Dado un punto (x1, y1, z1) en 3D y los coeficientes de la ecuación del plano, tenemos que encontrar el pie de la perpendicular de un punto en un plano 3D.
Ejemplos:
Entrada: a = 1, b = -2, c = 0, d = 0, x = -1, y = 3, z = 4
Salida: x2 = 0,4 y2 = 0,2 z2 = 4,0
Entrada: a = 2, b = -1, c = 1, d = 3, x = 1, y = 3, z = 4
Salida: x2 = -1,0 y2 = 4,0 z2 = 3,0
Enfoque: La ecuación del plano se da como ax + by + cz + d = 0. Por lo tanto, las relaciones de dirección de la normal al plano son (a, b, c) . Sea N el pie de la perpendicular desde el punto dado al plano dado, entonces, la línea PN tiene razones dirigidas (a, b, c) y pasa por P(x1, y1, z1).
La ecuación de la línea PN será como: –
(x – x1) / a = (y – y1) / b = (z – z1) / c = k
Por lo tanto, cualquier punto en la línea PN se puede escribir como: –
x = un * k + x1
y = segundo * k + y1
z = c * k + z1
dado que N se encuentra tanto en la línea como en el plano, satisfará (ax + by + cz + d = 0).
=>a * (a * k + x1) + b * (b * k + y1) + c * (c * k + z1) + d = 0.
=>a * a * k + a * x1 + b * b * k + b * y1 + c * c * k + c * z1 + d = 0.
=>(a * a + b * b + c * c)k = -a * x1 – b * y1 – c * z1-d.
=>k = (-a * x1 – b * y1 – c * z1 – d) / (a * a + b * b + c * c).
Ahora, las coordenadas del Punto N en términos de k serán:
x2 = a * k + x1
y2 = segundo * k + y1
z2 = c * k + z1
A continuación se muestra la implementación de lo anterior:
C++
// C++ program to find
// foot of perpendicular
// of a point in a 3 D plane.
#include <bits/stdc++.h>
#include <iomanip>
#include <iostream>
#include <math.h>
using namespace std;
// Function to find foot of perpendicular
void foot(float a, float b,
float c, float d,
float x1, float y1,
float z1)
{
float k = (-a * x1 - b * y1 - c * z1 - d) / (float)(a * a + b * b + c * c);
float x2 = a * k + x1;
float y2 = b * k + y1;
float z2 = c * k + z1;
std::cout << std::fixed;
std::cout << std::setprecision(1);
cout << " x2 = " << x2;
cout << " y2 = " << y2;
cout << " z2 = " << z2;
}
// Driver Code
int main()
{
float a = 1;
float b = -2;
float c = 0;
float d = 0;
float x1 = -1;
float y1 = 3;
float z1 = 4;
// function call
foot(a, b, c, d, x1, y1, z1);
return 0;
}
// This code is contributed by Amber_Saxena.
Java
// Java program to find
// foot of perpendicular
// of a point in a 3 D plane.
import java.util.*;
import java.text.*;
class solution
{
// Function to find foot of perpendicular
static void foot(float a, float b,
float c, float d,
float x1, float y1,
float z1)
{
float k = (-a * x1 - b * y1 - c * z1 - d) / (float)(a * a + b * b + c * c);
float x2 = a * k + x1;
float y2 = b * k + y1;
float z2 = c * k + z1;
DecimalFormat form = new DecimalFormat("0.0");
System.out.print(" x2 = " +form.format(x2));
System.out.print(" y2 = " +form.format(y2));
System.out.print( " z2 = " +form.format(z2));
}
// Driver Code
public static void main(String arr[])
{
float a = 1;
float b = -2;
float c = 0;
float d = 0;
float x1 = -1;
float y1 = 3;
float z1 = 4;
// function call
foot(a, b, c, d, x1, y1, z1);
}
}
Python3
# Python3 program to find
# foot of perpendicular
# of a point in a 3 D plane.
# Function to find foot of perpendicular
def foot(a, b, c, d, x1, y1, z1) :
k = (-a * x1 - b * y1 - c * z1 - d) / (a * a + b * b + c * c);
x2 = a * k + x1;
y2 = b * k + y1;
z2 = c * k + z1;
print("x2 =",round(x2,1))
print("y2 =",round(y2,1))
print("z2 =",round(z2,1))
# Driver Code
if __name__ == "__main__" :
a = 1
b = -2
c = 0
d = 0
x1 = -1
y1 = 3
z1 = 4
# function call
foot(a, b, c, d, x1, y1, z1)
# This code is contributed by Ryuga
C#
// C# program to find
// foot of perpendicular
// of a point in a 3 D plane.
using System;
using System.Globalization;
class GFG
{
// Function to find foot of perpendicular
static void foot(float a, float b,
float c, float d,
float x1, float y1,
float z1)
{
float k = (-a * x1 - b * y1 - c * z1 - d) /
(float)(a * a + b * b + c * c);
float x2 = a * k + x1;
float y2 = b * k + y1;
float z2 = c * k + z1;
NumberFormatInfo form = new NumberFormatInfo();
form.NumberDecimalSeparator = ".";
Console.Write(" x2 = " + x2.ToString(form));
Console.Write(" y2 = " + y2.ToString(form));
Console.Write( " z2 = " + z2.ToString(form));
}
// Driver Code
public static void Main(String []arr)
{
float a = 1;
float b = -2;
float c = 0;
float d = 0;
float x1 = -1;
float y1 = 3;
float z1 = 4;
// function call
foot(a, b, c, d, x1, y1, z1);
}
}
// This code contributed by Rajput-Ji
PHP
<?php
// PHP program to find foot of perpendicular
// of a point in a 3 D plane.
// Function to find foot of perpendicular
function foot($a, $b, $c, $d, $x1, $y1, $z1)
{
$k = (-$a * $x1 - $b * $y1 - $c * $z1 - $d) /
($a * $a + $b * $b + $c * $c);
$x2 = $a * $k + $x1;
$y2 = $b * $k + $y1;
$z2 = $c * $k + $z1;
echo "x2 = " . round($x2, 1);
echo " y2 = " . round($y2, 1);
echo " z2 = " . round($z2, 1);
}
// Driver Code
$a = 1; $b = -2; $c = 0; $d = 0;
$x1 = -1; $y1 = 3; $z1 = 4;
// function call
foot($a, $b, $c, $d, $x1, $y1, $z1);
// This code is contributed by ita_c
?>
Javascript
<script>
// JavaScript program to find
// foot of perpendicular
// of a point in a 3 D plane.
// Function to find foot of perpendicular
function foot(a, b, c, d, x1, y1, z1) {
var k = (-a * x1 - b * y1 - c * z1 - d) / (a * a + b * b + c * c);
var x2 = a * k + x1;
var y2 = b * k + y1;
var z2 = c * k + z1;
document.write("x2 =" + x2.toFixed(1) + " ");
document.write("y2 =" + y2.toFixed(1) + " ");
document.write("z2 =" + z2.toFixed(1) + " ");
}
// Driver Code
var a = 1;
var b = -2;
var c = 0;
var d = 0;
var x1 = -1;
var y1 = 3;
var z1 = 4;
// function call
foot(a, b, c, d, x1, y1, z1);
</script>
x2 = 0.4 y2 = 0.2 z2 = 4.0
Publicación traducida automáticamente
Artículo escrito por Amber_Saxena y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA