrt((X1 - X3) * (X1 - X3)
    + (Y1 - Y3) * (Y1 - Y3));

  // Find the distance between the
  // centers of circle 2 & circle 3
  let d23 = Math.sqrt((X2 - X3) * (X2 - X3)
    + (Y2 - Y3) * (Y2 - Y3));

  // If sum of radius is less than
  // the distance between their
  // centers then false
  if ((R1 + R2 < d12) || (R1 + R3 < d13)
    || (R2 + R3 < d23)) {

    return false;
  }
  else {

    // If circle 1 lies within
    // circle 2 or if circle
    // 2 lies within circle 1
    if (Math.abs(R1 - R2) >= d12) {

      // If circle 1 lies
      // within circle 2
      if (R1 < R2) {

        // Check whether R1
        // (common area between
        // R1 and R2) and
        // R3 intersect
        return R1 + R3 >= d13;
      }
      else {

        // Check whether R2
        //(common area between
        // R1 and R2) and
        // R3 intersect
        return R2 + R3 >= d23;
      }
    }

    // If circle 1 lies within
    // circle 3 or if circle
    // 3 lies within circle 1
    else if (Math.abs(R1 - R3) >= d13) {

      // If circle 1 lies
      // within circle 3
      if (R1 < R3) {

        // Check whether R1
        // (common area between
        // R1 and R3) and
        // R2 intersect
        return R1 + R2 >= d12;
      }
      else {

        // Check whether R3
        //(common area between
        // R1 and R3) and
        // R2 intersect
        return R2 + R3 >= d23;
      }
    }

    // If circle 2 lies within
    // circle 3 or if circle
    // 3 lies within circle 2
    else if (Math.abs(R2 - R3) >= d23) {

      // If circle 2
      // lies within circle 3
      if (R2 < R3) {

        // Check whether R2
        // (common area between
        // R2 and R3) and
        // R1 intersect
        return R1 + R2 >= d12;
      }
      else {

        // Check whether R3
        // (common area between
        // R2 and R3) and
        // R1 intersect
        return R1 + R3 >= d13;
      }
    }
    else {

      let x121, y121, x122,
        y122, x131, y131,
        x132, y132, x231,
        y231, x232, y232,
        a, b;

      // Find the point of
      // intersection for
      // circle 1 & circle 2
      a = (R1 * R1 - R2 * R2)
        / (2 * d12 * d12);

      b = Math.sqrt(2 * (R1 * R1 + R2 * R2)
        / (d12 * d12)
        - (R1 * R1 - R2 * R2)
        * (R1 * R1 - R2 * R2)
        / (Math.pow(d12, 4))
        - 1)
        / 2;

      // First point of
      // intersection (x121, y121)
      x121 = (X1 + X2) / 2.0 + a * (X2 - X1)
        + b * (Y2 - Y1);
      y121 = (Y1 + Y2) / 2.0 + a * (Y2 - Y1)
        + b * (X1 - X2);

      // Check whether the point
      // of intersection lies
      // within circle 3 or not
      if (R3 >= Math.sqrt(
        (x121 - X3) * (x121 - X3)
        + (y121 - Y3) * (y121 - Y3))) {

        return true;
      }

      // Second point of
      // intersection(x122, y122)
      x122 = (X1 + X2) / 2.0 + a * (X2 - X1)
        - b * (Y2 - Y1);

      y122 = (Y1 + Y2) / 2.0 + a * (Y2 - Y1)
        - b * (X1 - X2);

      // Check whether the point
      // of intersection lies
      // within circle 3 or not
      if (R3 >= Math.sqrt(
        (x122 - X3) * (x122 - X3)
        + (y122 - Y3) * (y122 - Y3))) {

        return true;
      }

      // Find the point of
      // intersection for
      // circle 1 & circle 3
      a = (R1 * R1 - R3 * R3) / (2 * d13 * d13);
      b = Math.sqrt(2 * (R1 * R1 + R3 * R3)
        / (d13 * d13)
        - (R1 * R1 - R3 * R3)
        * (R1 * R1 - R3 * R3)
        / (Math.pow(d13, 4))
        - 1)
        / 2;

      // First point of
      // intersection(x131, y131)
      x131 = (X1 + X3) / 2.0 + a * (X3 - X1)
        + b * (Y3 - Y1);
      y131 = (Y1 + Y3) / 2.0 + a * (Y3 - Y1)
        + b * (X1 - X3);

      // Check whether the point
      // of intersection lies
      // within circle 2 or not
      if (R2 >= Math.sqrt(
        (x131 - X2) * (x131 - X2)
        + (y131 - Y2) * (y131 - Y2))) {

        return true;
      }

      // Second point of
      // intersection(x132, y132)
      x132 = (X1 + X3) / 2.0 + a * (X3 - X1)
        - b * (Y3 - Y1);
      y132 = (Y1 + Y3) / 2.0 + a * (Y3 - Y1)
        - b * (X1 - X3);

      // Check whether the point
      // of intersection lies
      // within circle 2 or not
      if (R2 >= Math.sqrt(
        (x132 - X2) * (x132 - X2)
        + (y132 - Y2) * (y132 - Y2))) {

        return true;
      }

      // Find the point of
      // intersection for
      // circle 2 & circle 3
      a = (R2 * R2 - R3 * R3) / (2 * d23 * d23);
      b = Math.sqrt(2 * (R2 * R2 + R3 * R3)
        / (d23 * d23)
        - (R2 * R2 - R3 * R3)
        * (R2 * R2 - R3 * R3)
        / (Math.pow(d23, 4))
        - 1)
        / 2;

      // First point of
      // intersection(x231, y231)
      x231 = (X2 + X3) / 2.0 + a * (X3 - X2)
        + b * (Y3 - Y2);
      y231 = (Y2 + Y3) / 2.0 + a * (Y3 - Y2)
        + b * (X2 - X3);

      // Check whether the point
      // of intersection lies
      // within circle 1 or not
      if (R1 >= Math.sqrt(
        (x231 - X1) * (x231 - X1)
        + (y231 - Y1) * (y231 - Y1))) {
        return true;
      }

      // Second point of
      // intersection(x232, y232)
      x232 = (X2 + X3) / 2.0 + a * (X3 - X2)
        - b * (Y3 - Y2);
      y232 = (Y2 + Y3) / 2.0 + a * (Y3 - Y2)
        - b * (X2 - X3);

      // Check whether the point
      // of intersection lies
      // within circle 1 or not
      return R1 >= Math.sqrt(
        (x232 - X1) * (x232 - X1)
        + (y232 - Y1) * (y232 - Y1));
    }
  }
}

// Function to check if there is
// a common area between N
// circles
function isGood(t, N, X, Y, V) {
  if (N >= 3) {

    // Check for a common area
    // for all combination
    // of 3 circles
    for (let i = 0; i < N; i++) {
      for (let j = i + 1; j < N; j++) {
        for (let k = j + 1; k < N; k++) {

          // t * V give the
          // radius of the circle
          if (!intersection(
            X[i], Y[i], t * V[i], X[j],
            Y[j], t * V[j], X[k], Y[k],
            t * V[k]))
            return false;
        }
      }
    }

    return true;
  }

  // For N = 2
  else {

    //(x2 - x1) ^ 2 + (y2 - y1) ^ 2 <=
    //(r1 + r2) ^ 2 for 2 circles
    // to intersect
    return Math.sqrt((X[0] - X[1])
      * (X[0] - X[1])
      + (Y[0] - Y[1])
      * (Y[0] - Y[1]))
      <= t * (V[0] + V[1]);
  }
}

// Function to find minimum time
function binarySearch(N, X, Y, V) {
  // Time depends on the area
  // of the 2D plane
  // Area =(Max(X) - Min(X))*
  // (Max(Y) - Min(Y))
  let tl = 0.0, tu = 100000.0, t;

  // Number of iteration
  // depends on the precision
  for (let i = 0; i < 1000; i++) {

    t = (tl + tu) / 2.0;

    // If there is a common area
    // between N circles
    // for time t
    if (isGood(t, N, X, Y, V)) {
      tu = t;
    }

    // If there is no common area
    // between N circles
    // for time t
    else {
      tl = t;
    }
  }

  // Print the minimum time
  document.write(tu);
}

// Driver Code

let N = 4;
let X = [1, -3, -1, 2];
let Y = [2, 4, -2, -2];
let V = [3, 2, 4, 5];

// Function Call
binarySearch(N, X, Y, V);

// This code is contributed by saurabh_jaiswal.
</script>

Output:

1.1157499537009508

Time Complexity: O(N³)
Auxiliary Space: O(1)