How to calculate the point where two lines intersect

Swift version: 5.1

Paul Hudson @twostraws May 28th 2019

Finding where two lines cross can be done by calculating their cross product. The code below returns an optional tuple containing the X and Y intersection points, or nil if they don’t cross at all.

Note: Core Graphics doesn’t give us a CGLine type, so you’ll need pass this four points: where the first line starts and ends, and where the second line starts and ends.

func linesCross(start1: CGPoint, end1: CGPoint, start2: CGPoint, end2: CGPoint) -> (x: CGFloat, y: CGFloat)? {
    // calculate the differences between the start and end X/Y positions for each of our points
    let delta1x = end1.x - start1.x
    let delta1y = end1.y - start1.y
    let delta2x = end2.x - start2.x
    let delta2y = end2.y - start2.y

    // create a 2D matrix from our vectors and calculate the determinant
    let determinant = delta1x * delta2y - delta2x * delta1y

    if abs(determinant) < 0.0001 {
        // if the determinant is effectively zero then the lines are parallel/colinear
        return nil
    }

    // if the coefficients both lie between 0 and 1 then we have an intersection
    let ab = ((start1.y - start2.y) * delta2x - (start1.x - start2.x) * delta2y) / determinant

    if ab > 0 && ab < 1 {
        let cd = ((start1.y - start2.y) * delta1x - (start1.x - start2.x) * delta1y) / determinant

        if cd > 0 && cd < 1 {
            // lines cross – figure out exactly where and return it
            let intersectX = start1.x + ab * delta1x
            let intersectY = start1.y + ab * delta1y
            return (intersectX, intersectY)
        }
    }

    // lines don't cross
    return nil
}

Note: this code is adapted from “Intersection of Two Lines in Three-Space”, which is a one-page chapter by Ronald Goodman in the book Graphics Gems. For more on how cross products work, I can highly recomend the book “Essential Mathematics for Games and Interactive Applications” by James M. Van Verth and Lars M. Bishop.