Line Intersecting a Plane

One of the next tasks that I set myself was to write a little demo whereby the intersection of a line with a plane is detected and responded to.

In order to test this, I first used OpenGL to draw a line in space as well as a triangle that could move along the line with the use of the keyboard.

All points on the triangle are coplanar with the plane that is being tested to see whether the line intersects. The goal is to have the colour of the line turn to red once the line is intersecting the plane that the triangle lies on.

After scouring the web, the proved to be a fairly simple process but one that required a few mathematical calculations. The main key to solving the problem is through the use of the equation of a plane:

Where (A,B,C) represents the Normal vector for the plane, (x,y,z) are the coordinates for any point on the plane and D is the distance of the plane from the origin. This equation will evaluate to 0 for any point that lies on the plane. A point not on the plane will yield a result that represents the nearest distance from the point to the plane. If the value is positive, it means the point lies ahead of the plane (in the direction the plane normal points) and if the value is negative, it means the point is behind the plane.

In order to know whether the line is intersecting the plane, we plug the coordinates for the endpoints of the line and plug them into the equation of the plane. By inspecting the distance from the plane of each point, we can determine whether it is intersecting the plane. In order to do this, we need to be able to use the equation for the plane. In order to do that, we need two pieces of information:

• The normal of the plane
• The distance of the plane from the origin (D).

Calculating the Normal of the Plane

The normal of a plane is a vector that is perpendicular to the plane. To make calculations easier, we also normalize the vector, that is making it of length 1. The image below (credit: Wikipedia), shows two normals for a polygon.

In order to calculate the normal for a polygon or plane, we make use of the cross product operation for vectors which takes 2 vectors and returns a vector which is orthogonal to both. This is exactly what we require. Having performed the cross product (and normalizing!), we now have the normal to the plane the coordinates of which are the A, B and C values used in the general equation for a plane. Now we only need D.

Calculating the Distance from the Origin (D)

Once the normal for the plane has been calculated, we need D which is the distance the plane lies from the origin. This is done by simply rearranging the general equation for the plane.

For the (x,y,z) vector, we can use any point on the plane as it would evaluate to 0, allowing us to perform this rearrangement. For the demo, any points used to draw the triangle would suffice.

Checking Line Intersection

We can now calculate a point's distance from a plane using the general equation. How can we use this to check whether a line is intersecting the plane? Well, if the line is intersecting the plane, one of the points will be in front of the plane and one behind. If this is the case, one of the values for the general equation will be positive and the other negative. In order to check whether this is the case, simply substituting the coordinates for each endpoint of the line as the (x,y,z) vector in the general equation and then multiplying the results together. If the line is indeed intersecting the plane, the multiplication will result in a negative number.

Checking for this intersection and drawing the line in the appropriate colour means we are finished!