En este artículo, se explica cómo encontrar las intersecciones X, Y y Z de un avión. Hay dos formas de encontrar las intersecciones:
Caso 1: Cuando se da la ecuación general del plano.
Ejemplos:
Input: A = -6, B = 5, C = -3, D = 9 Output: 1.5 -1.8 3.0
Input: A = 7, B = 4, C = 5, D = -7 Output: 1.0 1.75 1.4
Acercarse:
El resultado se puede calcular siguiendo los siguientes pasos:
- Convertir ecuación general de un plano Ax + By + Cz + D = 0 a Ax + By + Cz = – D
- Dividir por -D en ambos lados de la ecuación
- Entonces, la ecuación se convierte en x/(-D/A) + y/(-D/B) + z(-D/C) = 1
- La ecuación anterior está en la forma de intersección de un plano. Por lo tanto,
las intersecciones en X serán = -D/A Las
intersecciones en Y serán = -D/B
Las intersecciones en Z serán = -D/C
A continuación se muestra la implementación del enfoque anterior:
C++
// C++ program to find the
// X, Y and Z intercepts of a plane
#include<iostream>
using namespace std;
float * XandYandZintercept(float A, float B,
float C, float D)
{
static float rslt[3];
// For finding the x-intercept
// put y = 0 and z = 0
float x = -D / A ;
// For finding the y-intercept
// put x = 0 and z = 0
float y = -D / B ;
// For finding the z-intercept
// put x = 0 and y = 0
float z = -D / C ;
rslt[0] = x;
rslt[1] = y;
rslt[2] = z;
return rslt;
}
// Driver code
int main ()
{
int A = 2 ;
int B = 5 ;
int C = 7;
int D = 8 ;
float *rslt = XandYandZintercept(A, B, C, D);
for(int i = 0; i < 3 ; i++)
{
cout << rslt[i] << " ";
}
}
// This code is contributed by ANKITKUMAR34
Java
// Java program to find the
// X, Y and Z intercepts of a plane
class GFG
{
static float[] XandYandZintercept(float A,
float B, float C, float D)
{
float rslt[] = new float[3];
// For finding the x-intercept
// put y = 0 and z = 0
float x = -D / A ;
// For finding the y-intercept
// put x = 0 and z = 0
float y = -D / B ;
// For finding the z-intercept
// put x = 0 and y = 0
float z = -D / C ;
rslt[0] = x;
rslt[1] = y;
rslt[2] = z;
return rslt;
}
// Driver code
public static void main (String[] args)
{
int A = 2 ;
int B = 5 ;
int C = 7;
int D = 8 ;
float rslt[] = XandYandZintercept(A, B, C, D);
for(int i = 0; i < 3 ; i++)
{
System.out.print(rslt[i] + " ");
}
}
}
// This code is contributed by AnkitRai01
Python3
# Python program to find the # X, Y and Z intercepts of a plane def XandYandZintercept(A, B, C, D): # For finding the x-intercept # put y = 0 and z = 0 x = -D / A # For finding the y-intercept # put x = 0 and z = 0 y = -D / B # For finding the z-intercept # put x = 0 and y = 0 z = -D / C return [x, y, z] # Driver code A = 2 B = 5 C = 7 D = 8 print(XandYandZintercept(A, B, C, D))
C#
// C# program to find the
// X, Y and Z intercepts of a plane
using System;
class GFG
{
static float[] XandYandZintercept(float A,
float B, float C, float D)
{
float []rslt = new float[3];
// For finding the x-intercept
// put y = 0 and z = 0
float x = -D / A ;
// For finding the y-intercept
// put x = 0 and z = 0
float y = -D / B ;
// For finding the z-intercept
// put x = 0 and y = 0
float z = -D / C ;
rslt[0] = x;
rslt[1] = y;
rslt[2] = z;
return rslt;
}
// Driver code
public static void Main()
{
int A = 2 ;
int B = 5 ;
int C = 7;
int D = 8 ;
float []rslt = XandYandZintercept(A, B, C, D);
for(int i = 0; i < 3 ; i++)
{
Console.Write(rslt[i] + " ");
}
}
}
// This code is contributed by AnkitRai01
Javascript
<script>
// JavaScript program to find the
// X, Y and Z intercepts of a plane
function XandYandZintercept(A, B, C, D) {
// For finding the x-intercept
// put y = 0 and z = 0
var x = -D / A;
// For finding the y-intercept
// put x = 0 and z = 0
var y = -D / B;
// For finding the z-intercept
// put x = 0 and y = 0
var z = -D / C;
return [x, y, z];
}
function equation_plane(p, q, r) {
var x1 = p[0];
var y1 = p[1];
var z1 = p[2];
var x2 = q[0];
var y2 = q[1];
var z2 = q[2];
var x3 = r[0];
var y3 = r[1];
var z3 = r[2];
// For Finding value of A, B, C, D
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;
// Calling the first created function
var [x, y, z] = XandYandZintercept(A, B, C, D);
document.write(x + " " + y + " " + z);
}
// 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;
var p = [x1, y1, z1];
var q = [x2, y2, z2];
var r = [x3, y3, z3];
equation_plane(p, q, r);
</script>
Producción:
[-4.0, -1.6, -1.1428571428571428]
Complejidad de tiempo: O(1)
Espacio Auxiliar: O(1)
Caso 2: Cuando se dan 3 puntos no colineales.
Ejemplos:
Input: A = (3, 17, 2), B = (4, 8, 5), C = (1, 8, 3) Output: 1.5 -1.8 3.0
Input: A = (2, 11, 4), B = (7, 8, 3), C = (9, 18, 23) Output: 1.0 1.75 1.4
Acercarse:
La idea es encontrar la forma cartesiana de la ecuación usando tres puntos.
- Cuando se dan tres puntos (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), el valor determinante de la siguiente array da la forma cartesiana.
- | (x – x1) (y – y1) (z – z1) |
| (x2 – x1) (y2 – y1) (z2 – z1)| = 0
| (x3 – x1) (y3 – y1) (z3 – z1)| - Una vez que se encuentra el determinante anterior, las intersecciones se pueden encontrar utilizando el primer enfoque mencionado.
A continuación se muestra la implementación del enfoque anterior:
C++
// C++ program to find the
// X, Y and Z intercepts of a plane
#include<bits/stdc++.h>
using namespace std;
float * XandYandZintercept( float A, float B,
float C, float D)
{
static float rslt[3];
// For finding the x-intercept
// put y = 0 and z = 0
float x = -D / A;
// For finding the y-intercept
// put x = 0 and z = 0
float y = -D / B ;
// For finding the z-intercept
// put x = 0 and y = 0
float z = -D / C;
rslt[0] = x;
rslt[1] = y;
rslt[2] = z;
return rslt;
}
void equation_plane(int p[], int q[], int r[])
{
int x1 = p[0];
int y1 = p[1];
int z1 = p[2];
int x2 = q[0];
int y2 = q[1];
int z2 = q[2];
int x3 = r[0];
int y3 = r[1];
int z3 = r[2];
// For Finding value of A, B, C, D
int a1 = x2 - x1;
int b1 = y2 - y1;
int c1 = z2 - z1;
int a2 = x3 - x1;
int b2 = y3 - y1;
int c2 = z3 - z1;
int A = b1 * c2 - b2 * c1;
int B = a2 * c1 - a1 * c2;
int C = a1 * b2 - b1 * a2;
int D = (- A * x1 - B * y1 - C * z1);
// Calling the first created function
float * rslt=XandYandZintercept(A, B, C, D);
for(int i = 0; i < 3; i++)
{
cout << rslt[i] << " ";
}
}
// Driver Code
int main()
{
int x1 =-1;
int y1 = 2;
int z1 = 1;
int x2 = 0;
int y2 =-3;
int z2 = 2;
int x3 = 1;
int y3 = 1;
int z3 =-4;
int p[3] = {x1, y1, z1};
int q[3] = {x2, y2, z2};
int r[3] = {x3, y3, z3};
equation_plane(p, q, r);
}
// This code is contributed by chitranayal
Java
// Java program to find the
// X, Y and Z intercepts of a plane
import java.util.*;
class solution{
static double[] XandYandZintercept( double A, double B,
double C, double D)
{
double []rslt = new double[3];
// For finding the x-intercept
// put y = 0 and z = 0
double x = -D / A;
// For finding the y-intercept
// put x = 0 and z = 0
double y = -D / B ;
// For finding the z-intercept
// put x = 0 and y = 0
double z = -D / C;
rslt[0] = x;
rslt[1] = y;
rslt[2] = z;
return rslt;
}
static void equation_plane(int []p, int []q, int []r)
{
int x1 = p[0];
int y1 = p[1];
int z1 = p[2];
int x2 = q[0];
int y2 = q[1];
int z2 = q[2];
int x3 = r[0];
int y3 = r[1];
int z3 = r[2];
// For Finding value of A, B, C, D
int a1 = x2 - x1;
int b1 = y2 - y1;
int c1 = z2 - z1;
int a2 = x3 - x1;
int b2 = y3 - y1;
int c2 = z3 - z1;
int A = b1 * c2 - b2 * c1;
int B = a2 * c1 - a1 * c2;
int C = a1 * b2 - b1 * a2;
int D = (- A * x1 - B * y1 - C * z1);
// Calling the first created function
double []rslt = XandYandZintercept(A, B, C, D);
for(int i = 0; i < 3; i++)
{
System.out.printf(rslt[i]+" ");
}
}
// Driver Code
public static void main(String args[])
{
int x1 =-1;
int y1 = 2;
int z1 = 1;
int x2 = 0;
int y2 =-3;
int z2 = 2;
int x3 = 1;
int y3 = 1;
int z3 =-4;
int []p = {x1, y1, z1};
int []q = {x2, y2, z2};
int []r = {x3, y3, z3};
equation_plane(p, q, r);
}
}
// This code is contributed by Surendra_Gangwar
Python3
# Python program to find the # X, Y and Z intercepts of a plane def XandYandZintercept(A, B, C, D): # For finding the x-intercept # put y = 0 and z = 0 x = -D / A # For finding the y-intercept # put x = 0 and z = 0 y = -D / B # For finding the z-intercept # put x = 0 and y = 0 z = -D / C return [x, y, z] def equation_plane(p, q, r): x1 = p[0] y1 = p[1] z1 = p[2] x2 = q[0] y2 = q[1] z2 = q[2] x3 = r[0] y3 = r[1] z3 = r[2] # For Finding value of A, B, C, D 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) # Calling the first created function print(XandYandZintercept(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))
C#
// C# program to find the
// X, Y and Z intercepts of a plane
using System;
class GFG{
static double[] XandYandZintercept(double A,
double B,
double C,
double D)
{
double[] rslt = new double[3];
// For finding the x-intercept
// put y = 0 and z = 0
double x = -D / A;
// For finding the y-intercept
// put x = 0 and z = 0
double y = -D / B ;
// For finding the z-intercept
// put x = 0 and y = 0
double z = -D / C;
rslt[0] = x;
rslt[1] = y;
rslt[2] = z;
return rslt;
}
static void equation_plane(int[] p,
int[] q,
int[] r)
{
int x1 = p[0];
int y1 = p[1];
int z1 = p[2];
int x2 = q[0];
int y2 = q[1];
int z2 = q[2];
int x3 = r[0];
int y3 = r[1];
int z3 = r[2];
// For Finding value
// of A, B, C, D
int a1 = x2 - x1;
int b1 = y2 - y1;
int c1 = z2 - z1;
int a2 = x3 - x1;
int b2 = y3 - y1;
int c2 = z3 - z1;
int A = (b1 * c2 -
b2 * c1);
int B = (a2 * c1 -
a1 * c2);
int C = (a1 * b2 -
b1 * a2);
int D = (- A * x1 -
B * y1 -
C * z1);
// Calling the first
// created function
double[] rslt = XandYandZintercept(A, B,
C, D);
for(int i = 0; i < 3; i++)
{
Console.Write(rslt[i] + " ");
}
}
// Driver code
static void Main()
{
int x1 =-1;
int y1 = 2;
int z1 = 1;
int x2 = 0;
int y2 =-3;
int z2 = 2;
int x3 = 1;
int y3 = 1;
int z3 =-4;
int[] p = {x1, y1, z1};
int[] q = {x2, y2, z2};
int[] r = {x3, y3, z3};
equation_plane(p, q, r);
}
}
// This code is contributed by divyeshrabadiya07
Javascript
<script>
// Javascript program to find the
// X, Y and Z intercepts of a plane
function XandYandZintercept(A, B, C, D)
{
let rslt = new Array(3);
// For finding the x-intercept
// put y = 0 and z = 0
let x = -D / A;
// For finding the y-intercept
// put x = 0 and z = 0
let y = -D / B ;
// For finding the z-intercept
// put x = 0 and y = 0
let z = -D / C;
rslt[0] = x;
rslt[1] = y;
rslt[2] = z;
return rslt;
}
function equation_plane(p, q, r)
{
let x1 = p[0];
let y1 = p[1];
let z1 = p[2];
let x2 = q[0];
let y2 = q[1];
let z2 = q[2];
let x3 = r[0];
let y3 = r[1];
let z3 = r[2];
// For Finding value of A, B, C, D
let a1 = x2 - x1;
let b1 = y2 - y1;
let c1 = z2 - z1;
let a2 = x3 - x1;
let b2 = y3 - y1;
let c2 = z3 - z1;
let A = b1 * c2 - b2 * c1;
let B = a2 * c1 - a1 * c2;
let C = a1 * b2 - b1 * a2;
let D = (- A * x1 - B * y1 - C * z1);
// Calling the first created function
let rslt = XandYandZintercept(A, B, C, D);
for(let i = 0; i < 3; i++)
{
document.write(rslt[i] + " ");
}
}
// Driver Code
let x1 = -1;
let y1 = 2;
let z1 = 1;
let x2 = 0;
let y2 = -3;
let z2 = 2;
let x3 = 1;
let y3 = 1;
let z3 = -4;
let p = [ x1, y1, z1 ];
let q = [ x2, y2, z2 ];
let r = [ x3, y3, z3 ];
equation_plane(p, q, r);
// This code is contributed by rag2127
</script>
Producción:
[-0.11538461538461539, -0.42857142857142855, -0.3333333333333333]
Complejidad de tiempo: O(1), Complejidad de espacio: O(1)
Publicación traducida automáticamente
Artículo escrito por Atul_kumar_Shrivastava y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA