Checking if Line and Plane Intersect

The post is compiled with information on Wikipedia page on Line-plane Interaction and this notes on finding the normal to a plane.

What is the question?

This post is about checking if a line intersect with a plane, given 2 different coordinates on the line and 3 different coordinates on the plane.

How to solve it?

Suppose having a line with coordinates $\mathbf{a}$ and $\mathbf{b}$ on the line, the vector equation representing the line with the set of $\mathbf{p}$ consist the line is: \begin{equation} \label{eq:line} \mathbf{p} = \mathbf{a} + d \mathbf{(b - a)} \end{equation}

Suppose having a plane with coordinates $\mathbf{e}$, $\mathbf{f}$ and $\mathbf{g}$ on the plane, then the normal to the plane is $\mathbf{n = (f - e) \times (g - e)}$. If $\mathbf{p_0}$ is a point on the plane (e.g. $\mathbf{p_0}$ can be $\mathbf{e}$, $\mathbf{f}$ or $\mathbf{g}$), the plane can be expressed as the set of points $\mathbf{p}$ for which: \begin{equation} \label{eq:plane} \mathbf{(p - p_0) \cdot n} = 0 \end{equation}

The point(s) where the line and the plane intersect, the points have the same coordinates. Substitute \eqref{eq:line} into \eqref{eq:plane}:

$$\begin{align*} ((\mathbf{a} + d \mathbf{(b - a)}) - \mathbf{p_0}) \cdot \mathbf{n} &= 0\\ d \mathbf{(b - a)} \cdot \mathbf{n} + \mathbf{(a - p_0)} \cdot \mathbf{n} &= 0\\ d &= \frac{\mathbf{(p_0 - a)} \cdot \mathbf{n}}{\mathbf{(b - a)} \cdot \mathbf{n}} \end{align*}$$

If $\mathbf{(b - a)} \cdot \mathbf{n} = 0$ then the line and the plane are parallel. In this case, if $\mathbf{(p_0 - a)} \cdot \mathbf{n} = 0$ then the line is on the plane. Otherwise the line and plane have no intersection.

If $\mathbf{(b - a)} \cdot \mathbf{n} \neq 0$ there is a single point of intersection.