Title: Determine where two lines intersect in C#
This example determines whether two segments intersect and where the lines that contain them intersect. There are several ways you can approach this problem. This example uses lines defined by parametric equations where 0 <= t1, t2 <= 1. If the first segment has end points (x11, y11) and (x12, y12), and the second segment has end points (x21, y21) and (x22, y22), then the line segments are defined by the following functions:
X1(t) = x11 + dx1 * t1
Y1(t) = y11 + dy1 * t1
X2(t) = x21 + dx2 * t2
Y2(t) = y21 + dy2 * t2
In other words, as the value of t1 varies from 0 to 1, (X1(t), Y1(t)) gives the points along the first segment.
When the two segments intersect, the points (X1(t1), Y1(t1)) and (X2(t2), Y2(t2)) are the same. Setting the equations for the points equal gives:
x11 + dx1 * t1 = x21 + dx2 * t2
y11 + dy1 * t1 = y21 + dy2 * t2
You can rearrange those equations to get:
x11  x21 + dx1 * t1 = dx2 * t2
y11  y21 + dy1 * t1 = dy2 * t2
And then:
(x11  x21 + dx1 * t1) * dy2 = dx2 * t2 * dy2
(y11  y21 + dy1 * t1) * (dx2) = dy2 * t2 * (dx2)
Adding the equations gives:
(x11  x21) * dy2 + ( dx1 * dy2) * t1 +
(y21  y11) * dx2 + (dy1 * dx2) * t1 = 0
Now solving for t1 gives:
t1 * (dy1 * dx2  dx1 * dy2) = (x11  x21) * dy2 + (y21  y11) * dx2
t1 = ((x11  x21) * dy2 + (y21  y11) * dx2) / (dy1 * dx2  dx1 * dy2)
You can solve for t2 similarly.
Notes:
 If 0 <= t1 <= 1, then the point lies on the first segment.
 If 0 <= t2 <= 1, then the point lies on the second segment.
 If dy1 * dx2  dx1 * dy2 = 0, then you can't calculate t1 or t2 (it would require dividing by 0), and the lines are parallel.
 If the point of intersection is not on both segments, then this is almost certainly not the point where the two segments are closest.
The FindIntersection method shown in the following code finds the intersection between the lines that contain the segments p1 > p2 and p3 > p4.
// Find the point of intersection between
// the lines p1 > p2 and p3 > p4.
private void FindIntersection(
PointF p1, PointF p2, PointF p3, PointF p4,
out bool lines_intersect, out bool segments_intersect,
out PointF intersection,
out PointF close_p1, out PointF close_p2)
{
// Get the segments' parameters.
float dx12 = p2.X  p1.X;
float dy12 = p2.Y  p1.Y;
float dx34 = p4.X  p3.X;
float dy34 = p4.Y  p3.Y;
// Solve for t1 and t2
float denominator = (dy12 * dx34  dx12 * dy34);
float t1 =
((p1.X  p3.X) * dy34 + (p3.Y  p1.Y) * dx34)
/ denominator;
if (float.IsInfinity(t1))
{
// The lines are parallel (or close enough to it).
lines_intersect = false;
segments_intersect = false;
intersection = new PointF(float.NaN, float.NaN);
close_p1 = new PointF(float.NaN, float.NaN);
close_p2 = new PointF(float.NaN, float.NaN);
return;
}
lines_intersect = true;
float t2 =
((p3.X  p1.X) * dy12 + (p1.Y  p3.Y) * dx12)
/ denominator;
// Find the point of intersection.
intersection = new PointF(p1.X + dx12 * t1, p1.Y + dy12 * t1);
// The segments intersect if t1 and t2 are between 0 and 1.
segments_intersect =
((t1 >= 0) && (t1 <= 1) &&
(t2 >= 0) && (t2 <= 1));
// Find the closest points on the segments.
if (t1 < 0)
{
t1 = 0;
}
else if (t1 > 1)
{
t1 = 1;
}
if (t2 < 0)
{
t2 = 0;
}
else if (t2 > 1)
{
t2 = 1;
}
close_p1 = new PointF(p1.X + dx12 * t1, p1.Y + dy12 * t1);
close_p2 = new PointF(p3.X + dx34 * t2, p3.Y + dy34 * t2);
}
This method takes as parameters the points that define the segments and the following output parameters to return results:
 lines_intersect  True if the lines containing the segments intersect
 segments_intersect  True if the segments intersect
 intersection  The point where the lines intersect
 close_p1  The point on the first segment that is closest to the point of intersection
 close_p2  The point on the second segment that is closest to the point of intersection
First the code calculates dx12, dy12, and the other values that define the lines. It then plugs the values and the points' coordinates into the equation shown earlier to calculate t1. If the results is infinity, then the denominator is 0 so the lines are parallel.
Next the code uses the values of t1 and t2 to find the points of intersection between the two lines.
If t1 and t2 are both between 0 and 1, then the line segments intersect.
The code then adjusts t1 and t2 so they are between 0 and 1. Those values generate the points on the two segments that are closest to the point of intersection. Finally the code uses the adjusted values of t1 and t2 to find those closest points.
To use the example, click two points to define the first segment. Then click two more points to define the second segment. The program draws the segments. It draws the "close" points in blue and the point of intersection in red. (In the figure above, one of the close points is below the point of intersection so you can't see it.)
Download the example to experiment with it and to see additional details.
