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Check if a line touches or intersects a circle

Given coordinate of the center and radius > 1 of a circle and the equation of a line. The task is to check if the given line collide with the circle or not. There are three possibilities :

  1. Line intersect the circle.
  2. Line touches the circle.
  3. Line is outside the circle.

Note: General equation of a line is a*x + b*y + c = 0, so only constant a, b, c are given in the input.

Examples :

Input : radius = 5, center = (0, 0), 
        a = 1, b = -1, c = 0.
Output : Intersect

Input :  radius = 5, center = (0, 0), 
         a = 5, b = 0, c = 0.
Output : Touch

Input : radius = 5, center = (0, 0),
         a = 1, b = 1, c = -16.
Output : Outside



The idea is to compare the perpendicular distance between center of circle and line with the radius of the circle.

Algorithm:
1. Find the perpendicular (say p) between center of circle and given line.
2. Compare this distance p with radius r.
……a) If p > r, then line lie outside the circle.
……b) If p = r, then line touches the circle.
……c) If p < r, then line intersect the circle.

How to find the perpendicular distance ?
Distance of a line from a point can be computed using below formula:

frac{ax_{0}+by_{0}+c}{sqrt{a^{2}+b^{2}}}

Refer Wiki for details of above formula.

C++

// CPP program to check if a line touches or 
// intersects or outside a circle.
#include <bits/stdc++.h>
using namespace std;
  
void checkCollision(int a, int b, int c, 
                  int x, int y, int radius)
{
    // Finding the distance of line from center.
    int dist = (abs(a * x + b * y + c)) / 
                     sqrt(a * a + b * b);
  
    // Checking if the distance is less than, 
    // greater than or equal to radius.
    if (radius == dist)
        cout << "Touch" << endl;
    else if (radius > dist)
        cout << "Intersect" << endl;
    else
        cout << "Outside" << endl;
}
  
// Driven Program
int main()
{
    int radius = 5;
    int x = 0, y = 0;
    int a = 3, b = 4, c = 25;
    checkCollision(a, b, c, x, y, radius);
    return 0;
}

Java

// Java program to check if a line touches or 
// intersects or outside a circle.
  
import java.io.*;
  
class GFG {
      
    static void checkCollision(int a, int b, int c, 
                               int x, int y, int radius)
    {
        // Finding the distance of line from center.
        double dist = (Math.abs(a * x + b * y + c)) / 
                        Math.sqrt(a * a + b * b);
      
        // Checking if the distance is less than, 
        // greater than or equal to radius.
        if (radius == dist)
            System.out.println ( "Touch" );
        else if (radius > dist)
            System.out.println( "Intersect") ;
        else
            System.out.println( "Outside") ;
    }
      
    // Driven Program
    public static void main (String[] args) 
    {
        int radius = 5;
        int x = 0, y = 0;
        int a = 3, b = 4, c = 25;
        checkCollision(a, b, c, x, y, radius);
      
    }
}
  
// This article is contributed by vt_m.

Python3

# python program to check if a line
# touches or  intersects or outside
# a circle.
  
import math
  
def checkCollision(a, b, c, x, y, radius):
      
    # Finding the distance of line 
    # from center.
    dist = ((abs(a * x + b * y + c)) /
            math.sqrt(a * a + b * b))
  
    # Checking if the distance is less 
    # than, greater than or equal to radius.
    if (radius == dist):
        print("Touch")
    elif (radius > dist):
        print("Intersect")
    else:
        print("Outside")
  
# Driven Program
radius = 5
x = 0
y = 0
a = 3
b = 4
c = 25
checkCollision(a, b, c, x, y, radius)
  
# This code is contributed by Sam007

C#

// C# program to check if a line touches or 
// intersects or outside a circle.
using System;
  
class GFG {
      
    static void checkCollision(int a, int b, int c, 
                            int x, int y, int radius)
    {
        // Finding the distance of line from center.
        double dist = (Math.Abs(a * x + b * y + c)) / 
                        Math.Sqrt(a * a + b * b);
      
        // Checking if the distance is less than, 
        // greater than or equal to radius.
        if (radius == dist)
            Console.WriteLine ("Touch");
        else if (radius > dist)
            Console.WriteLine("Intersect");
        else
            Console.WriteLine("Outside");
    }
      
    // Driven Program
    public static void Main () 
    {
        int radius = 5;
        int x = 0, y = 0;
        int a = 3, b = 4, c = 25;
          
        checkCollision(a, b, c, x, y, radius);
      
    }
}
  
// This article is contributed by vt_m.

PHP

<?php
// PHP program to check if a line  
// touches or intersects or outside 
// a circle.
  
function checkCollision($a, $b, $c
                        $x, $y, $radius)
{
    // Finding the distance 
    // of line from center.
    $dist = (abs($a * $x + $b * $y + $c)) / 
                   sqrt($a * $a + $b * $b);
  
    // Checking if the distance is less than, 
    // greater than or equal to radius.
    if ($radius == $dist)
        echo "Touch";
    else if ($radius > $dist)
        echo "Intersect";
    else
        echo "Outside" ;
}
  
// Driver Code
$radius = 5;
$x = 0;
$y = 0;
$a = 3;
$b = 4;
$c = 25;
checkCollision($a, $b, $c, $x, $y, $radius);
  
// This code is contributed by Sam007
?>

Output :

Touch

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



This article is attributed to GeeksforGeeks.org

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