通过三维平面的点的镜像
给定三维中的一个点(x，y，z)和平面方程的系数，任务是通过给定的平面找到该点的镜像。 例:
输入: a = 1，b = -2，c = 0，d = 0，x = -1，y = 3，z = 4 输出:x3 = 1.799999999998，y3 =-2.599999999999996，z3 = 4.0 T6】输入: a = 2，b = -1，c = 1，d = 3，x = 4.0
逼近:平面方程为 ax + by + cz + d = 0。因此，法线与平面的方向比是 (a，b，c) 。假设 N 是从给定点到给定平面的垂线的脚，那么，线 PN 具有有向比(a，b，c)，并且它穿过 P(x1，y1，z1)。 PN 线的方程式如下:-
(x - x1) / a = (y - y1) / b = (z - z1) / c = k
因此，PN 线上的任何点都可以写成:-
x = a*k + x1
y = b*k + y1
z = c*k + z1
因为 N 位于直线和平面上，所以将满足(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).
现在，N 点在 k 方面的坐标将是:-
x2 = a * k + x1
y2 = b * k + y1
z2 = c * k + z1
由于点 N(x2，y2，z2)是点 P(x1，y1，z1)和点 Q(x3，y3，z3)的中点，点 Q 的坐标为:-
=> x3 = 2 * x2 - x1
=> y3 = 2 * y2 - y1
=> z3 = 2 * z2 - z1
C++
// C++ program to find
// Mirror of a point
// through a 3 D plane
#include <bits/stdc++.h>
#include<math.h>
#include <iostream>
#include <iomanip>
using namespace std;
// Function to mirror image
void mirror_point(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;
float x3 = 2 * x2 - x1;
float y3 = 2 * y2 - y1;
float z3 = 2 * z2 - z1;
std::cout << std::fixed;
std::cout << std::setprecision(1);
cout << " x3 = " << x3;
cout << " y3 = " << y3;
cout << " z3 = " << z3;
}
// 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
mirror_point(a, b, c, d,
x1, y1, z1);
return 0;
}
// This code is contributed
// by Amber_Saxena.
C
// C program to find
// Mirror of a point
// through a 3 D plane
#include<stdio.h>
// Function to mirror image
void mirror_point(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;
float x3 = 2 * x2 - x1;
float y3 = 2 * y2 - y1;
float z3 = 2 * z2 - z1;
printf("x3 = %.1f ", x3);
printf("y3 = %.1f ", y3);
printf("z3 = %.1f ", z3);
}
// 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
mirror_point(a, b, c, d,
x1, y1, z1);
}
// This code is contributed
// by Amber_Saxena.
Java 语言(一种计算机语言，尤用于创建网站)
// Java program to find
// Mirror of a point
// through a 3 D plane
import java.io.*;
class GFG
{
// Function to mirror image
static void mirror_point(int a, int b,
int c, int d,
int x1, int y1,
int 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;
float x3 = 2 * x2 - x1;
float y3 = 2 * y2 - y1;
float z3 = 2 * z2 - z1;
System.out.print("x3 = " + x3 + " ");
System.out.print("y3 = " + y3 + " ");
System.out.print("z3 = " + z3 + " ");
}
// Driver Code
public static void main(String[] args)
{
int a = 1;
int b = -2;
int c = 0;
int d = 0;
int x1 = -1;
int y1 = 3;
int z1 = 4;
// function call
mirror_point(a, b, c, d,
x1, y1, z1) ;
}
}
// This code is contributed
// by inder_verma
计算机编程语言
# Function to mirror image
def mirror_point(a, b, c, d, x1, y1, z1):
k =(-a * x1-b * y1-c * z1-d)/float((a * a + b * b + c * c))
x2 = a * k + x1
y2 = b * k + y1
z2 = c * k + z1
x3 = 2 * x2-x1
y3 = 2 * y2-y1
z3 = 2 * z2-z1
print "x3 =", x3,
print "y3 =", y3,
print "z3 =", z3,
# Driver Code
a = 1
b = -2
c = 0
d = 0
x1 = -1
y1 = 3
z1 = 4
# function call
mirror_point(a, b, c, d, x1, y1, z1)
C
// C# program to find Mirror of
// a point through a 3 D plane
using System;
class GFG
{
// Function to mirror image
static void mirror_point(int a, int b,
int c, int d,
int x1, int y1,
int 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;
float x3 = 2 * x2 - x1;
float y3 = 2 * y2 - y1;
float z3 = 2 * z2 - z1;
Console.Write("x3 = " + x3 + " ");
Console.Write("y3 = " + y3 + " ");
Console.Write("z3 = " + z3 + " ");
}
// Driver Code
static public void Main ()
{
int a = 1;
int b = -2;
int c = 0;
int d = 0;
int x1 = -1;
int y1 = 3;
int z1 = 4;
// function call
mirror_point(a, b, c, d,
x1, y1, z1);
}
}
// This code is contributed by jit_t
服务器端编程语言（Professional Hypertext Preprocessor 的缩写）
<?php
// PHP program to find Mirror of
// a point through a 3 D plane
// Function to mirror image
function mirror_point($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;
$x3 = 2 * $x2 - $x1;
$y3 = 2 * $y2 - $y1;
$z3 = 2 * $z2 - $z1;
echo sprintf("x3 = %.1f ", $x3);
echo sprintf("y3 = %.1f ", $y3);
echo sprintf("z3 = %.1f ", $z3);
}
// Driver Code
$a = 1;
$b = -2;
$c = 0;
$d = 0;
$x1 = -1;
$y1 = 3;
$z1 = 4;
// function call
mirror_point($a, $b, $c, $d,
$x1, $y1, $z1);
// This code is contributed
// by Amber_Saxena.
?>
java 描述语言
<script>
// JavaScript program to find
// Mirror of a point
// through a 3 D plane
// Function to mirror image
function mirror_point(a, b, c, d, x1, y1, z1)
{
var k = parseFloat(
(-a * x1 - b * y1 -
c * z1 - d) / parseFloat(a * a + b * b +
c * c)
);
var x2 = parseFloat(a * k + x1);
var y2 = parseFloat(b * k + y1);
var z2 = parseFloat(c * k + z1);
var x3 = parseFloat(2 * x2 - x1).toFixed(1);
var y3 = parseFloat(2 * y2 - y1).toFixed(1);
var z3 = parseFloat(2 * z2 - z1).toFixed(1);
document.write(" x3 = " + x3);
document.write(" y3 = " + y3);
document.write(" z3 = " + z3);
}
// 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
mirror_point(a, b, c, d, x1, y1, z1);
</script>
Output:
x3 = 1.8 y3 = -2.6 z3 = 4.0
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