3D中两个平面之间的角度

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。任务是找到这两个平面之间的角度在3D中。

例子：

Input: a1 = 1, b1 = 1, c1 = 2, d1 = 1, a2 = 2, b2 = -1, c2 = 1, d2 = -4
Output: Angle is 60.0 degree

Input: a1 = 2, b1 = 2, c1 = -3, d1 = -5, a2 = 3, b2 = -3, c2 = 5, d2 = -6
Output: Angle is 123.696598882 degree

方法：考虑给定两个平面的以下方程式：

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

其中a1，b1，c1和a2，b2，c2是垂直于平面P1和P2的方向比率。

两个平面之间的角度等于由平面的法线向量确定的角度。
这些平面之间的角度通过使用以下公式给出：
cos A = $\huge\frac{(a_{1}*a_{2}+b_{1}*b_{2}+c_{1}*c_{2})}{(\sqrt{a_{1}*a_{1}+b_{1}*b_{1}+c_{1}*c_{1}})*(\sqrt{a_{2}*a_{2}+b_{2}*b_{2}+c_{2}*c_{2}})}$

使用逆属性，我们得到：
A = $Cos^{-1}\left ( \frac{(a_{1}*a_{2}+b_{1}*b_{2}+c_{1}*c_{2})}{(\sqrt{a_{1}*a_{1}+b_{1}*b_{1}+c_{1}*c_{1}})*(\sqrt{a_{2}*a_{2}+b_{2}*b_{2}+c_{2}*c_{2}})} \right )$

下面是上述公式的实现：

C++

// C++ program to find 
// the Angle between
// two Planes in 3 D.
#include  
#include
  
using namespace std;
  
// Function to find Angle
void distance(float a1, float b1,
              float c1, float a2,
              float b2, float c2)
{
    float d = (a1 * a2 + b1 * 
               b2 + c1 * c2);
    float e1 = sqrt(a1 * a1 + b1 * 
                    b1 + c1 * c1);
    float e2 = sqrt(a2 * a2 + b2 * 
                    b2 + c2 * c2);
    d = d / (e1 * e2);
    float pi = 3.14159;
    float A = (180 / pi) * (acos(d));
    cout << "Angle is " 
         << A << " degree";
}
  
// Driver Code 
int main()
{
    float a1 = 1;
    float b1 = 1;
    float c1 = 2;
    float d1 = 1;
    float a2 = 2;
    float b2 = -1;
    float c2 = 1;
    float d2 = -4;
    distance(a1, b1, c1, 
             a2, b2, c2);
    return 0;
}
  
// This code is contributed 
// by Akanksha Rai(Abby_akku)

C

// C program to find 
// the Angle between
// two Planes in 3 D.
#include
#include
  
// Function to find Angle
void distance(float a1, float b1,
              float c1, float a2,
              float b2, float c2)
{
    float d = (a1 * a2 + b1 * 
               b2 + c1 * c2);
    float e1 = sqrt(a1 * a1 + b1 * 
                    b1 + c1 * c1);
    float e2 = sqrt(a2 * a2 + b2 * 
                    b2 + c2 * c2);
    d = d / (e1 * e2);
    float pi = 3.14159;
    float A = (180 / pi) * (acos(d));
    printf("Angle is %.2f degree", A);
}
  
// Driver Code 
int main()
{
    float a1 = 1;
    float b1 = 1;
    float c1 = 2;
    float d1 = 1;
    float a2 = 2;
    float b2 = -1;
    float c2 = 1;
    float d2 = -4;
    distance(a1, b1, c1, 
             a2, b2, c2);
    return 0;
}
  
// This code is contributed 
// by Amber_Saxena.

Java

// Java program to find 
// the Angle between
// two Planes in 3 D. 
import java .io.*; 
import java.lang.Math; 
  
class GFG 
{ 
      
// Function to find Angle
static void distance(float a1, float b1,
                     float c1, float a2,
                     float b2, float c2) 
{ 
      
    float d = (a1 * a2 + b1 * 
               b2 + c1 * c2);
    float e1 = (float)Math.sqrt(a1 * a1 + b1 * 
                                b1 + c1 * c1);
    float e2 = (float)Math.sqrt(a2 * a2 + b2 * 
                                b2 + c2 * c2);
    d = d / (e1 * e2);
    float pi = (float)3.14159;
    float A = (180 / pi) * (float)(Math.acos(d));
    System.out.println("Angle is "+ A +" degree");
} 
  
// Driver code 
public static void main(String[] args) 
{ 
    float a1 = 1;
    float b1 = 1;
    float c1 = 2;
    float d1 = 1;
    float a2 = 2;
    float b2 = -1;
    float c2 = 1;
    float d2 = -4;
    distance(a1, b1, c1, 
             a2, b2, c2);
} 
} 
  
// This code is contributed 
// by Amber_Saxena.

Python

# Python program to find the Angle between
# two Planes in 3 D.
  
import math
  
# Function to find Angle
def distance(a1, b1, c1, a2, b2, c2): 
      
    d = ( a1 * a2 + b1 * b2 + c1 * c2 )
    e1 = math.sqrt( a1 * a1 + b1 * b1 + c1 * c1)
    e2 = math.sqrt( a2 * a2 + b2 * b2 + c2 * c2)
    d = d / (e1 * e2)
    A = math.degrees(math.acos(d))
    print("Angle is"), A, ("degree")
  
# Driver Code 
a1 = 1
b1 = 1
c1 = 2
d1 = 1
a2 = 2
b2 = -1
c2 = 1
d2 = -4
distance(a1, b1, c1, a2, b2, c2)

C#

// C# program to find 
// the Angle between
// two Planes in 3 D. 
using System; 
  
class GFG 
{ 
      
// Function to find Angle
static void distance(float a1, float b1,
                     float c1, float a2,
                      float b2, float c2) 
{ 
      
    float d = (a1 * a2 + b1 * 
               b2 + c1 * c2);
    float e1 = (float)Math.Sqrt(a1 * a1 + b1 * 
                                b1 + c1 * c1);
    float e2 = (float)Math.Sqrt(a2 * a2 + b2 * 
                                b2 + c2 * c2);
    d = d / (e1 * e2);
    float pi = (float)3.14159;
    float A = (180 / pi) * (float)(Math.Acos(d));
    Console.Write("Angle is "+ A +" degree");
} 
  
// Driver code 
public static void Main() 
{ 
    float a1 = 1;
    float b1 = 1;
    float c1 = 2;
    float a2 = 2;
    float b2 = -1;
    float c2 = 1;
      
    distance(a1, b1, c1, 
            a2, b2, c2);
} 
} 
  
// This code is contributed 
// by ChitraNayal

输出：

Angle is 60.0 degree