Program to find equation of a plane passing through 3 points
Given three points (x1, y1, z1), (x2, y2, z2), (x3, y3, z3). The task is to find the equation of the plane passing through these 3 points.
Examples:
Input: x1 = -1 y1 = w z1 = 1
x2 = 0 y2 = -3 z2 = 2
x3 = 1 y3 = 1 z3 = -4
Output: equation of plane is 26 x + 7 y + 9 z + 3 = 0.
Input: x1 = 2, y1 = 1, z1 = -1, 1
x2 = 0, y2 = -2, z2 = 0
x3 = 1, y3 = -1, z3 = 2
Output: equation of plane is -7 x + 5 y + 1 z + 10 = 0.
Approach: Let P, Q and R be the three points with coordinates (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) respectively. Then the equation of plane is a * (x – x0) + b * (y – y0) + c * (z – z0) = 0, where a, b, c are direction ratios of normal to the plane and (x0, y0, z0) are co-ordinates of any point(i.e P, Q, or R) passing through the plane. For finding direction ratios of normal to the plane, take any two vectors in plane, let it be vector PQ, vector PR.
=> Vector PQ = (x2 - x1, y2 - y1, z2 - z1) = (a1, b1, c1).
=> Vector PR = (x3 - x1, y3 - y1, z3 - z1) = (a2, b2, c2).
Normal vector to this plane will be vector PQ x vector PR.
=> PQ X PR = (b1 * c2 - b2 * c1) i
+ (a2 * c1 - a1 * c2) j
+ (a1 * b2 - b1 *a2) k = ai + bj + ck.
Direction ratios of normal vector will be a, b, c. Taking any one point from P, Q, or R, let its co-ordinate be (x0, y0, z0). Then the equation of plane passing through a point(x0, y0, z0) and having direction ratios a, b, c will be
=> a * (x - x0) + b * (y - y0) + c * (z - z0) = 0.
=> a * x - a * x0 + b * y - b * y0 + c * z - c * z0 = 0.
=> a * x + b * y + c * z + (- a * x0 - b * y0 - c * z0) = 0.
Below is the implementation of the above approach:
C++
#include <bits/stdc++.h>
#include<math.h>
#include <iostream>
#include <iomanip>
using namespace std;
void equation_plane( float x1, float y1,
float z1, float x2,
float y2, float z2,
float x3, float y3, float z3)
{
float a1 = x2 - x1;
float b1 = y2 - y1;
float c1 = z2 - z1;
float a2 = x3 - x1;
float b2 = y3 - y1;
float c2 = z3 - z1;
float a = b1 * c2 - b2 * c1;
float b = a2 * c1 - a1 * c2;
float c = a1 * b2 - b1 * a2;
float d = (- a * x1 - b * y1 - c * z1);
std::cout << std::fixed;
std::cout << std::setprecision(2);
cout << "equation of plane is " << a << " x + " << b
<< " y + " << c << " z + " << d << " = 0." ;
}
int main()
{
float x1 =-1;
float y1 = 2;
float z1 = 1;
float x2 = 0;
float y2 =-3;
float z2 = 2;
float x3 = 1;
float y3 = 1;
float z3 =-4;
equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3);
return 0;
}
|
C
#include<stdio.h>
void equation_plane( float x1, float y1,
float z1, float x2,
float y2, float z2,
float x3, float y3, float z3)
{
float a1 = x2 - x1;
float b1 = y2 - y1;
float c1 = z2 - z1;
float a2 = x3 - x1;
float b2 = y3 - y1;
float c2 = z3 - z1;
float a = b1 * c2 - b2 * c1;
float b = a2 * c1 - a1 * c2;
float c = a1 * b2 - b1 * a2;
float d = (- a * x1 - b * y1 - c * z1);
printf ( "equation of plane is %.2f x + %.2f"
" y + %.2f z + %.2f = 0." ,a,b,c,d);
return ;
}
int main()
{
float x1 =-1;
float y1 = 2;
float z1 = 1;
float x2 = 0;
float y2 =-3;
float z2 = 2;
float x3 = 1;
float y3 = 1;
float z3 =-4;
equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3);
return 0;
}
|
Java
import java .io.*;
class GFG
{
static void equation_plane( float x1, float y1,
float z1, float x2,
float y2, float z2,
float x3, float y3,
float z3)
{
float a1 = x2 - x1;
float b1 = y2 - y1;
float c1 = z2 - z1;
float a2 = x3 - x1;
float b2 = y3 - y1;
float c2 = z3 - z1;
float a = b1 * c2 - b2 * c1;
float b = a2 * c1 - a1 * c2;
float c = a1 * b2 - b1 * a2;
float d = (- a * x1 - b * y1 - c * z1);
System.out.println( "equation of plane is " + a +
" x + " + b + " y + " + c +
" z + " + d + " = 0." );
}
public static void main(String[] args)
{
float x1 =- 1 ;
float y1 = 2 ;
float z1 = 1 ;
float x2 = 0 ;
float y2 =- 3 ;
float z2 = 2 ;
float x3 = 1 ;
float y3 = 1 ;
float z3 =- 4 ;
equation_plane(x1, y1, z1, x2,
y2, z2, x3, y3, z3);
}
}
|
Python
def equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3):
a1 = x2 - x1
b1 = y2 - y1
c1 = z2 - z1
a2 = x3 - x1
b2 = y3 - y1
c2 = z3 - z1
a = b1 * c2 - b2 * c1
b = a2 * c1 - a1 * c2
c = a1 * b2 - b1 * a2
d = ( - a * x1 - b * y1 - c * z1)
print "equation of plane is " ,
print a, "x +" ,
print b, "y +" ,
print c, "z +" ,
print d, "= 0."
x1 = - 1
y1 = 2
z1 = 1
x2 = 0
y2 = - 3
z2 = 2
x3 = 1
y3 = 1
z3 = - 4
equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3)
|
C#
using System;
class GFG
{
static void equation_plane( float x1, float y1,
float z1, float x2,
float y2, float z2,
float x3, float y3,
float z3)
{
float a1 = x2 - x1;
float b1 = y2 - y1;
float c1 = z2 - z1;
float a2 = x3 - x1;
float b2 = y3 - y1;
float c2 = z3 - z1;
float a = b1 * c2 - b2 * c1;
float b = a2 * c1 - a1 * c2;
float c = a1 * b2 - b1 * a2;
float d = (- a * x1 - b * y1 - c * z1);
Console.Write( "equation of plane is " + a +
"x + " + b + "y + " + c +
"z + " + d + " = 0" );
}
public static void Main()
{
float x1 =-1;
float y1 = 2;
float z1 = 1;
float x2 = 0;
float y2 =-3;
float z2 = 2;
float x3 = 1;
float y3 = 1;
float z3 =-4;
equation_plane(x1, y1, z1,
x2, y2, z2,
x3, y3, z3);
}
}
|
PHP
<?php
function equation_plane( $x1 , $y1 , $z1 ,
$x2 , $y2 , $z2 ,
$x3 , $y3 , $z3 )
{
$a1 = $x2 - $x1 ;
$b1 = $y2 - $y1 ;
$c1 = $z2 - $z1 ;
$a2 = $x3 - $x1 ;
$b2 = $y3 - $y1 ;
$c2 = $z3 - $z1 ;
$a = $b1 * $c2 - $b2 * $c1 ;
$b = $a2 * $c1 - $a1 * $c2 ;
$c = $a1 * $b2 - $b1 * $a2 ;
$d = (- $a * $x1 - $b * $y1 - $c * $z1 );
echo sprintf( "equation of the plane is %.2fx" .
" + %.2fy + %.2fz + %.2f = 0" ,
$a , $b , $c , $d );
}
$x1 =-1;
$y1 = 2;
$z1 = 1;
$x2 = 0;
$y2 =-3;
$z2 = 2;
$x3 = 1;
$y3 = 1;
$z3 =-4;
equation_plane( $x1 , $y1 , $z1 , $x2 ,
$y2 , $z2 , $x3 , $y3 , $z3 );
?>
|
Javascript
<script>
function equation_plane(x1 , y1 , z1 , x2 ,
y2 , z2 , x3 , y3, z3) {
var a1 = x2 - x1;
var b1 = y2 - y1;
var c1 = z2 - z1;
var a2 = x3 - x1;
var b2 = y3 - y1;
var c2 = z3 - z1;
var a = b1 * c2 - b2 * c1;
var b = a2 * c1 - a1 * c2;
var c = a1 * b2 - b1 * a2;
var d = (-a * x1 - b * y1 - c * z1);
document.write( "equation of plane is " + a + " x + "
+ b + " y + " + c + " z + " + d + " = 0." );
}
var x1 = -1;
var y1 = 2;
var z1 = 1;
var x2 = 0;
var y2 = -3;
var z2 = 2;
var x3 = 1;
var y3 = 1;
var z3 = -4;
equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3);
</script>
|
Output:
equation of plane is 26 x + 7 y + 9 z + 3 = 0.
Time Complexity: O(1)
Auxiliary Space: O(1)
Last Updated :
09 Jun, 2022
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