确定两点相对于 3D 平面的位置

给定四个整数a 、 b 、 c和d ，它们表示平面ax + by + cz + d = 0的方程系数和两个整数坐标(x1, y1, z1)和(x2, y2, z2) ，任务是找出两个点是否位于同一侧，或不同侧，或在平面的表面上。

例子：

Input: a = 1, b = 2, c = 3, d = 4, x1 = -2, y1 = -2, z1 = 1, x2 = -4, y2 = 11, z2 = -1
Output: On same side
Explanation: On applying (x1, y1, z1) and (x2, y2, z2) on ax+by+cz+d=0 gives 1 and 19 respectively, both of which have the same sign, hence both the point lies on the same side of the plane.

Input: a = 4, b = 2, c = 1, d = 3, x1 = -2, y1 = -2, z1 = 1, x2 = -4, y2 = 11, z2 = -1
Output: On different sides

方法：这个想法是基于这样一个事实，如果应用于方程的两个点具有相同的奇偶性（符号），那么它们将位于平面的同一侧，如果它们具有不同的奇偶性，则它们将位于平面的同一侧飞机的不同侧面。请按照以下步骤解决问题：

• 将给定点的坐标放入平面方程并将值存储在变量P1和P2 中。
• 检查获得的值的符号：
  • 如果P1和P2具有相同的奇偶校验，则它们在平面的同一侧。
  • 如果P1和P2具有不同的奇偶性，则它们位于平面的相对两侧。
  • 如果P1和P2为零，则它们位于平面上。

下面是上述方法的实现：

C++

// C++ program for the above approach
#include 
using namespace std;
 
// Function to check position of two
// points with respect to a plane in 3D
void check_position(int a, int b, int c, int d,
                    int x1, int y1, int z1,
                    int x2, int y2, int z2)
{
    // Put coordinates in plane equation
    int value_1 = a * x1 + b * y1 + c * z1 + d;
    int value_2 = a * x2 + b * y2 + c * z2 + d;
 
    // If both values have same sign
    if ((value_1 > 0 && value_2 > 0)
        || (value_1 < 0 && value_2 < 0))
        cout << "On same side";
 
    // If both values have different sign
    if ((value_1 > 0 && value_2 < 0)
        || (value_1 < 0 && value_2 > 0))
        cout << "On different sides";
 
    // If both values are zero
    if (value_1 == 0 && value_2 == 0)
        cout << "Both on the plane";
 
    // If either of the two values is zero
    if (value_1 == 0 && value_2 != 0)
        cout << "Point 1 on the plane";
    if (value_1 != 0 && value_2 == 0)
        cout << "Point 2 on the plane";
}
 
// Driver Code
int main()
{
 
    // Given Input
    int a = 1, b = 2, c = 3, d = 4;
 
    // Coordinates of points
    int x1 = -2, y1 = -2, z1 = 1;
    int x2 = -4, y2 = 11, z2 = -1;
 
    // Function Call
    check_position(a, b, c, d,
                   x1, y1, z1,
                   x2, y2, z2);
 
    return 0;
}

Java

// Java program for the above approach
public class GFG {
 
    // Function to check position of two
    // points with respect to a plane in 3D
    static void check_position(int a, int b, int c, int d,
                               int x1, int y1, int z1,
                               int x2, int y2, int z2)
    {
        // Put coordinates in plane equation
        int value_1 = a * x1 + b * y1 + c * z1 + d;
        int value_2 = a * x2 + b * y2 + c * z2 + d;
 
        // If both values have same sign
        if ((value_1 > 0 && value_2 > 0)
            || (value_1 < 0 && value_2 < 0))
            System.out.print("On same side");
 
        // If both values have different sign
        if ((value_1 > 0 && value_2 < 0)
            || (value_1 < 0 && value_2 > 0))
            System.out.print("On different sides");
 
        // If both values are zero
        if (value_1 == 0 && value_2 == 0)
            System.out.print("Both on the plane");
 
        // If either of the two values is zero
        if (value_1 == 0 && value_2 != 0)
            System.out.print("Point 1 on the plane");
        if (value_1 != 0 && value_2 == 0)
            System.out.print("Point 2 on the plane");
    }
 
    // Driver code
    public static void main(String[] args)
    {
        // Given Input
        int a = 1, b = 2, c = 3, d = 4;
 
        // Coordinates of points
        int x1 = -2, y1 = -2, z1 = 1;
        int x2 = -4, y2 = 11, z2 = -1;
 
        // Function Call
        check_position(a, b, c, d, x1, y1, z1, x2, y2, z2);
    }
}
 
// This code is contributed by sk944795.

Python3

# Python3 program for the above approach
 
# Function to check position of two
# points with respect to a plane in 3D
def check_position(a, b, c, d, x1, y1,
                       z1, x2, y2, z2):
                        
    # Put coordinates in plane equation
    value_1 = a * x1 + b * y1 + c * z1 + d
    value_2 = a * x2 + b * y2 + c * z2 + d
 
    # If both values have same sign
    if ((value_1 > 0 and value_2 > 0) or
        (value_1 < 0 and value_2 < 0)):
        print("On same side")
 
    # If both values have different sign
    if ((value_1 > 0 and value_2 < 0) or
        (value_1 < 0 and value_2 > 0)):
        print("On different sides")
 
    # If both values are zero
    if (value_1 == 0 and value_2 == 0):
        print("Both on the plane")
 
    # If either of the two values is zero
    if (value_1 == 0 and value_2 != 0):
        print("Point 1 on the plane")
    if (value_1 != 0 and value_2 == 0):
        print("Point 2 on the plane")
 
# Driver Code
if __name__ == '__main__':
 
    # Given Input
    a, b, c, d = 1, 2, 3, 4
 
    # Coordinates of points
    x1, y1, z1 = -2, -2, 1
    x2, y2, z2 = -4, 11, -1
 
    # Function Call
    check_position(a, b, c, d, x1,
                   y1, z1, x2, y2, z2)
 
# This code is contributed by mohit kumar 29

C#

// C# program for the above approach
using System;
 
class GFG{
     
    // Function to check position of two
    // points with respect to a plane in 3D
    static void check_position(int a, int b, int c, int d,
                               int x1, int y1, int z1,
                               int x2, int y2, int z2)
    {
        // Put coordinates in plane equation
        int value_1 = a * x1 + b * y1 + c * z1 + d;
        int value_2 = a * x2 + b * y2 + c * z2 + d;
  
        // If both values have same sign
        if ((value_1 > 0 && value_2 > 0)
            || (value_1 < 0 && value_2 < 0))
            Console.Write("On same side");
  
        // If both values have different sign
        if ((value_1 > 0 && value_2 < 0)
            || (value_1 < 0 && value_2 > 0))
            Console.Write("On different sides");
  
        // If both values are zero
        if (value_1 == 0 && value_2 == 0)
            Console.Write("Both on the plane");
  
        // If either of the two values is zero
        if (value_1 == 0 && value_2 != 0)
           Console.Write("Point 1 on the plane");
        if (value_1 != 0 && value_2 == 0)
            Console.Write("Point 2 on the plane");
    }
 
// Driver code
static void Main()
{
    // Given Input
        int a = 1, b = 2, c = 3, d = 4;
  
        // Coordinates of points
        int x1 = -2, y1 = -2, z1 = 1;
        int x2 = -4, y2 = 11, z2 = -1;
  
        // Function Call
        check_position(a, b, c, d, x1, y1, z1, x2, y2, z2);
     
}
}
 
// This code is contributed by sanjoy_62.

Javascript

 
// JavaScript program for the above approach
 
// Function to check position of two
// points with respect to a plane in 3D
function check_position(a , b , c , d,
                           x1 , y1 , z1,
                           x2 , y2 , z2)
{
    // Put coordinates in plane equation
    var value_1 = a * x1 + b * y1 + c * z1 + d;
    var value_2 = a * x2 + b * y2 + c * z2 + d;
 
    // If both values have same sign
    if ((value_1 > 0 && value_2 > 0)
        || (value_1 < 0 && value_2 < 0))
        document.write("On same side");
 
    // If both values have different sign
    if ((value_1 > 0 && value_2 < 0)
        || (value_1 < 0 && value_2 > 0))
        document.write("On different sides");
 
    // If both values are zero
    if (value_1 == 0 && value_2 == 0)
        document.write("Both on the plane");
 
    // If either of the two values is zero
    if (value_1 == 0 && value_2 != 0)
        document.write("Povar 1 on the plane");
    if (value_1 != 0 && value_2 == 0)
        document.write("Povar 2 on the plane");
}
 
// Driver code
 
// Given Input
var a = 1, b = 2, c = 3, d = 4;
 
// Coordinates of points
var x1 = -2, y1 = -2, z1 = 1;
var x2 = -4, y2 = 11, z2 = -1;
 
// Function Call
check_position(a, b, c, d, x1, y1, z1, x2, y2, z2);
 
// This code is contributed by 29AjayKumar
 

输出：

On same side

时间复杂度： O(1)
辅助空间： O(1)

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