Angle between two Planes in 3D

Given two planes P1: a1 * x + b1 * y + c1 * z + d1 = 0 and P2: a2 * x + b2 * y + c2 * z + d2 = 0. The task is to find the angle between these two planes in 3D.

Examples:

Input: a1 = 1, b1 = 1, c1 = 2, d1 = 1, a2 = 2, b2 = -1, c2 = 1, d2 = -4
Output: Angle is is 60.0 degree

Input: a1 = 2, b1 = 2, c1 = -3, d1 = -5, a2 = 3, b2 = -3, c2 = 5, d2 = -6
Output: Angle is 123.696598882 degree



Approach: Consider the below equations of given two planes:

P1 : a1 * x + b1 * y + c1 * z + d1 = 0 and,
P2 : a2 * x + b2 * y + c2 * z + d2 = 0, 

where a1, b1, c1, and a2, b2, c2 are direction ratios of normal to the plane P1 and P2.

The angle between two planes is equal to the angle determined by the normal vectors of the planes.
Angle between these planes is given by using the following formula:-
Cos A = \huge\frac{(a_{1}*a_{2}+b_{1}*b_{2}+c_{1}*c_{2})}{(\sqrt{a_{1}*a_{1}+b_{1}*b_{1}+c_{1}*c_{1}})*(\sqrt{a_{2}*a_{2}+b_{2}*b_{2}+c_{2}*c_{2}})}

Using inverse property, we get:
A = Cos^{-1}\left ( \frac{(a_{1}*a_{2}+b_{1}*b_{2}+c_{1}*c_{2})}{(\sqrt{a_{1}*a_{1}+b_{1}*b_{1}+c_{1}*c_{1}})*(\sqrt{a_{2}*a_{2}+b_{2}*b_{2}+c_{2}*c_{2}})} \right )

Below is the implementation of the above formulae:

C++

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// C++ program to find 
// the Angle between
// two Planes in 3 D.
#include <bits/stdc++.h> 
#include<math.h>
  
using namespace std;
  
// Function to find Angle
void distance(float a1, float b1,
              float c1, float a2,
              float b2, float c2)
{
    float d = (a1 * a2 + b1 * 
               b2 + c1 * c2);
    float e1 = sqrt(a1 * a1 + b1 * 
                    b1 + c1 * c1);
    float e2 = sqrt(a2 * a2 + b2 * 
                    b2 + c2 * c2);
    d = d / (e1 * e2);
    float pi = 3.14159;
    float A = (180 / pi) * (acos(d));
    cout << "Angle is " 
         << A << " degree";
}
  
// Driver Code 
int main()
{
    float a1 = 1;
    float b1 = 1;
    float c1 = 2;
    float d1 = 1;
    float a2 = 2;
    float b2 = -1;
    float c2 = 1;
    float d2 = -4;
    distance(a1, b1, c1, 
             a2, b2, c2);
    return 0;
}
  
// This code is contributed 
// by Akanksha Rai(Abby_akku) 

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C

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// C program to find 
// the Angle between
// two Planes in 3 D.
#include<stdio.h>
#include<math.h>
  
// Function to find Angle
void distance(float a1, float b1,
              float c1, float a2,
              float b2, float c2)
{
    float d = (a1 * a2 + b1 * 
               b2 + c1 * c2);
    float e1 = sqrt(a1 * a1 + b1 * 
                    b1 + c1 * c1);
    float e2 = sqrt(a2 * a2 + b2 * 
                    b2 + c2 * c2);
    d = d / (e1 * e2);
    float pi = 3.14159;
    float A = (180 / pi) * (acos(d));
    printf("Angle is %.2f degree", A);
}
  
// Driver Code 
int main()
{
    float a1 = 1;
    float b1 = 1;
    float c1 = 2;
    float d1 = 1;
    float a2 = 2;
    float b2 = -1;
    float c2 = 1;
    float d2 = -4;
    distance(a1, b1, c1, 
             a2, b2, c2);
    return 0;
}
  
// This code is contributed 
// by Amber_Saxena.

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Java

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// Java program to find 
// the Angle between
// two Planes in 3 D. 
import java .io.*; 
import java.lang.Math; 
  
class GFG 
      
// Function to find Angle
static void distance(float a1, float b1,
                     float c1, float a2,
                     float b2, float c2) 
      
    float d = (a1 * a2 + b1 * 
               b2 + c1 * c2);
    float e1 = (float)Math.sqrt(a1 * a1 + b1 * 
                                b1 + c1 * c1);
    float e2 = (float)Math.sqrt(a2 * a2 + b2 * 
                                b2 + c2 * c2);
    d = d / (e1 * e2);
    float pi = (float)3.14159;
    float A = (180 / pi) * (float)(Math.acos(d));
    System.out.println("Angle is "+ A +" degree");
  
// Driver code 
public static void main(String[] args) 
    float a1 = 1;
    float b1 = 1;
    float c1 = 2;
    float d1 = 1;
    float a2 = 2;
    float b2 = -1;
    float c2 = 1;
    float d2 = -4;
    distance(a1, b1, c1, 
             a2, b2, c2);
  
// This code is contributed 
// by Amber_Saxena.

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Python

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# Python program to find the Angle between
# two Planes in 3 D.
  
import math
  
# Function to find Angle
def distance(a1, b1, c1, a2, b2, c2): 
      
    d = ( a1 * a2 + b1 * b2 + c1 * c2 )
    e1 = math.sqrt( a1 * a1 + b1 * b1 + c1 * c1)
    e2 = math.sqrt( a2 * a2 + b2 * b2 + c2 * c2)
    d = d / (e1 * e2)
    A = math.degrees(math.acos(d))
    print("Angle is"), A, ("degree")
  
# Driver Code 
a1 = 1
b1 = 1
c1 = 2
d1 = 1
a2 = 2
b2 = -1
c2 = 1
d2 = -4
distance(a1, b1, c1, a2, b2, c2)  

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C#

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// C# program to find 
// the Angle between
// two Planes in 3 D. 
using System; 
  
class GFG 
      
// Function to find Angle
static void distance(float a1, float b1,
                     float c1, float a2,
                      float b2, float c2) 
      
    float d = (a1 * a2 + b1 * 
               b2 + c1 * c2);
    float e1 = (float)Math.Sqrt(a1 * a1 + b1 * 
                                b1 + c1 * c1);
    float e2 = (float)Math.Sqrt(a2 * a2 + b2 * 
                                b2 + c2 * c2);
    d = d / (e1 * e2);
    float pi = (float)3.14159;
    float A = (180 / pi) * (float)(Math.Acos(d));
    Console.Write("Angle is "+ A +" degree");
  
// Driver code 
public static void Main() 
    float a1 = 1;
    float b1 = 1;
    float c1 = 2;
    float a2 = 2;
    float b2 = -1;
    float c2 = 1;
      
    distance(a1, b1, c1, 
            a2, b2, c2);
  
// This code is contributed 
// by ChitraNayal

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PHP

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<?php 
// PHP program to find the Angle 
// between two Planes in 3 D.
  
// Function to find Angle
function distance($a1, $b1
                  $c1, $a2
                  $b2, $c2
    $d = ($a1 * $a2 + $b1
          $b2 + $c1 * $c2); 
    $e1 = sqrt($a1 * $a1 + $b1
               $b1 + $c1 * $c1); 
    $e2 = sqrt($a2 * $a2 + $b2
               $b2 + $c2 * $c2); 
    $d = $d / ($e1 * $e2); 
    $pi = 3.14159; 
    $A = (180 / $pi) * (acos($d)); 
    echo sprintf("Angle is %.2f degree", $A); 
  
// Driver Code 
$a1 = 1; 
$b1 = 1; 
$c1 = 2; 
$d1 = 1; 
$a2 = 2; 
$b2 = -1; 
$c2 = 1; 
$d2 = -4; 
distance($a1, $b1, $c1
         $a2, $b2, $c2);    
  
// This code is contributed 
// by Amber_Saxena.
?> 

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Output:

Angle is 60.0 degree


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