Line-plane intersection

In analytic geometry, the intersection of a line and a plane can be the empty set, a point, or a line. Distinguishing these cases, and determining equations for the point and line in the latter cases have use, for example, in computer graphics, motion planning, and collision detection.

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

$\mathbf{l}_a %2B (\mathbf{l}_b - \mathbf{l}_a)t, \quad t\in \mathbb{R},$

where $\mathbf{l}_a=(x_a, y_a, z_a)$ and $\mathbf{l}_b=(x_b, y_b, z_b)$ are two distinct points along the line.

Similarly a general point on a plane can be represented as

$\mathbf{p}_0 %2B (\mathbf{p}_1-\mathbf{p}_0)u %2B (\mathbf{p}_2-\mathbf{p}_0)v, \quad u,v\in\mathbb{R}$

where $\mathbf{p}_k=(x_k,y_k,z_k)$ , $k=0,1,2$ 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:

$\mathbf{l}_a %2B (\mathbf{l}_b - \mathbf{l}_a)t = \mathbf{p}_0 %2B (\mathbf{p}_1-\mathbf{p}_0)u %2B (\mathbf{p}_2-\mathbf{p}_0)v$

This can be simplified to

$\mathbf{l}_a - \mathbf{p}_0 = (\mathbf{l}_a - \mathbf{l}_b)t %2B (\mathbf{p}_1-\mathbf{p}_0)u %2B (\mathbf{p}_2-\mathbf{p}_0)v,$

which can be expressed in matrix form as:

$\begin{bmatrix} x_a - x_0 \\ y_a - y_0 \\ z_a - z_0 \end{bmatrix} = \begin{bmatrix} x_a - x_b & x_1 - x_0 & x_2 - x_0 \\ y_a - y_b & y_1 - y_0 & y_2 - y_0 \\ z_a - z_b & z_1 - z_0 & z_2 - z_0 \end{bmatrix} \begin{bmatrix} t \\ u \\ v \end{bmatrix}$

The point of intersection is then equal to

$\mathbf{l}_a %2B (\mathbf{l}_b - \mathbf{l}_a)t$

If the line is parallel to the plane then the vectors $\mathbf{l}_b - \mathbf{l}_a$ , $\mathbf{p}_1-\mathbf{p}_0$ , and $\mathbf{p}_2-\mathbf{p}_0$ 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 $t \in [0,1],$ , then the intersection point is on the line between $\mathbf{l}_a$ and $\mathbf{l}_b$ .

If the solution satisfies

$u,v \in [0,1], \;\;\; (u%2Bv) \leq 1,$

then the intersection point is in the plane inside the triangle spanned by the three points $\mathbf{p}_0$ , $\mathbf{p}_1$ and $\mathbf{p}_2$ .

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

$\begin{bmatrix} t \\ u \\ v \end{bmatrix} = \begin{bmatrix} x_a - x_b & x_1 - x_0 & x_2 - x_0 \\ y_a - y_b & y_1 - y_0 & y_2 - y_0 \\ z_a - z_b & z_1 - z_0 & z_2 - z_0 \end{bmatrix}^{-1} \begin{bmatrix} x_a - x_0 \\ y_a - y_0 \\ z_a - z_0 \end{bmatrix}.$

Algebraic form

In vector notation, a plane can be expressed as the set of points $\mathbf{p}$ for which

$(\mathbf{p}-\mathbf{p_0})\cdot\mathbf{n} = 0$

where $\mathbf{n}$ is a normal vector to the plane and $\mathbf{p_0}$ is a point on the plane. The vector equation for a line is

$\mathbf{p} = d\mathbf{l} %2B \mathbf{l_0}$

where $\mathbf{l}$ is a vector in the direction of the line and $\mathbf{l_0}$ is a point on the line. Substitute the line into the plane equation to get

$(d \mathbf{l} %2B \mathbf{l_0} - \mathbf{p_0})\cdot\mathbf{n} = 0$

Distribute $\mathbf{n}$ to get

$d \mathbf{l}\cdot\mathbf{n} %2B (\mathbf{l_0}-\mathbf{p_0})\cdot\mathbf{n} = 0$

And solve for $d$

$d = {(\mathbf{p_0}-\mathbf{l_0})\cdot\mathbf{n} \over \mathbf{l}\cdot\mathbf{n}}$

If the line starts outside the plane and is parallel to the plane, there is no intersection. In this case, the above denominator will be zero and the numerator will be non-zero. If the line starts inside the plane and is parallel to the plane, the line intersects the plane everywhere. In this case, both the numerator and denominator above will be zero. In all other cases, the line intersects the plane once and $d$ represents the intersection as the distance along the line from $\mathbf{l_0}$ .

Uses

In the ray tracing method of computer graphics a surface can be represented as a set of pieces of planes. The intersection of a ray of light with each plane is used to produce an image of the surface. In vision-based 3D reconstruction, a subfield of computer vision, depth values are commonly measured by so-called triangulation method, which finds the intersection between light plane and ray reflected toward camera.

The algorithm can be generalised to cover intersection with other planar figures, in particular, the intersection of a polyhedron with a line.