In this note I will explain how to find the intersection point P between two line segments. Note that this method will also calculate intersections on extended line segments.

As a reminder the cross product is the area of the parallelogram enclosed by the two As a reminder, the cross product is the area of the parallelogram enclosed by the two vectors. In 2D graphics, we will calculate only the z-component of the cross-vector, which will be called the cross-product in this note.

It can be calculated using the following formula:

Calculating the actual intersection:

As an example we will calculate the intersection point of line segments and as shown in image 1.

We will consider the line segments as vectors, this gives us the following vectors.

,

To calculate the intersection point we will first calculate the area of the parallelogram formed by AB and CD as shown in image 2.

The area can be calculated using the cross product of

Calculating the offset on segment

We will now calculate the area below vector as seen in image 3

It can be seen the offset on segment on will be equal to this area divided by the total area calculated earlier. Fortunately we can easily calculate the area by using the area shown in image 4.

The areas shown in image 3 and image 4 are the same. Note that image 4 shows the parallelogram formed by and , we will use the cross-product to calculate it.

We will introduce vector for this.

Now we are almost done. We have both areas (9 and 36) so we can create the offset

which can be simplified to

If we multiply the offset with we find the point on the vector .

Since the line segment does not start on the actual point needs to be moved by

Let’s check whether it is correct

As a general formula:

The offset on can be negative or larger then one. In that case the intersection is on the extension of line segment . This is an advantage of this method.

If vector P1,P2 has been normalized (length 1) the division by the length is obviously not necessary. If not the result is divided the square length. Although divided by the length (square root) the result will equal that of a normalized vector. Keep in mind that the dot project is a projection on the vector thus requiring a additional division by the length!

Calculating the point on the vector can be done using