Check if a point is inside, outside or on the ellipse

Given an ellipse centered at (h, k), with semi-major axis a, semi-minor axis b, both aligned with the Cartesian plane. The task is to determine if the point (x, y) is within the area bounded by the ellipse.

Examples:

Input: h = 0, k = 0, x = 2, y = 1, a = 4, b = 5 
Output: Inside

Input: h = 1, k = 2, x = 200, y = 100, a = 6, b = 5
Output: Outside


Approach: We have to solve the equation of ellipse for the given point (x, y),



(x-h)^2/a^2 + (y-k)^2/b^2 <= 1

If in the inequation, results comes less than 1 then the point lies within, else if it comes exact 1 then the point lies on the ellipse, and if the inequation is unsatisfied then point lies outside of the ellipse.

Below is the implementation of the above approach:

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ Program to check if the point
// lies within the ellipse or not
#include <bits/stdc++.h>
using namespace std;
  
// Function to check the point
int checkpoint(int h, int k, int x, int y, int a, int b)
{
  
    // checking the equation of
    // ellipse with the given point
    int p = (pow((x - h), 2) / pow(a, 2))
            + (pow((y - k), 2) / pow(b, 2));
  
    return p;
}
  
// Driver code
int main()
{
    int h = 0, k = 0, x = 2, y = 1, a = 4, b = 5;
  
    if (checkpoint(h, k, x, y, a, b) > 1)
        cout << "Outside" << endl;
  
    else if (checkpoint(h, k, x, y, a, b) == 1)
        cout << "On the ellipse" << endl;
  
    else
        cout << "Inside" << endl;
  
    return 0;
}

chevron_right






                        filter_none









Java














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















// Java Program to check if the point 


// lies within the ellipse or not 


import java.util.*; 


  


class solution 


{ 


   


// Function to check the point 


static int checkpoint(int h, int k, int x, int y, int a, int b) 


{ 


  


    // checking the equation of 


    // ellipse with the given point 


    int p = ((int)Math.pow((x - h), 2) / (int)Math.pow(a, 2)) 


            + ((int)Math.pow((y - k), 2) / (int)Math.pow(b, 2)); 


  


    return p; 


} 


  


//Driver code 


public static void main(String arr[]) 


{ 


    


    int h = 0, k = 0, x = 2, y = 1, a = 4, b = 5; 


  


    if (checkpoint(h, k, x, y, a, b) > 1) 


       System.out.println("Outside"); 


  


    else if (checkpoint(h, k, x, y, a, b) == 1) 


        System.out.println("On the ellipse"); 


  


    else


       System.out.println("Inside"); 


  


} 


} 


  


//This code is contributed by Surendra_Gangwar 



















                        chevron_right







                        filter_none









Python 3














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















# Python 3 Program to check if  


# the point lies within the  


# ellipse or not 


import math  


  


# Function to check the point 


def checkpoint( h, k, x, y, a, b): 


  


    # checking the equation of 


    # ellipse with the given point 


    p = ((math.pow((x - h), 2) // math.pow(a, 2)) + 


         (math.pow((y - k), 2) // math.pow(b, 2))) 


  


    return p 


  


# Driver code 


if __name__ == "__main__": 


  


    h = 0


    k = 0


    x = 2


    y = 1


    a = 4


    b = 5


  


    if (checkpoint(h, k, x, y, a, b) > 1): 


        print ("Outside") 


  


    elif (checkpoint(h, k, x, y, a, b) == 1): 


        print("On the ellipse") 


  


    else: 


        print("Inside") 


  


# This code is contributed 


# by ChitraNayal 



















                        chevron_right







                        filter_none









C#














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















// C# Program to check if the point 


// lies within the ellipse or not 


using System; 


  


class GFG 


{ 


  


// Function to check the point 


static int checkpoint(int h, int k, int x, 


                      int y, int a, int b) 


{ 


  


    // checking the equation of 


    // ellipse with the given point 


    int p = ((int)Math.Pow((x - h), 2) /  


             (int)Math.Pow(a, 2)) +  


            ((int)Math.Pow((y - k), 2) /  


             (int)Math.Pow(b, 2)); 


  


    return p; 


} 


  


// Driver code 


public static void Main() 


{ 


    int h = 0, k = 0, x = 2, 


        y = 1, a = 4, b = 5; 


  


    if (checkpoint(h, k, x, y, a, b) > 1) 


    Console.WriteLine("Outside"); 


  


    else if (checkpoint(h, k, x, y, a, b) == 1) 


        Console.WriteLine("On the ellipse"); 


  


    else


    Console.WriteLine("Inside"); 


} 


} 


  


// This code is contributed by inder_verma 



















                        chevron_right







                        filter_none









PHP














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















<?php 


// PHP Program to check if the point  


// lies within the ellipse or not  


  


// Function to check the point  


function checkpoint($h, $k, $x,  


                    $y, $a, $b)  


{  


  


    // checking the equation of  


    // ellipse with the given point  


    $p = (pow(($x - $h), 2) / pow($a, 2)) +  


         (pow(($y - $k), 2) / pow($b, 2));  


  


    return $p;  


}  


  


// Driver code  


$h = 0;  


$k = 0; 


$x = 2; 


$y = 1; 


$a = 4; 


$b = 5;  


  


if (checkpoint($h, $k, $x, $y, $a, $b) > 1)  


    echo ("Outside");  


  


else if (checkpoint($h, $k, $x, $y, $a, $b) == 1)  


    echo("On the ellipse" );  


  


else


    echo ("Inside") ;  


      


// This code is contributed by Shivi_Aggarwal 


?> 



















                        chevron_right







                        filter_none











Output:


Inside







GeeksforGeeks has prepared a complete interview preparation course with premium videos, theory, practice problems, TA support and many more features. Please refer Placement 100 for details 





    

        My Personal Notes
        arrow_drop_up
    

    

        
        

            
            
        

    


Recommended Posts:
IshwarGupta
Check out this Author's contributed articles.
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on  the "Improve Article" button below.