How do you check for intersection between a line segment and a line ray emanating from a point at an angle from horizontal?

Question

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?

Answer 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.

Answer 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;     }