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

Given an ellipse centered at (h, k), with semi-major axis a, and 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 come to less than 1 then the point lies within, else if it comes to exactly 1 then the point lies on the ellipse, and if the inequation is unsatisfied then the point lies outside of the ellipse.

Below is the implementation of the above approach: 

C++

// 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
double checkpoint(double h, double k, double x, double y, double a, double b)
{

    // checking the equation of
    // ellipse with the given point
    double p = (pow((x - h), 2) / pow(a, 2))
            + (pow((y - k), 2) / pow(b, 2));

    return p;
}

// Driver code
int main()
{
    double 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;
}

Java

// Java Program to check if the point
// lies within the ellipse or not
import java.util.*;

class solution {

    // Function to check the point
    static double checkpoint(double h, double k, double x,
                             double y, double a, double b)
    {

        // checking the equation of
        // ellipse with the given point
        double p = ((double)Math.pow((x - h), 2)
                    / (double)Math.pow(a, 2))
                   + ((double)Math.pow((y - k), 2)
                      / (double)Math.pow(b, 2));

        return p;
    }

    // Driver code
    public static void main(String arr[])
    {

        double 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

Python 3

# 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

C#

// C# Program to check if the point
// lies within the ellipse or not
using System;

class GFG {

    // Function to check the point
    static double checkpoint(double h, double k, double x,
                             double y, double a, double b)
    {

        // checking the equation of
        // ellipse with the given point
        double p = ((double)Math.Pow((x - h), 2)
                    / (double)Math.Pow(a, 2))
                   + ((double)Math.Pow((y - k), 2)
                      / (double)Math.Pow(b, 2));

        return p;
    }

    // Driver code
    public static void Main()
    {
        double 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

Javascript

<script>

// javascript 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
    var p = Math.pow((x - h), 2) / Math.pow(a, 2)
            + Math.pow((y - k), 2) / Math.pow(b, 2);

    return p;
}

// Driver code

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

    if (checkpoint(h, k, x, y, a, b) > 1)
       document.write("Outside");

    else if (checkpoint(h, k, x, y, a, b) == 1)
        document.write("On the ellipse");

    else
       document.write("Inside");

// This code is contributed by 29AjayKumar

</script>

PHP

<?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
?>

Output

Inside

Time complexity: O(1)
Auxiliary Space: O(1)

 Distance Formula Approach

To determine if a point is inside, outside or on an ellipse, we can use the following formula based on the distance between the point and the center of the ellipse:

((x – h) * cos(theta) + (y – k) * sin(theta))^2 / a^2 + ((x – h) * sin(theta) – (y – k) * cos(theta))^2 / b^2 <= 1

where (h, k) is the center of the ellipse, a is the semi-major axis, b is the semi-minor axis, and theta is the angle between the x-axis and the major axis of the ellipse.

If the result of this formula is less than or equal to 1, then the point is inside the ellipse. If it is greater than 1, then the point is outside the ellipse. If it is equal to 1, then the point is on the ellipse.

Below are the steps for the above approach:

1. Calculate the value of theta using the formula: theta = atan2(b * (y – k), a * (x – h))
2. Calculate the value of the distance from the formula mentioned above.
3. Compare the result with 1 to determine the position of the point.

Below is the code for the above approach:

C++

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

// Function to check whether the given point
// lies inside, outside, or on the ellipse
string check_point_position(int h, int k, int x, int y,
                            int a, int b)
{

  // Find the value of angle theta
  double theta = atan2(b * (y - k), a * (x - h));

  // Find the value of distance
  double distance
    = pow((x - h) * cos(theta) + (y - k) * sin(theta),
          2)
    / pow(a, 2)
    + pow((x - h) * sin(theta) - (y - k) * cos(theta),
          2)
    / pow(b, 2);

  if (distance < 1)
    return "Inside";
  else if (distance > 1)
    return "Outside";
  else
    return "On";
}

// Driver Code
int main()
{
  cout << check_point_position(0, 0, 2, 1, 4, 5)
    << endl; // Output: Inside

  cout << check_point_position(1, 2, 200, 100, 6, 5)
    << endl; // Output: Outside

  return 0;
}

Java

import java.util.*;

public class Main {
    // Function to check whether the given point
    // lies inside, outside, or on the ellipse
    public static String checkPointPosition(int h, int k, int x, int y, int a, int b) {
        // Find the value of angle theta
        double theta = Math.atan2(b * (y - k), a * (x - h));

        // Find the value of distance
        double distance = Math.pow((x - h) * Math.cos(theta) +
                                   (y - k) * Math.sin(theta), 2) / Math.pow(a, 2)
                + Math.pow((x - h) * Math.sin(theta) - (y - k) *
                           Math.cos(theta), 2) / Math.pow(b, 2);

        if (distance < 1) {
            return "Inside";
        } else if (distance > 1) {
            return "Outside";
        } else {
            return "On";
        }
    }

    // Driver Code
    public static void main(String[] args) {
        System.out.println(checkPointPosition(0, 0, 2, 1, 4, 5)); // Output: Inside
        System.out.println(checkPointPosition(1, 2, 200, 100, 6, 5)); // Output: Outside
    }
}

Python3

# Python program for the above approach
import math

# Function to check whether the given point
# lies inside, outside, or on the ellipse


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

    # Find the value of angle theta
    theta = math.atan2(b * (y - k), a * (x - h))

    # Find the value of distance
    distance = ((x - h) * math.cos(theta) + (y - k)
                * math.sin(theta))**2 / a**2
    + ((x - h) * math.sin(theta) - (y - k) * math.cos(theta))**2 / b**2

    if distance < 1:
        return "Inside"
    elif distance > 1:
        return "Outside"
    else:
        return "On"


# Driver Code
print(check_point_position(0, 0, 2, 1, 4, 5))  # Output: Inside

print(check_point_position(1, 2, 200, 100, 6, 5))  # Output: Outside

C#

using System;

public class Program {
    // Function to check whether the given point
    // lies inside, outside, or on the ellipse
    public static string
    CheckPointPosition(double h, double k, double x,
                       double y, double a, double b)
    {
        // Find the value of angle theta
        double theta = Math.Atan2(b * (y - k), a * (x - h));

        // Find the value of distance
        double distance
            = Math.Pow((x - h) * Math.Cos(theta)
                           + (y - k) * Math.Sin(theta),
                       2)
                  / Math.Pow(a, 2)
              + Math.Pow((x - h) * Math.Sin(theta)
                             - (y - k) * Math.Cos(theta),
                         2)
                    / Math.Pow(b, 2);

        if (distance < 1) {
            return "Inside";
        }
        else if (distance > 1) {
            return "Outside";
        }
        else {
            return "On";
        }
    }

    public static void Main()
    {
        Console.WriteLine(CheckPointPosition(
            0, 0, 2, 1, 4, 5)); // Output: Inside
        Console.WriteLine(CheckPointPosition(
            1, 2, 200, 100, 6, 5)); // Output: Outside
    }
}

Javascript

function checkPointPosition(h, k, x, y, a, b) {
    // Find the value of angle theta
    let theta = Math.atan2(b * (y - k), a * (x - h));

    // Find the value of distance
    let distance = (Math.pow((x - h) * Math.cos(theta) + (y - k) * Math.sin(theta), 2) / Math.pow(a, 2)) +
                   (Math.pow((x - h) * Math.sin(theta) - (y - k) * Math.cos(theta), 2) / Math.pow(b, 2));

    if (distance < 1) {
        return "Inside";
    } else if (distance > 1) {
        return "Outside";
    } else {
        return "On";
    }
}

// Driver Code
console.log(checkPointPosition(0, 0, 2, 1, 4, 5)); // Output: Inside
console.log(checkPointPosition(1, 2, 200, 100, 6, 5)); // Output: Outside

Output

Inside
Outside

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