在 3D 中查找三角形的所有角度

给定 3D 中三角形的 3 个顶点的坐标，即 A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3)。任务是找出由上述坐标形成的三角形的所有角度。
例子：

Input: 
x1 = -1, y1 = 3, z1 = 2
x2 = 2, y2 = 3, z2 = 5
x3 = 3, y3 = 5, z3 = -2

Output:
angle A =  90.0 degree
angle B =  54.736 degree
angle C =  35.264 degree

方法：
为了找到角度 A 找出 AB 和 AC 的方向比：
AB 的方向比 = x2-x1, y2-y1, z2-z1
AC 的方向比 = x3-x1, y3-y1, z3-z1
那么角度 A = $cos^{-1}\left(\frac{(x_{2}-x_{1})*(x_{3}-x_{1})+(y_{2}-y_{1})*(y_{3}-y_{1})+(z_{2}-z_{1})*(z_{3}-z_{1})}{(\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}})*(\sqrt{(x_{3}-x_{1})^{2}+(y_{3}-y_{1})^{2}+(z_{3}-z_{1})^{2}})}\right)$
为了找到角度 B 找出 BA 和 BC 的方向比：
BA 的方向比 = x1-x2, y1-y2, z1-z2
BC 的方向比 = x3-x2, y3-y2, z3-z2
那么角度 B = $cos^{-1}\left(\frac{(x_{1}-x_{2})*(x_{3}-x_{2})+(y_{1}-y_{2})*(y_{3}-y_{2})+(z_{1}-z_{2})*(z_{3}-z_{2})}{(\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}})*(\sqrt{(x_{3}-x_{2})^{2}+(y_{3}-y_{2})^{2}+(z_{3}-z_{2})^{2}})}\right)$
为了找到角度 C，找出 CB 和 CA 的方向比：
CB 的方向比 = x2-x3, y2-y3, z2-z3
CA 的方向比 = x1-x3, y1-y3, z1-z3
那么角度 C = $cos^{-1}\left(\frac{(x_{2}-x_{3})*(x_{1}-x_{3})+(y_{2}-y_{3})*(y_{1}-y_{3})+(z_{2}-z_{3})*(z_{1}-z_{3})}{(\sqrt{(x_{2}-x_{3})^{2}+(y_{2}-y_{3})^{2}+(z_{2}-z_{3})^{2}})*(\sqrt{(x_{1}-x_{3})^{2}+(y_{1}-y_{3})^{2}+(z_{1}-z_{3})^{2}})}\right)$
下面是上述方法的实现：

C++

//CPP program for finding all angles of a triangle
#include
#include
using namespace std;
 
// function for finding the angle
float angle_triangle(int x1, int x2, int x3,
            int y1, int y2, int y3, int z1, int z2, int z3)
{
 
    int num = (x2-x1)*(x3-x1)+(y2-y1)*(y3-y1)+(z2-z1)*(z3-z1) ;
 
    float den = sqrt(pow((x2-x1),2)+pow((y2-y1),2)+pow((z2-z1),2))*\
                sqrt(pow((x3-x1),2)+pow((y3-y1),2)+pow((z3-z1),2)) ;
 
    float angle = acos(num / den)*(180.0/3.141592653589793238463) ;
 
    return angle ;
}
 
// Driver code
int main()
{
int x1 = -1;
int y1 = 3;
int z1 = 2;
int x2 = 2;
int y2 = 3;
int z2 = 5;
int x3 = 3;
int y3 = 5;
int z3 = -2;
float angle_A = angle_triangle(x1, x2, x3, y1, y2,
                                y3, z1, z2, z3);
float angle_B = angle_triangle(x2, x3, x1, y2, y3,
                                y1, z2, z3, z1);
float angle_C = angle_triangle(x3, x2, x1, y3,
                            y2, y1, z3, z2, z1);
cout<<"Angles are :"<

---

Java

//Java program for finding all angles of a triangle
 
class GFG{
// function for finding the angle
static double angle_triangle(int x1, int x2, int x3,
            int y1, int y2, int y3, int z1, int z2, int z3)
{
 
    int num = (x2-x1)*(x3-x1)+(y2-y1)*(y3-y1)+(z2-z1)*(z3-z1) ;
 
    double den = Math.sqrt(Math.pow((x2-x1),2)+
                Math.pow((y2-y1),2)+Math.pow((z2-z1),2))*
                Math.sqrt(Math.pow((x3-x1),2)+
                Math.pow((y3-y1),2)+Math.pow((z3-z1),2)) ;
 
    double angle = Math.acos(num / den)*(180.0/3.141592653589793238463) ;
 
    return angle ;
}
 
// Driver code
public static void main(String[] args)
{
int x1 = -1;
int y1 = 3;
int z1 = 2;
int x2 = 2;
int y2 = 3;
int z2 = 5;
int x3 = 3;
int y3 = 5;
int z3 = -2;
double angle_A = angle_triangle(x1, x2, x3, y1,
                            y2, y3, z1, z2, z3);
double angle_B = angle_triangle(x2, x3, x1,
                            y2, y3, y1, z2, z3, z1);
double angle_C = angle_triangle(x3, x2, x1,
                            y3, y2, y1, z3, z2, z1);
System.out.println("Angles are :");
System.out.println("angle A = "+angle_A+" degree");
System.out.println("angle B = "+angle_B+" degree");
System.out.println("angle C = "+angle_C+" degree");
}
}
// This code is contributed by mits

Python3

# Python Code for finding all angles of a triangle
import math
 
# function for finding the angle
def angle_triangle(x1, x2, x3, y1, y2, y3, z1, z2, z3):
 
    num = (x2-x1)*(x3-x1)+(y2-y1)*(y3-y1)+(z2-z1)*(z3-z1)
 
    den = math.sqrt((x2-x1)**2+(y2-y1)**2+(z2-z1)**2)*\
                math.sqrt((x3-x1)**2+(y3-y1)**2+(z3-z1)**2)
 
    angle = math.degrees(math.acos(num / den))
 
    return round(angle, 3)
 
# driver code    
x1 = -1
y1 = 3
z1 = 2
x2 = 2
y2 = 3
z2 = 5
x3 = 3
y3 = 5
z3 = -2
angle_A = angle_triangle(x1, x2, x3, y1, y2, y3, z1, z2, z3)
angle_B = angle_triangle(x2, x3, x1, y2, y3, y1, z2, z3, z1)
angle_C = angle_triangle(x3, x2, x1, y3, y2, y1, z3, z2, z1)
print("Angles are :")
print("angle A = ", angle_A, "degree")
print("angle B = ", angle_B, "degree")
print("angle C = ", angle_C, "degree")

C#

// C# program for finding all
// angles of a triangle
using System;
                     
class GFG
{
     
// function for finding the angle
static double angle_triangle(int x1, int x2, int x3,
                             int y1, int y2, int y3,
                             int z1, int z2, int z3)
{
 
    int num = (x2 - x1) * (x3 - x1) +
              (y2 - y1) * (y3 - y1) +
              (z2 - z1) * (z3 - z1);
 
    double den = Math.Sqrt(Math.Pow((x2 - x1), 2) +
                           Math.Pow((y2 - y1), 2) +
                           Math.Pow((z2 - z1), 2)) *
                 Math.Sqrt(Math.Pow((x3 - x1), 2) +
                           Math.Pow((y3 - y1), 2) +
                           Math.Pow((z3 - z1), 2));
 
    double angle = Math.Acos(num / den) *
                   (180.0/3.141592653589793238463);
 
    return angle ;
}
 
// Driver code
public static void Main()
{
    int x1 = -1, y1 = 3, z1 = 2;
    int x2 = 2, y2 = 3, z2 = 5;
    int x3 = 3, y3 = 5, z3 = -2;
    double angle_A = angle_triangle(x1, x2, x3,
                                    y1, y2, y3,
                                    z1, z2, z3);
    double angle_B = angle_triangle(x2, x3, x1,
                                    y2, y3, y1,
                                    z2, z3, z1);
    double angle_C = angle_triangle(x3, x2, x1,
                                    y3, y2, y1,
                                    z3, z2, z1);
    Console.WriteLine("Angles are :");
    Console.WriteLine("angle A = " + angle_A +
                                   " degree");
    Console.WriteLine("angle B = " + angle_B +
                                   " degree");
    Console.WriteLine("angle C = " + angle_C +
                                   " degree");
}
}
 
// This code is contributed by 29AjayKumar

Javascript

//Javascript program for finding all angles of a triangle
 
// function for finding the angle
function angle_triangle(x1,x2,x3,y1,y2,y3,z1,z2,z3)
{
    let num = (x2-x1)*(x3-x1)+(y2-y1)*(y3-y1)+(z2-z1)*(z3-z1) ;
   
    let den = Math.sqrt(Math.pow((x2-x1),2)+
                Math.pow((y2-y1),2)+Math.pow((z2-z1),2))*
                Math.sqrt(Math.pow((x3-x1),2)+
                Math.pow((y3-y1),2)+Math.pow((z3-z1),2)) ;
   
    let angle = Math.acos(num / den)*(180.0/3.141592653589793238463) ;
   
    return angle ;
}
 
// Driver code
let x1 = -1;
let y1 = 3;
let z1 = 2;
let x2 = 2;
let y2 = 3;
let z2 = 5;
let x3 = 3;
let y3 = 5;
let z3 = -2;
let angle_A = angle_triangle(x1, x2, x3, y1,
                            y2, y3, z1, z2, z3);
let angle_B = angle_triangle(x2, x3, x1,
                            y2, y3, y1, z2, z3, z1);
let angle_C = angle_triangle(x3, x2, x1,
                            y3, y2, y1, z3, z2, z1);
document.write("Angles are :<br>");
document.write("angle A = "+angle_A+" degree<br>");
document.write("angle B = "+angle_B.toFixed(3)+" degree<br>");
document.write("angle C = "+angle_C.toFixed(3)+" degree<br>");
 
// This code is contributed by rag2127

输出：

Angles are :
angle A =  90.0 degree
angle B =  54.736 degree
angle C =  35.264 degree

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