Problem Definition
The problem is to determine the direction of line segment AB given two points A(x1, y1) and B(x2, y2). The direction of the line segment reflects how the x-axis and y-axis change as it goes from A to B. We need to determine whether the direction of AB is upward, downward, leftward, or rightward.
The problem is as follows:
Given four integers x1, y1, x2, y2, determine the direction of line segment AB.
- 0: Horizontal line (y1 == y2)
- 1: Upward (y1 < y2)
- -1: Downward (y1 > y2)
- 2: Rightward (x1 < x2)
- -2: Leftward (x1 > x2)
Problem Analysis
To find the direction from point A to point B, we need to compare the x-coordinates and y-coordinates of the two points. The following cases are considered.
- If the y-coordinates are the same, the line segment is considered horizontal (y1 == y2).
- If A’s y-coordinate is less than B’s y-coordinate, the line segment is directed upward (y1 < y2).
- If A’s y-coordinate is greater than B’s y-coordinate, the line segment is directed downward (y1 > y2).
- If A’s x-coordinate is less than B’s x-coordinate, the line segment is directed rightward (x1 < x2).
- If A’s x-coordinate is greater than B’s x-coordinate, the line segment is directed leftward (x1 > x2).
These conditions can be used to resolve the problem.
Algorithm Design
To solve the problem, we design the following algorithm:
- Input the coordinates of two points A(x1, y1) and B(x2, y2).
- Compare y1 and y2 to determine the result in three cases (horizontal, upward, downward).
- Compare x1 and x2 to define the cases for rightward or leftward directions.
- Output the result.
Implementation
Below is the code implemented in Swift based on the above algorithm:
import Foundation
func determineDirection(x1: Int, y1: Int, x2: Int, y2: Int) -> String {
if y1 == y2 {
return "Horizontal line"
} else if y1 < y2 {
return "Upward"
} else {
return "Downward"
}
}
func determineHorizontalDirection(x1: Int, x2: Int) -> String {
if x1 < x2 {
return "Rightward"
} else if x1 > x2 {
return "Leftward"
} else {
return "Vertical line"
}
}
let x1 = 1, y1 = 2, x2 = 3, y2 = 4
print(determineDirection(x1: x1, y1: y1, x2: x2, y2: y2))
print(determineHorizontalDirection(x1: x1, x2: x2))
In the above Swift code, two functions are used to determine the direction of the line segment. The first function determineDirection assesses the direction based on the y-coordinates, while the second function determineHorizontalDirection assesses it based on the x-coordinates. Each case returns the appropriate string.
Test Cases
Now let’s look at a few test cases:
- Case 1: Line segment from A(1, 2) to B(3, 4)
- Case 2: Horizontal line from A(1, 3) to B(1, 3)
- Case 3: Line segment from A(5, 6) to B(2, 5)
We validate the algorithm through the results of each test case:
// Case 1: A(1, 2), B(3, 4)
let x1_case1 = 1, y1_case1 = 2, x2_case1 = 3, y2_case1 = 4
print(determineDirection(x1: x1_case1, y1: y1_case1, x2: x2_case1, y2: y2_case1)) // Upward
print(determineHorizontalDirection(x1: x1_case1, x2: x2_case1)) // Rightward
// Case 2: A(1, 3), B(1, 3)
let x1_case2 = 1, y1_case2 = 3, x2_case2 = 1, y2_case2 = 3
print(determineDirection(x1: x1_case2, y1: y1_case2, x2: x2_case2, y2: y2_case2)) // Horizontal line
print(determineHorizontalDirection(x1: x1_case2, x2: x2_case2)) // Vertical line
// Case 3: A(5, 6), B(2, 5)
let x1_case3 = 5, y1_case3 = 6, x2_case3 = 2, y2_case3 = 5
print(determineDirection(x1: x1_case3, y1: y1_case3, x2: x2_case3, y2: y2_case3)) // Downward
print(determineHorizontalDirection(x1: x1_case3, x2: x2_case3)) // Leftward
Conclusion
In this post, we examined the process of designing and implementing an algorithm to determine the direction of a line segment given two points. This algorithm allows for a straightforward determination of directionality through the comparison of the given two coordinates.
The problem of determining line segment direction requires a basic geometric approach, and it is important to clearly understand each condition and write the corresponding logic. Through such problems, one can develop algorithmic thinking and improve their skills.