Find Corners of Rectangle using mid points

3.5

Consider a rectangle ABCD, we’re given the co-ordinates of the mid points of side AD and BC (p and q respectively) along with their length L (AD = BC = L). Now given the parameters, we need to print the co-ordinates of the 4 points A, B, C and D.

Rectangle

Examples:

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)

From the problem statement 3 cases can arise :

  1. The Rectangle is horizontal i.e., AD and BC are parallel to X-axis
  2. The Rectangle is vertical i.e., AD and BC are parallel to Y-axis
  3. The Rectangle is inclined at a certain angle with the axes

The first two cases are trivial and can easily be solved using basic geometry. For the third case we need to apply some mathematical concepts to find the points.

Consider the above diagram for clarity. We have the co-ordinates of p and q. Thus we can find the slope of AD and BC (As pq is perpendicular to AD). Once we have the slope of AD, we can find the equation of straight line passing through AD. Now we can apply distance formula to obtain the displacements along X and Y axes.

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))

Now we can simply find the co-ordinates of 4 corners by simply adding and subtracting the displacements obtained accordingly.

Below is the implementation in C++.

// C++ program to find corner points of
// a rectangle using given length and middle
// points.
#include <bits/stdc++.h>
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;
}

Output:

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

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

Reference:
StackOverflow

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