使用中点找到矩形的角

考虑一个矩形 ABCD，我们给出了边 AD 和 BC（分别为 p 和 q）的中点的坐标以及它们的长度 L（AD = BC = L）。现在给定参数，我们需要打印 4 个点 A、B、C 和 D 的坐标。

例子：

Input : p = (1, 0)
        q = (1, 2)
        L = 2
Output : (0, 0), (0, 2), (2, 2), (2, 0)
Explanation:
The printed points form a rectangle which
satisfy the input constraints.

Input : p = (1, 1)
        q = (-1, -1)
        L = 2*sqrt(2)
Output : (0, 2), (-2, 0), (0, -2), (2, 0)

从问题陈述中可以出现 3 种情况：

1. 矩形是水平的，即 AD 和 BC 平行于 X 轴
2. 矩形是垂直的，即 AD 和 BC 平行于 Y 轴
3. 矩形与轴成一定角度倾斜

前两种情况是微不足道的，可以使用基本几何轻松解决。对于第三种情况，我们需要应用一些数学概念来找到点。

为清楚起见，请考虑上图。我们有 p 和 q 的坐标。因此我们可以找到 AD 和 BC 的斜率（因为 pq 垂直于 AD）。一旦我们有了 AD 的斜率，我们就可以找到通过 AD 的直线方程。现在我们可以应用距离公式来获得沿 X 和 Y 轴的位移。

If slope of AD = m, then
m = (p.x- q.x)/(q.y - p.y)

and displacement along X axis, dx =  
   L/(2*sqrt(1+m*m))

Similarly, dy = m*L/(2*sqrt(1+m*m))

现在我们可以通过简单地添加和减去相应获得的位移来简单地找到 4 个角的坐标。

下面是实现。

C++

// C++ program to find corner points of
// a rectangle using given length and middle
// points.
#include 
using namespace std;
 
// Structure to represent a co-ordinate point
struct Point
{
    float x, y;
    Point()
    {
        x = y = 0;
    }
    Point(float a, float b)
    {
        x = a, y = b;
    }
};
 
// This function receives two points and length
// of the side of rectangle and prints the 4
// corner points of the rectangle
void printCorners(Point p, Point q, float l)
{
    Point a, b, c, d;
 
    // horizontal rectangle
    if (p.x == q.x)
    {
        a.x = p.x - (l/2.0);
        a.y = p.y;
 
        d.x = p.x + (l/2.0);
        d.y = p.y;
 
        b.x = q.x - (l/2.0);
        b.y = q.y;
 
        c.x = q.x + (l/2.0);
        c.y = q.y;
    }
 
    // vertical rectangle
    else if (p.y == q.y)
    {
        a.y = p.y - (l/2.0);
        a.x = p.x;
 
        d.y = p.y + (l/2.0);
        d.x = p.x;
 
        b.y = q.y - (l/2.0);
        b.x = q.x;
 
        c.y = q.y + (l/2.0);
        c.x = q.x;
    }
 
    // slanted rectangle
    else
    {
        // calculate slope of the side
        float m = (p.x-q.x)/float(q.y-p.y);
 
        // calculate displacements along axes
        float dx = (l /sqrt(1+(m*m))) *0.5 ;
        float dy = m*dx;
 
        a.x = p.x - dx;
        a.y = p.y - dy;
 
        d.x = p.x + dx;
        d.y = p.y + dy;
 
        b.x = q.x - dx;
        b.y = q.y - dy;
 
        c.x = q.x + dx;
        c.y = q.y + dy;
    }
 
    cout << a.x << ", " << a.y << " n"
         << b.x << ", " << b.y << "n";
         << c.x << ", " << c.y << " n"
         << d.x << ", " << d.y << "nn";
}
 
// Driver code
int main()
{
    Point p1(1, 0), q1(1, 2);
    printCorners(p1, q1, 2);
 
    Point p(1, 1), q(-1, -1);
    printCorners(p, q, 2*sqrt(2));
 
    return 0;
}

Java

// Java program to find corner points of
// a rectangle using given length and middle
// points.
 
class GFG
{
 
    // Structure to represent a co-ordinate point
    static class Point
    {
 
        float x, y;
 
        Point()
        {
            x = y = 0;
        }
 
        Point(float a, float b)
        {
            x = a;
            y = b;
        }
    };
 
    // This function receives two points and length
    // of the side of rectangle and prints the 4
    // corner points of the rectangle
    static void printCorners(Point p, Point q, float l)
    {
        Point a = new Point(), b = new Point(),
                c = new Point(), d = new Point();
 
        // horizontal rectangle
        if (p.x == q.x)
        {
            a.x = (float) (p.x - (l / 2.0));
            a.y = p.y;
 
            d.x = (float) (p.x + (l / 2.0));
            d.y = p.y;
 
            b.x = (float) (q.x - (l / 2.0));
            b.y = q.y;
 
            c.x = (float) (q.x + (l / 2.0));
            c.y = q.y;
        }
        // vertical rectangle
        else if (p.y == q.y)
        {
            a.y = (float) (p.y - (l / 2.0));
            a.x = p.x;
 
            d.y = (float) (p.y + (l / 2.0));
            d.x = p.x;
 
            b.y = (float) (q.y - (l / 2.0));
            b.x = q.x;
 
            c.y = (float) (q.y + (l / 2.0));
            c.x = q.x;
        }
        // slanted rectangle
        else
        {
            // calculate slope of the side
            float m = (p.x - q.x) / (q.y - p.y);
 
            // calculate displacements along axes
            float dx = (float) ((l / Math.sqrt(1 + (m * m))) * 0.5);
            float dy = m * dx;
 
            a.x = p.x - dx;
            a.y = p.y - dy;
 
            d.x = p.x + dx;
            d.y = p.y + dy;
 
            b.x = q.x - dx;
            b.y = q.y - dy;
 
            c.x = q.x + dx;
            c.y = q.y + dy;
        }
 
        System.out.print((int)a.x + ", " + (int)a.y + " \n"
                + (int)b.x + ", " + (int)b.y + "\n"
                + (int)c.x + ", " + (int)c.y + " \n"
                + (int)d.x + ", " + (int)d.y + "\n");
    }
 
    // Driver code
    public static void main(String[] args)
    {
        Point p1 = new Point(1, 0), q1 = new Point(1, 2);
        printCorners(p1, q1, 2);
 
        Point p = new Point(1, 1), q = new Point(-1, -1);
        printCorners(p, q, (float) (2 * Math.sqrt(2)));
    }
}
 
// This code contributed by Rajput-Ji

Python3

# Python3 program to find corner points of
# a rectangle using given length and middle
# points.
import math
 
# Structure to represent a co-ordinate point
class Point:
     
    def __init__(self, a = 0, b = 0):
         
        self.x = a
        self.y = b
   
# This function receives two points and length
# of the side of rectangle and prints the 4
# corner points of the rectangle
def printCorners(p, q, l):
     
    a, b, c, d = Point(), Point(), Point(), Point()
     
    # Horizontal rectangle
    if (p.x == q.x):
        a.x = p.x - (l / 2.0)
        a.y = p.y
         
        d.x = p.x + (l / 2.0)
        d.y = p.y
         
        b.x = q.x - (l / 2.0)
        b.y = q.y
         
        c.x = q.x + (l / 2.0)
        c.y = q.y
         
    # Vertical rectangle
    elif (p.y == q.y):
        a.y = p.y - (l / 2.0)
        a.x = p.x
         
        d.y = p.y + (l / 2.0)
        d.x = p.x
         
        b.y = q.y - (l / 2.0)
        b.x = q.x
         
        c.y = q.y + (l / 2.0)
        c.x = q.x
     
    # Slanted rectangle
    else:
         
        # Calculate slope of the side
        m = (p.x - q.x) / (q.y - p.y)
         
        # Calculate displacements along axes
        dx = (l / math.sqrt(1 + (m * m))) * 0.5
        dy = m * dx
         
        a.x = p.x - dx
        a.y = p.y - dy
         
        d.x = p.x + dx
        d.y = p.y + dy
         
        b.x = q.x - dx
        b.y = q.y - dy
         
        c.x = q.x + dx
        c.y = q.y + dy
         
    print(int(a.x), ", ", int(a.y), sep = "")
    print(int(b.x), ", ", int(b.y), sep = "")
    print(int(c.x), ", ", int(c.y), sep = "")
    print(int(d.x), ", ", int(d.y), sep = "")
    print()
     
# Driver code
p1 = Point(1, 0)
q1 = Point(1, 2)
printCorners(p1, q1, 2)
 
p = Point(1, 1)
q = Point(-1, -1)
printCorners(p, q, 2 * math.sqrt(2))
 
# This code is contributed by shubhamsingh10

C#

// C# program to find corner points of
// a rectangle using given length and middle
// points.
using System;
 
class GFG
{
 
    // Structure to represent a co-ordinate point
    public class Point
    {
 
        public float x, y;
 
        public Point()
        {
            x = y = 0;
        }
 
        public Point(float a, float b)
        {
            x = a;
            y = b;
        }
    };
 
    // This function receives two points and length
    // of the side of rectangle and prints the 4
    // corner points of the rectangle
    static void printCorners(Point p, Point q, float l)
    {
        Point a = new Point(), b = new Point(),
                c = new Point(), d = new Point();
 
        // horizontal rectangle
        if (p.x == q.x)
        {
            a.x = (float) (p.x - (l / 2.0));
            a.y = p.y;
 
            d.x = (float) (p.x + (l / 2.0));
            d.y = p.y;
 
            b.x = (float) (q.x - (l / 2.0));
            b.y = q.y;
 
            c.x = (float) (q.x + (l / 2.0));
            c.y = q.y;
        }
         
        // vertical rectangle
        else if (p.y == q.y)
        {
            a.y = (float) (p.y - (l / 2.0));
            a.x = p.x;
 
            d.y = (float) (p.y + (l / 2.0));
            d.x = p.x;
 
            b.y = (float) (q.y - (l / 2.0));
            b.x = q.x;
 
            c.y = (float) (q.y + (l / 2.0));
            c.x = q.x;
        }
         
        // slanted rectangle
        else
        {
            // calculate slope of the side
            float m = (p.x - q.x) / (q.y - p.y);
 
            // calculate displacements along axes
            float dx = (float) ((l / Math.Sqrt(1 + (m * m))) * 0.5);
            float dy = m * dx;
 
            a.x = p.x - dx;
            a.y = p.y - dy;
 
            d.x = p.x + dx;
            d.y = p.y + dy;
 
            b.x = q.x - dx;
            b.y = q.y - dy;
 
            c.x = q.x + dx;
            c.y = q.y + dy;
        }
 
        Console.Write((int)a.x + ", " + (int)a.y + " \n"
                + (int)b.x + ", " + (int)b.y + "\n"
                + (int)c.x + ", " + (int)c.y + " \n"
                + (int)d.x + ", " + (int)d.y + "\n");
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        Point p1 = new Point(1, 0), q1 = new Point(1, 2);
        printCorners(p1, q1, 2);
 
        Point p = new Point(1, 1), q = new Point(-1, -1);
        printCorners(p, q, (float) (2 * Math.Sqrt(2)));
    }
}
 
// This code has been contributed by 29AjayKumar

Javascript

// Javascript program to find corner points of
// a rectangle using given length and middle
// points.
 
// Structure to represent a co-ordinate point
class Point
{
    constructor(a,b)
    {
        this.x=a;
        this.y=b;
    }
}
 
// This function receives two points and length
    // of the side of rectangle and prints the 4
    // corner points of the rectangle
function printCorners(p,q,l)
{
    let a = new Point(), b = new Point(),
                c = new Point(), d = new Point();
   
        // horizontal rectangle
        if (p.x == q.x)
        {
            a.x =  (p.x - (l / 2.0));
            a.y = p.y;
   
            d.x =  (p.x + (l / 2.0));
            d.y = p.y;
   
            b.x =  (q.x - (l / 2.0));
            b.y = q.y;
   
            c.x =  (q.x + (l / 2.0));
            c.y = q.y;
        }
        // vertical rectangle
        else if (p.y == q.y)
        {
            a.y = (p.y - (l / 2.0));
            a.x = p.x;
   
            d.y = (p.y + (l / 2.0));
            d.x = p.x;
   
            b.y = (q.y - (l / 2.0));
            b.x = q.x;
   
            c.y = (q.y + (l / 2.0));
            c.x = q.x;
        }
        // slanted rectangle
        else
        {
            // calculate slope of the side
            let m = (p.x - q.x) / (q.y - p.y);
   
            // calculate displacements along axes
            let dx =  ((l / Math.sqrt(1 + (m * m))) * 0.5);
            let dy = m * dx;
   
            a.x = p.x - dx;
            a.y = p.y - dy;
   
            d.x = p.x + dx;
            d.y = p.y + dy;
   
            b.x = q.x - dx;
            b.y = q.y - dy;
   
            c.x = q.x + dx;
            c.y = q.y + dy;
        }
   
        document.write(a.x + ", " + a.y + " <br>"
                + b.x + ", " + b.y + "<br>"
                + c.x + ", " + c.y + " <br>"
                + d.x + ", " + d.y + "<br>");
}
 
// Driver code
let p1 = new Point(1, 0), q1 = new Point(1, 2);
printCorners(p1, q1, 2);
 
let p = new Point(1, 1), q = new Point(-1, -1);
printCorners(p, q,  (2 * Math.sqrt(2)));
 
// This code is contributed by rag2127

输出：

0, 0 
0, 2
2, 2 
2, 0

0, 2 
-2, 0
0, -2 
2, 0

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