Line-plane intersection

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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.

[edit] Equations in 3D Euclidean space

[edit] Parametric form

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

$\mathbf{p}_a + (\mathbf{p}_b - \mathbf{p}_a)t, \quad t\in \mathbb{R},$

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

Similarly a plane can be represented as

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

where $\mathbf{p}_k=(x_k,y_k,z_k)$ , $k = 0,1,2$ are three points in the plane.

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

$\mathbf{p}_a + (\mathbf{p}_b - \mathbf{p}_a)t = \mathbf{p}_0 + (\mathbf{p}_1-\mathbf{p}_0)u + (\mathbf{p}_2-\mathbf{p}_0)v$

This can be simplified to

$\mathbf{p}_a - \mathbf{p}_0 = (\mathbf{p}_a - \mathbf{p}_b)t + (\mathbf{p}_1-\mathbf{p}_0)u + (\mathbf{p}_2-\mathbf{p}_0)v,$

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

$\mathbf{p}_a + (\mathbf{p}_b - \mathbf{p}_a)t$

[edit] Usage

If the solution satisfies the condition $t \in [0,1],$ , then the intersection point is on the line between $\mathbf{p}_a$ and $\mathbf{p}_b$ .

If the solution satisfies

$u,v \in [0,1], \;\;\; (u+v) \leq 1,$

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:

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