3D 中点与平面之间的距离

给定一个点 (x1, y1, z1) 和一个平面 a * x + b * y + c * z + d = 0。任务是找到该点与给定平面之间的垂直（最短）距离。

例子 ：

Input: x1 = 4, y1 = -4, z1 = 3, a = 2, b = -2, c = 5, d = 8 
Output: Perpendicular distance is 6.78902858227
Input: x1 = 2, y1 = 8, z1 = 5, a = 1, b = -2, c = -2, d = -1 
Output: Perpendicular distance is 8.33333333333 

方法：给定点到平面的垂直距离（即最短距离）是该点到给定平面的垂直距离。设给定点的坐标为 (x1, y1, z1)
和平面方程由方程a * x + b * y + c * z + d = 0 给出，其中a、b 和c 是实常数。
3-D 中点和平面之间的距离公式由下式给出：

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

下面是上述公式的实现：

C++

// C++ program to find the
// Perpendicular(shortest)
// distance between a point
// and a Plane in 3 D.
#include
#include
 
using namespace std;
 
// Function to find distance
void shortest_distance(float x1, float y1,
                       float z1, float a,
                       float b, float c,
                       float d)
{
    d = fabs((a * x1 + b * y1 +
              c * z1 + d));
    float e = sqrt(a * a + b *
                   b + c * c);
    cout << "Perpendicular distance is "
         << (d / e);
        return;
}
 
// Driver Code
int main()
{
    float x1 = 4;
    float y1 = -4;
    float z1 = 3;
    float a = 2;
    float b = -2;
    float c = 5;
    float d = 8;
 
    // Function call
    shortest_distance(x1, y1, z1,
                      a, b, c, d);
}
 
// This code is contributed
// by Akanksha Rai(Abby_akku)

C

// C program to find the Perpendicular(shortest)
// distance between a point and a Plane in 3 D.
 
#include
#include
 
// Function to find distance
void shortest_distance(float x1, float y1, float z1,
                    float a, float b, float c, float d)
{
    d = fabs((a * x1 + b * y1 + c * z1 + d));
    float e = sqrt(a * a + b * b + c * c);
    printf("Perpendicular distance is %f", d/e);
        return;
}
 
// Driver Code
int main()
{
    float x1 = 4;
    float y1 = -4;
    float z1 = 3;
    float a = 2;
    float b = -2;
    float c = 5;
    float d = 8;
 
    // Function call
    shortest_distance(x1, y1, z1, a, b, c, d);
}
// This code is contributed
// by Amber_Saxena.

Java

// Java program to find the
// Perpendicular(shortest)
// distance between a point
// and a Plane in 3 D.
import java .io.*;
 
class GFG
{
     
// Function to find distance
static void shortest_distance(float x1, float y1,
                              float z1, float a,
                              float b, float c,
                              float d)
{
    d = Math.abs((a * x1 + b *
                 y1 + c * z1 + d));
    float e = (float)Math.sqrt(a * a + b *
                               b + c * c);
    System.out.println("Perpendicular distance " +
                                   "is " + d / e);
}
 
// Driver code
public static void main(String[] args)
{
    float x1 = 4;
    float y1 = -4;
    float z1 = 3;
    float a = 2;
    float b = -2;
    float c = 5;
    float d = 8;
 
    // Function call
    shortest_distance(x1, y1, z1,
                      a, b, c, d);
}
}
 
// This code is contributed
// by Amber_Saxena.

Python

# Python program to find the Perpendicular(shortest)
# distance between a point and a Plane in 3 D.
 
import math
 
# Function to find distance
def shortest_distance(x1, y1, z1, a, b, c, d):
     
    d = abs((a * x1 + b * y1 + c * z1 + d))
    e = (math.sqrt(a * a + b * b + c * c))
    print("Perpendicular distance is"), d/e
     
 
# Driver Code
x1 = 4
y1 = -4
z1 = 3
a = 2
b = -2
c = 5
d = 8
 
# Function call
shortest_distance(x1, y1, z1, a, b, c, d)

C#

// C# program to find the
// Perpendicular(shortest)
// distance between a point
// and a Plane in 3 D.
using System;
 
class GFG
{
     
// Function to find distance
static void shortest_distance(float x1, float y1,
                              float z1, float a,
                              float b, float c,
                              float d)
{
    d = Math.Abs((a * x1 + b *
                   y1 + c * z1 + d));
    float e = (float)Math.Sqrt(a * a + b *
                               b + c * c);
    Console.Write("Perpendicular distance " +
                              "is " + d / e);
}
 
// Driver code
public static void Main()
{
    float x1 = 4;
    float y1 = -4;
    float z1 = 3;
    float a = 2;
    float b = -2;
    float c = 5;
    float d = 8;
 
    // Function call
    shortest_distance(x1, y1, z1,
                      a, b, c, d);
}
}
 
// This code is contributed
// by ChitraNayal

Javascript

 
 
// Javascript program to find the
// Perpendicular(shortest)
// distance between a point
// and a Plane in 3 D.
 
// Function to find distance
function shortest_distance( x1,  y1, z1,  a,
                        b,  c, d)
{
    d = Math.abs((a * x1 + b * y1 +
              c * z1 + d));
    let e = Math.sqrt(a * a + b *
                   b + c * c);
    document.write("Perpendicular distance is "
         + (d / e));
        return;
}
 
    // driver code    
    let x1 = 4;
    let y1 = -4;
    let z1 = 3;
    let a = 2;
    let b = -2;
    let c = 5;
    let d = 8;
 
    // Function call
    shortest_distance(x1, y1, z1,
                      a, b, c, d);
     
 // This code is contributed by jana_sayantan.  

输出：

Perpendicular distance is 6.78902858227