The number of horizontal or vertical line segments required to connect 3 points

Suppose that given three different points (or coordinates), you want to find the number of horizontal or vertical line segments that can be formed by connecting these three points. Such line segments are also called polylines. In order to solve this problem, you need the concept of computational geometry. In this article, we will discuss various ways to solve this problem in C.

Input and output scenarios

Assume that c1, c2 and c3 are the coordinates of 3 points on the Cartesian plane. The number of horizontal or vertical line segments connecting these 3 points will be as shown below.

Input: c1 = (-1, -1), c2 = (-2, 3), c3 = (4, 3)
Output: 1
Input: c1 = (1, -1), c2 = (1, 3), c3 = (4, 3)
Output: 2
Input: c1 = (1, 1), c2 = (2, 6), c3 = (5, 2)
Output: 3

Note − Horizontal and vertical line segments must be aligned with the coordinate axes.

Use If statement

We can use if statement to check if there is a horizontal or vertical line between these three points.

• Create a function by combining c1.x with c2.x, c1.x with c3.x and c2.x and c3.x. If either condition is met, it means there is a horizontal line segment and the count is incremented.

• Similarly, this function combines c1.y with c2.y, c1.y with c3.y and c2 .y and c3.y. If any of the conditions is met, the vertical line segment does exist. The count increases again.

Example

#include <iostream>
using namespace std;

struct Coordinate {
   int x;
   int y;
};
int countLineSegments(Coordinate c1, Coordinate c2, Coordinate c3) {
   int count = 0;
   // Check for the horizontal segment
   if (c1.x == c2.x || c1.x == c3.x || c2.x == c3.x)
      count++; 
   // Check for the vertical segment
   if (c1.y == c2.y || c1.y == c3.y || c2.y == c3.y)
      count++; 
   return count;
}

int main() {
   Coordinate c1, c2, c3;
   c1.x = -1; c1.y = -5;
   c2.x = -2; c2.y = 3;
   c3.x = 4; c3.y = 3;

   int numSegments = countLineSegments(c1, c2, c3);

   std::cout << "Number of horizontal or vertical line segments: " << numSegments << std::endl;

   return 0;
}

Output

Number of horizontal or vertical line segments: 1

Note− If all three points are on the same axis of the Cartesian plane, that is, the X axis or the Y axis, the number of line segments required to connect them is 1. If the points form an L shape, the result is 2, otherwise the result is 3.

Use auxiliary functions

• Here, we can use auxiliary functions (horizontalLine and verticalLine) to calculate line segments.

• We exploit the fact that in the Cartesian system all points of a horizontal line lie on the same y-coordinate. horizontalLineThe function checks whether two points can form a horizontal line segment by comparing their y coordinates. If the y-coordinates are the same, there is a horizontal line.

• Similarly, all points of a vertical line lie at the same x-coordinate. verticalLineThe function checks whether two points can form a vertical line segment by comparing their x-coordinates. If the x-coordinates are the same, there is a vertical line.

• Next, we have the countLineSegments function, which is used to count the number of horizontal and vertical line segments. If there are horizontal or vertical line segments, the count is incremented after each iteration.

Example

#include <iostream>
using namespace std;

struct Coordinate {
   int x;
   int y;
};

// Helper functions
bool horizontalLine(Coordinate c1, Coordinate c2)  {
   return c1.y == c2.y;
}

bool verticalLine(Coordinate c1, Coordinate c2)  {
   return c1.x == c2.x;
}

int countLineSegments(Coordinate c1, Coordinate c2, Coordinate c3)  {
   int count = 0;
   // Check for horizontal segment
   if (horizontalLine(c1, c2) || horizontalLine(c1, c3) || horizontalLine(c2, c3))
      count++; 
   // Check for vertical segment
   if (verticalLine(c1, c2) || verticalLine(c1, c3) || verticalLine(c2, c3))
      count++; 
   return count;
}

int main() {
   Coordinate c1, c2, c3;
   c1.x = -1; c1.y = -5;
   c2.x = -2; c2.y = 3;
   c3.x = 4; c3.y = 3;

   int numSegments = countLineSegments(c1, c2, c3);

   std::cout << "Number of horizontal or vertical line segments: " << numSegments << std::endl;

   return 0;
}

Output

Number of horizontal or vertical line segments: 1

in conclusion

In this article, we explore various methods using C to find the number of horizontal and vertical lines that can connect 3 different points in the Cartesian plane. We have discussed the if statement approach to solving this problem. However, due to the large number of iterations, the time complexity also increases. We can effectively solve this problem by using auxiliary functions, which reduces the number of iterations and thereby reduces the time complexity.

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