How do you check for intersection between a line s

问题:

Given a line segment, that is two points (x1,y1) and (x2,y2), one point P(x,y) and an angle theta. How do we find if this line segment and the line ray that emanates from P at an angle theta from horizontal intersects or not? If they do intersect, how to find the point of intersection?

回答1:

Let's label the points q = (x1, y1) and q + s = (x2, y2). Hence s = (x2 − x1, y2 − y1). Then the problem looks like this:

Let r = (cos θ, sin θ). Then any point on the ray through p is representable as p + t r (for a scalar parameter 0 ≤ t) and any point on the line segment is representable as q + u s (for a scalar parameter 0 ≤ u ≤ 1).

The two lines intersect if we can find t and u such that p + t r = q + u s:

See this answer for how to find this point (or determine that there is no such point).

Then your line segment intersects the ray if 0 ≤ t and 0 ≤ u ≤ 1.

回答2:

Here is a C# code for the algorithm given in other answers:

    /// <summary>
    /// Returns the distance from the ray origin to the intersection point or null if there is no intersection.
    /// </summary>
    public double? GetRayToLineSegmentIntersection(Point rayOrigin, Vector rayDirection, Point point1, Point point2)
    {
        var v1 = rayOrigin - point1;
        var v2 = point2 - point1;
        var v3 = new Vector(-rayDirection.Y, rayDirection.X);


        var dot = v2 * v3;
        if (Math.Abs(dot) < 0.000001)
            return null;

        var t1 = Vector.CrossProduct(v2, v1) / dot;
        var t2 = (v1 * v3) / dot;

        if (t1 >= 0.0 && (t2 >= 0.0 && t2 <= 1.0))
            return t1;

        return null;
    }

回答3:

Thanks Gareth for a great answer. Here is the solution implemented in Python. Feel free to remove the tests and just copy paste the actual function. I have followed the write-up of the methods that appeared here, https://rootllama.wordpress.com/2014/06/20/ray-line-segment-intersection-test-in-2d/.

import numpy as np

def magnitude(vector):
   return np.sqrt(np.dot(np.array(vector),np.array(vector)))

def norm(vector):
   return np.array(vector)/magnitude(np.array(vector))

def lineRayIntersectionPoint(rayOrigin, rayDirection, point1, point2):
    """
    >>> # Line segment
    >>> z1 = (0,0)
    >>> z2 = (10, 10)
    >>>
    >>> # Test ray 1 -- intersecting ray
    >>> r = (0, 5)
    >>> d = norm((1,0))
    >>> len(lineRayIntersectionPoint(r,d,z1,z2)) == 1
    True
    >>> # Test ray 2 -- intersecting ray
    >>> r = (5, 0)
    >>> d = norm((0,1))
    >>> len(lineRayIntersectionPoint(r,d,z1,z2)) == 1
    True
    >>> # Test ray 3 -- intersecting perpendicular ray
    >>> r0 = (0,10)
    >>> r1 = (10,0)
    >>> d = norm(np.array(r1)-np.array(r0))
    >>> len(lineRayIntersectionPoint(r0,d,z1,z2)) == 1
    True
    >>> # Test ray 4 -- intersecting perpendicular ray
    >>> r0 = (0, 10)
    >>> r1 = (10, 0)
    >>> d = norm(np.array(r0)-np.array(r1))
    >>> len(lineRayIntersectionPoint(r1,d,z1,z2)) == 1
    True
    >>> # Test ray 5 -- non intersecting anti-parallel ray
    >>> r = (-2, 0)
    >>> d = norm(np.array(z1)-np.array(z2))
    >>> len(lineRayIntersectionPoint(r,d,z1,z2)) == 0
    True
    >>> # Test ray 6 --intersecting perpendicular ray
    >>> r = (-2, 0)
    >>> d = norm(np.array(z1)-np.array(z2))
    >>> len(lineRayIntersectionPoint(r,d,z1,z2)) == 0
    True
    """
    # Convert to numpy arrays
    rayOrigin = np.array(rayOrigin, dtype=np.float)
    rayDirection = np.array(norm(rayDirection), dtype=np.float)
    point1 = np.array(point1, dtype=np.float)
    point2 = np.array(point2, dtype=np.float)

    # Ray-Line Segment Intersection Test in 2D
    # http://bit.ly/1CoxdrG
    v1 = rayOrigin - point1
    v2 = point2 - point1
    v3 = np.array([-rayDirection[1], rayDirection[0]])
    t1 = np.cross(v2, v1) / np.dot(v2, v3)
    t2 = np.dot(v1, v3) / np.dot(v2, v3)
    if t1 >= 0.0 and t2 >= 0.0 and t2 <= 1.0:
        return [rayOrigin + t1 * rayDirection]
    return []

if __name__ == "__main__":
    import doctest
    doctest.testmod()