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Number of common tangents between two circles if their centers and radius is given

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Given two circles with a given radius and centers. The task is to find the number of common tangents between these circles.
Examples: 
 

Input: x1 = -10, y1 = 8, x2 = 14, y2 = -24, r1 = 30, r2 = 10
Output: 3

Input: x1 = 40, y1 = 8, x2 = 14, y2 = 54, r1 = 39, r2 = 51
Output: 2


Approach
 

  • First of all we will check whether the circles touch each other externally, intersect each other or do not touch each other at all.(Please refer here)

  • Then if the circles do not touch each other externally, then obviously they will have 4 common tangents, two direct and two transverse.

  • If the circles touch each other externally, then they will have 3 common tangents, two direct and one transverse. 
    The tangent in between can be thought of as the transverse tangents coinciding together.

  • If the circles intersect each other, then they will have 2 common tangents, both of them will be direct.

    •  

  • If one circle is inside another circle, then they will have only one common tangent

Algorithm:

Step 1: Create the “circle” function, which has six inputs: x1, y1, x2, y2, r1, and r2.
Step 2: Use the following formula to determine the separation between the centers of the two circles: (x1 – x2)^2 + (y1 – y2)^2. Use the formula                    (r1 + r2)^2 to determine the total of the two circles’ radii. 
Step 3: To ascertain the relationship between the two circles, compare the computed distance and the sum of radii:                                                                  a. If the distance is equal to the sum of radii, return 1 as there is only one common tangent.
             b. If the distance is greater than the sum of radii, return -1 as there are four common tangents.
             c. If the distance is less than the sum of radii, return 0 as there are two common tangents.

Below is the implementation of the above approach:

C++

// C++ program to find
// the number of common tangents
// between the two circles
 
#include <bits/stdc++.h>
using namespace std;
 
int circle(int x1, int y1, int x2,
        int y2, int r1, int r2)
{
 
    int distSq = (x1 - x2) * (x1 - x2)
                + (y1 - y2) * (y1 - y2);
 
    int radSumSq = (r1 + r2) * (r1 + r2);
 
    if (distSq == radSumSq)
        return 1;
    else if (distSq > radSumSq)
        return -1;
    else
        return 0;
}
 
// Driver code
int main()
{
    int x1 = -10, y1 = 8;
    int x2 = 14, y2 = -24;
    int r1 = 30, r2 = 10;
    int t = circle(x1, y1, x2,
                y2, r1, r2);
    if (t == 1)
        cout << "There are 3 common tangents"
            << " between the circles.";
    else if (t < 0)
        cout << "There are 4 common tangents"
            << " between the circles.";
    else
        cout << "There are 2 common tangents"
            << " between the circles.";
 
    return 0;
}

                    

Java

// Java program to find
// the number of common tangents
// between the two circles
import java.io.*;
 
class GFG
{
     
static int circle(int x1, int y1, int x2,
        int y2, int r1, int r2)
{
 
    int distSq = (x1 - x2) * (x1 - x2)
                + (y1 - y2) * (y1 - y2);
 
    int radSumSq = (r1 + r2) * (r1 + r2);
 
    if (distSq == radSumSq)
        return 1;
    else if (distSq > radSumSq)
        return -1;
    else
        return 0;
}
 
// Driver code
public static void main (String[] args)
{
 
    int x1 = -10, y1 = 8;
    int x2 = 14, y2 = -24;
    int r1 = 30, r2 = 10;
    int t = circle(x1, y1, x2,
                y2, r1, r2);
    if (t == 1)
        System.out.println ("There are 3 common tangents"+
                    " between the circles.");
    else if (t < 0)
        System.out.println ("There are 4 common tangents"+
            " between the circles.");
    else
        System.out.println ("There are 2 common tangents" +
                " between the circles.");
 
     
}
}
 
// This code is contributed by ajit.

                    

Python3

# Python3 program to find
# the number of common tangents
# between the two circles
 
def circle(x1, y1, x2,y2, r1, r2):
 
 
    distSq = (x1 - x2) * (x1 - x2)+ (y1 - y2) * (y1 - y2)
 
    radSumSq = (r1 + r2) * (r1 + r2)
 
    if (distSq == radSumSq):
        return 1
    elif (distSq > radSumSq):
        return -1
    else:
        return 0
 
 
# Driver code
x1,y1 = -10,8;
x2,y2 = 14,-24;
r1,r2 = 30,10;
 
t = circle(x1, y1, x2,y2, r1, r2);
 
if (t == 1):
    print("There are 3 common tangents between the circles.")
elif (t < 0):
    print("There are 4 common tangents between the circles.")
else:
    print("There are 2 common tangents between the circles.")
 
# This code is contributed by mohit kumar 29

                    

C#

// C# program to find
// the number of common tangents
// between the two circles
using System;
 
class GFG
{
     
static int circle(int x1, int y1, int x2,
        int y2, int r1, int r2)
{
 
    int distSq = (x1 - x2) * (x1 - x2)
                + (y1 - y2) * (y1 - y2);
 
    int radSumSq = (r1 + r2) * (r1 + r2);
 
    if (distSq == radSumSq)
        return 1;
    else if (distSq > radSumSq)
        return -1;
    else
        return 0;
}
 
// Driver code
public static void Main (String []args)
{
 
    int x1 = -10, y1 = 8;
    int x2 = 14, y2 = -24;
    int r1 = 30, r2 = 10;
    int t = circle(x1, y1, x2,
                y2, r1, r2);
    if (t == 1)
        Console.WriteLine ("There are 3 common tangents"+
                    " between the circles.");
    else if (t < 0)
        Console.WriteLine ("There are 4 common tangents"+
            " between the circles.");
    else
        Console.WriteLine ("There are 2 common tangents" +
                " between the circles.");
 
}
}
 
// This code is contributed by Arnab Kundu

                    

PHP

<?php
// PHP program to find
// the number of common tangents
// between the two circles
 
function circle($x1, $y1, $x2,
        $y2, $r1, $r2)
{
 
    $distSq = ($x1 - $x2) * ($x1 - $x2)
                + ($y1 - $y2) * ($y1 - $y2);
 
    $radSumSq = ($r1 + $r2) * ($r1 + $r2);
 
    if ($distSq == $radSumSq)
        return 1;
    else if ($distSq > $radSumSq)
        return -1;
    else
        return 0;
}
 
    // Driver code
    $x1 = -10; $y1 = 8;
    $x2 = 14; $y2 = -24;
    $r1 = 30; $r2 = 10;
    $t = circle($x1, $y1, $x2,
                $y2, $r1, $r2);
    if ($t == 1)
        echo "There are 3 common tangents"
                ," between the circles.";
    else if ($t < 0)
        echo "There are 4 common tangents"
            , " between the circles.";
    else
        echo "There are 2 common tangents"
            , " between the circles.";
             
    // This code is contributed by AnkitRai01
?>

                    

Javascript

<script>
 
// Javascript program to find
// the number of common tangents
// between the two circles
 
function circle(x1, y1, x2,
        y2, r1, r2)
{
 
    var distSq = (x1 - x2) * (x1 - x2)
                + (y1 - y2) * (y1 - y2);
 
    var radSumSq = (r1 + r2) * (r1 + r2);
 
    if (distSq == radSumSq)
        return 1;
    else if (distSq > radSumSq)
        return -1;
    else
        return 0;
}
 
// Driver code
    var x1 = -10, y1 = 8;
    var x2 = 14, y2 = -24;
    var r1 = 30, r2 = 10;
    var t = circle(x1, y1, x2,
                y2, r1, r2);
    if (t == 1)
        document.write(
"There are 3 common tangents between the circles."
);
    else if (t < 0)
        document.write(
"There are 4 common tangents between the circles."
);
    else
        document.write(
"There are 2 common tangents between the circles."
);
 
</script>

                    

Output:

There are 3 common tangents between the circles.

Time Complexity: O(1)

Auxiliary Space: O(1)



Last Updated : 25 Mar, 2023
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