Swift Coding Test Course, Determining Line Segment Intersection

Problem Description

Given two line segments AB and CD, the problem is to determine whether these two segments intersect. Here, line segment AB is composed of point A(x1, y1) and point B(x2, y2), and line segment CD is composed of point C(x3, y3) and point D(x4, y4). Your goal is to verify whether AB and CD intersect and to return the result as a Boolean value.

The conditions for line segments to cross each other are as follows:

• When distinct segments block each other
• When endpoints lie on the other segment

Approach to the Problem

To solve this problem, we need to utilize geometric properties. Generally, to determine whether two segments intersect, we can use linear methods and cross-verification. The basic idea is to have arrays of points that define each segment and ensure that the segments can be expressed by equations.

The fundamental principle for determining line intersection is to use the direction of vectors to check whether each segment blocks the other. After defining each segment, we need to create a function to determine whether they intersect.

Algorithm Design

First, let’s figure out how to compute the direction of the two segments. Assume we have two given points A(x1, y1), B(x2, y2) and C(x3, y3), D(x4, y4). We can use the cross product to find the relative positions of the two points.

The cross product is defined as follows:

                (B - A) × (D - A) 
            

Through the cross product of the vectors, we can determine the orientation of the two segments. The next step is to check whether the segments intersect.

The function to implement is as follows:

1. Calculate the cross product of the two points.
2. Determine the intersection of the segments based on the results.
3. Return a Boolean value according to each case.

Swift Code Example

Now, based on the above algorithm, let’s write the code in Swift.

                
                struct Point {
                    var x: Double
                    var y: Double
                }

                func orientation(p: Point, q: Point, r: Point) -> Int {
                    let val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y)
                    if val == 0 { return 0 // Collinear
                    } else if val > 0 { return 1 // Clockwise
                    } else { return 2 // Counterclockwise
                    }
                }

                func doIntersect(p1: Point, q1: Point, p2: Point, q2: Point) -> Bool {
                    let o1 = orientation(p: p1, q: q1, r: p2)
                    let o2 = orientation(p: p1, q: q1, r: q2)
                    let o3 = orientation(p: p2, q: q2, r: p1)
                    let o4 = orientation(p: p2, q: q2, r: q1)

                    if o1 != o2 && o3 != o4 {
                        return true
                    }

                    // Collinear cases
                    return false
                }

                // Example
                let p1 = Point(x: 1, y: 1)
                let q1 = Point(x: 10, y: 1)
                let p2 = Point(x: 1, y: 2)
                let q2 = Point(x: 10, y: 2)

                if doIntersect(p1: p1, q1: q1, p2: p2, q2: q2) {
                    print("They intersect.")
                } else {
                    print("They do not intersect.")
                }
                
            

The above code determines whether the given two segments intersect and outputs the results. The method used is to calculate the direction using the orientation function and check for intersection in the doIntersect function.

Optimization of the Algorithm

Below are optimization techniques to confirm the intersection of line segments:

• Implement a relatively simple linear search algorithm to quickly determine intersection
• Add additional logic to handle cases where points coincide with the endpoints of segments
• Use sorted points to improve numerical efficiency in most cases

Conclusion

In this tutorial, we solved an algorithmic problem of determining the intersection of line segments using Swift. The line intersection problem can be useful in various situations and contributes to improving problem-solving skills in algorithms. In actual interviews, you may often encounter similar problems. Therefore, it is important to practice different strategies through such problems.

In the next post, we will cover an algorithm problem on a different topic. Your attention is greatly appreciated!