Line-plane Intersection - Parametric Form

Parametric Form

A line is described by all points that are a given direction from a point. Thus a general point on a line can be represented as

where and are two distinct points along the line.

Similarly a general point on a plane can be represented as

where, are three points in the plane which are not co-linear.

The point at which the line intersects the plane is therefore described by setting the point on the line equal to the point on the plane, giving the parametric equation:

This can be simplified to

which can be expressed in matrix form as:

The point of intersection is then equal to

If the line is parallel to the plane then the vectors, and will be linearly dependent and the matrix will be singular. This situation will also occur when the line lies in the plane.

If the solution satisfies the condition, then the intersection point is on the line between and .

If the solution satisfies

then the intersection point is in the plane inside the triangle spanned by the three points, and .

This problem is typically solved by expressing it in matrix form, and inverting it: