Dados tres puntos (x1, y1, z1), (x2, y2, z2), (x3, y3, z3). La tarea es encontrar la ecuación del plano que pasa por estos 3 puntos.
Ejemplos:
Entrada: x1 = -1 y1 = w z1 = 1
x2 = 0 y2 = -3 z2 = 2
x3 = 1 y3 = 1 z3 = -4
Salida: la ecuación del plano es 26 x + 7 y + 9 z + 3 = 0
Entrada: x1 = 2, y1 = 1, z1 = -1, 1
x2 = 0, y2 = -2, z2 = 0
x3 = 1, y3 = -1, z3 = 2
Salida: la ecuación del plano es -7 x + 5 y + 1 z + 10 = 0.
Planteamiento: Sean P, Q y R los tres puntos con coordenadas (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) respectivamente. Entonces la ecuación del plano es a * (x – x0) + b * (y – y0) + c * (z – z0) = 0 , donde a, b, c son relaciones de dirección normal al plano y (x0, y0, z0) son las coordenadas de cualquier punto (es decir, P, Q o R) que pasa por el plano. Para encontrar las relaciones de dirección de la normal al plano, tome dos vectores cualesquiera en el plano, sea el vector PQ, el vector PR.
=> Vector PQ = (x2 - x1, y2 - y1, z2 - z1) = (a1, b1, c1). => Vector PR = (x3 - x1, y3 - y1, z3 - z1) = (a2, b2, c2).
El vector normal a este plano será 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.
Las relaciones de dirección del vector normal serán a, b, c. Tomando cualquier punto de P, Q o R, sea su coordenada (x0, y0, z0). Entonces la ecuación del plano que pasa por un punto (x0, y0, z0) y que tiene relaciones de dirección a, b, c será
=> 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.
A continuación se muestra la implementación del enfoque anterior:
C++
// C++ program to find equation of a plane
// passing through given 3 points.
#include <bits/stdc++.h>
#include<math.h>
#include <iostream>
#include <iomanip>
using namespace std;
// Function to find equation of plane.
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.";
}
// Driver Code
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;
}
// This code is contributed
// by Amber_Saxena.
C
// C program to find equation of a plane
// passing through given 3 points.
#include<stdio.h>
// Function to find equation of plane.
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;
}
// Driver Code
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;
}
// This code is contributed
// by Amber_Saxena.
Java
// Java program to find equation
// of a plane passing through
// given 3 points.
import java .io.*;
class GFG
{
// Function to find equation of plane.
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.");
}
// Driver code
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);
}
}
// This code is contributed
// by Amber_Saxena.
Python
# Python program to find equation of a plane # passing through given 3 points. # Function to find equation of plane. 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." # Driver Code 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#
// C# program to find equation
// of a plane passing through
// given 3 points.
using System;
class GFG
{
// Function to find equation of plane.
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");
}
// Driver code
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);
}
}
// This code is contributed
// by ChitraNayal
PHP
<?php
// PHP program to find equation
// of a plane passing through
// given 3 points.
// Function to find equation of plane.
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);
}
// Driver Code
$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);
// This code is contributed
// by Amber_Saxena.
?>
Javascript
<script>
// javascript program to find equation
// of a plane passing through
// given 3 points.
// Function to find equation of plane.
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.");
}
// Driver code
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);
// This code is contributed by Rajput-Ji
</script>
Producción:
equation of plane is 26 x + 7 y + 9 z + 3 = 0.
Complejidad de Tiempo : O(1)
Espacio Auxiliar : O(1)
Publicación traducida automáticamente
Artículo escrito por anamika9988 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA