3-D 中两个平行平面之间的距离

给定两个平面P1 ： a1 * x + b1 * y + c1 * z + d1 = 0和P2 ： a2 * x + b2 * y + c2 * z + d2 = 0 。任务是编写一个程序来查找这两个平面之间的距离。

例子 ：

Input: a1 = 1, b1 = 2, c1 = -1, d1 = 1, a2 = 3, b2 = 6, c2 = -3, d2 = -4
Output: Distance is 0.952579344416

Input: a1 = 1, b1 = 2, c1 = -1, d1 = 1, a2 = 1, b2 = 6, c2 = -3, d2 = -4
Output: Planes are not parallel 

方法：考虑两个平面由等式给出：-

P1 : a1 * x + b1 * y + c1 * z + d1 = 0, where a1, b1 and c1, d1 are real constants and 
P2 : a2 * x + b2 * y + c2 * z + d2 = 0, where a2, b2 and c2, d2 are real constants.

两个平面平行的条件是：-

=> a1 / a2 = b1 / b2 = c1 / c2

在任何一个平面中找到一个点，使得从该点到另一个平面的距离就是这两个平面之间的距离。可以使用以下公式计算距离：

Distance = (| a*x1 + b*y1 + c*z1 + d |) / (sqrt( a*a + b*b + c*c))

设平面 P1 中的一个点为 P(x1, y1, z1)，
将 x = y = 0 放入等式 a1 * x + b1 * y + c1 * z + d1 = 0 并找到 z。
=> z = -d1 / c1
现在我们有了 P(0, 0, z) = P(x1, y1, z1) 的坐标。
点 P 到平面 P2 的距离为：-

Distance = (| a2*x1 + b2*y1 + c2*z1 + d2 |) / (sqrt( a2*a2 + b2*b2 + c2*c2)) 
= (| a2*0 + b2*0 + c2*z1 + d2 |) / (sqrt( a2*a2 + b2*b2 + c2*c2)) 
= (| c2*z1 + d2 |) / (sqrt( a2*a2 + b2*b2 + c2*c2)) 
 

下面是上述公式的实现：

C++

// C++ program to find the Distance
// between two parallel Planes in 3 D.
#include 
#include
 
using namespace std;
 
// Function to find distance
void distance(float a1, float b1,
              float c1, float d1,
              float a2, float b2,
              float c2, float d2)
{
    float x1, y1, z1, d;
    if (a1 / a2 == b1 / b2 &&
        b1 / b2 == c1 / c2)
    {
        x1 = y1 = 0;
        z1 = -d1 / c1;
        d = fabs(( c2 * z1 + d2)) /
           (sqrt(a2 * a2 + b2 *
                 b2 + c2 * c2));
        cout << "Perpendicular distance is "
             << d << endl;
    }
    else
        cout << "Planes are not parallel";
    return;
}
 
// Driver Code
int main()
{
    float a1 = 1;
    float b1 = 2;
    float c1 = -1;
    float d1 = 1;
    float a2 = 3;
    float b2 = 6;
    float c2 = -3;
    float d2 = -4;
    distance(a1, b1, c1, d1,
             a2, b2, c2, d2); // Fxn cal
    return 0;
}
 
// This code is contributed
// by Akanksha Rai(Abby_akku)

C

// C program to find the Distance between
// two parallel Planes in 3 D.
  
#include 
#include
  
// Function to find distance
void distance(float a1, float b1, float c1,
            float d1, float a2, float b2,
            float c2, float d2)
{
    float x1,y1,z1,d;
    if (a1 / a2 == b1 / b2 && b1 / b2 == c1 / c2)
    {
        x1 = y1 = 0;
        z1 =-d1 / c1;
        d = fabs(( c2 * z1 + d2)) / (sqrt(a2 * a2 + b2 * b2 + c2 * c2));
        printf("Perpendicular distance is %f\n", d);
    }
    else
        printf("Planes are not parallel");
    return;
}
   
// Driver Code
int main()
{
    float a1 = 1;
    float b1 = 2;
    float c1 = -1;
    float d1 = 1;
    float a2 = 3;
    float b2 = 6;
    float c2 = -3;
    float d2 = -4;
    distance(a1, b1, c1, d1, a2, b2, c2, d2);     // Fxn cal
    return 0;
}
   
// This code is contributed 
// by Amber_Saxena.

Java

// Java program to find the Distance
// between two parallel Planes in 3 D.
import java .io.*;
import java.lang.Math;
 
class GFG
{
     
// Function to find distance
static void distance(float a1, float b1, float c1,
                     float d1, float a2, float b2,
                     float c2, float d2)
{
     
    float x1,y1,z1,d;
    if (a1 / a2 == b1 / b2 &&
        b1 / b2 == c1 / c2)
    {
        x1 = y1 = 0;
        z1 =-d1 / c1;
        d = Math.abs(( c2 * z1 + d2)) /
            (float)(Math.sqrt(a2 * a2 + b2 *
                              b2 + c2 * c2));
        System.out.println("Perpendicular distance is "+ d);
    }
    else
        System.out.println("Planes are not parallel");
}
 
// Driver code
public static void main(String[] args)
{
    float a1 = 1;
    float b1 = 2;
    float c1 = -1;
    float d1 = 1;
    float a2 = 3;
    float b2 = 6;
    float c2 = -3;
    float d2 = -4;
    distance(a1, b1, c1, d1,
             a2, b2, c2, d2);// Fxn cal
}
}
 
// This code is contributed
// by Amber_Saxena.

Python

# Python program to find the Distance between
# two parallel Planes in 3 D.
 
import math
 
# Function to find distance
def distance(a1, b1, c1, d1, a2, b2, c2, d2):
     
    if (a1 / a2 == b1 / b2 and b1 / b2 == c1 / c2):
        x1 = y1 = 0
        z1 =-d1 / c1
        d = abs(( c2 * z1 + d2)) / (math.sqrt(a2 * a2 + b2 * b2 + c2 * c2))
        print("Perpendicular distance is"), d
    else:
        print("Planes are not parallel")
 
# Driver Code
a1 = 1
b1 = 2
c1 = -1
d1 = 1
a2 = 3
b2 = 6
c2 = -3
d2 = -4
distance(a1, b1, c1, d1, a2, b2, c2, d2)     # Fxn cal

C#

// C# program to find the Distance
// between two parallel Planes in 3 D.
using System;
 
class GFG
{
     
// Function to find distance
static void distance(float a1, float b1,
                     float c1, float d1,
                     float a2, float b2,
                     float c2, float d2)
{
    float z1, d;
    if (a1 / a2 == b1 / b2 &&
        b1 / b2 == c1 / c2)
    {
         
        z1 =-d1 / c1;
        d = Math.Abs((c2 * z1 + d2)) /
            (float)(Math.Sqrt(a2 * a2 + b2 *
                              b2 + c2 * c2));
        Console.Write("Perpendicular distance is " + d);
    }
    else
        Console.Write("Planes are not parallel");
}
 
// Driver code
public static void Main()
{
    float a1 = 1;
    float b1 = 2;
    float c1 = -1;
    float d1 = 1;
    float a2 = 3;
    float b2 = 6;
    float c2 = -3;
    float d2 = -4;
    distance(a1, b1, c1, d1,
             a2, b2, c2, d2);// Fxn cal
}
}
 
// This code is contributed
// by ChitraNayal

Javascript

 
// Javascript program to find the Distance
// between two parallel Planes in 3 D.
 
// Function to find distance
function distance(a1, b1, c1,
                     d1,  a2, b2,
                     c2, d2)
{
     
    let x1,y1,z1,d;
    if (a1 / a2 == b1 / b2 &&
        b1 / b2 == c1 / c2)
    {
        x1 = y1 = 0;
        z1 =-d1 / c1;
        d = Math.abs(( c2 * z1 + d2)) /
            (Math.sqrt(a2 * a2 + b2 *
                              b2 + c2 * c2));
        document.write("Perpendicular distance is "+ d);
    }
    else
        document.write("Planes are not parallel");
}
 
// Driver Code
 
     let a1 = 1;
    let b1 = 2;
    let c1 = -1;
    let d1 = 1;
    let a2 = 3;
    let b2 = 6;
    let c2 = -3;
    let d2 = -4;
    distance(a1, b1, c1, d1,
             a2, b2, c2, d2);// Fxn cal
            

输出：

Perpendicular distance is 0.952579344416

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