Equation for a line through a plane in homogeneous coordinates.

Definition

You are right, 3D points and planes are described with 4 homogeneous coordinates. A point at $\vec{r}$ is $P=(\vec{r};1)$ and a plane $W=(\vec{n};-d)=(\vec{n};-\vec{r}\cdot\vec{n})$ with normal $\vec{n}$ through point $\vec{r}$, or with minimum distance to origin $d$.

A line needs 6 coordinates (plücker coordinates) describing the direction and moment about the axis. A line along $\vec{e}$ through a point $\vec{r}$ has coordinates $L=[\vec{e};\vec{r}\times\vec{e}]$. Given a line $L=[\vec{l};\vec{m}]$, the direction is recovered by $\vec{e}=\frac{\vec{l}}{|\vec{l}|}$ and the position by $\vec{r} = \frac{\vec{l}\times\vec{m}}{|\vec{l}|^2}$

Now derive the point $P=(\vec{r};1)$ where line $L=[\vec{l};\vec{m}]$ meets plane $W=(\vec{w};\epsilon)$ as follows:

• See that for the point to be on the plane you must have $\epsilon = - \vec{r}\cdot \vec{w}$
• For the point to be on the line you must have $\vec{m} = \vec{r} \times \vec{l}$
• Use the vector triple product to get $$ \vec{w} \times \vec{m} = \vec{w} \times \left( \vec{r} \times \vec{l} \right) = \vec{r} (\vec{w}\cdot\vec{l})-\vec{l}(\vec{w}\cdot\vec{r}) $$

$$ \vec{w} \times \vec{m} = \vec{r} (\vec{w}\cdot\vec{l}) - \vec{l}(-\epsilon) $$

$$ \vec{r} = \frac{\vec{w}\times\vec{m}-\epsilon \vec{l}}{\vec{w}\cdot\vec{l}} $$

• Define the line-plane meet operator as

$$ \begin{aligned} P & = [W\times] L \\ \begin{pmatrix} \vec{p} \\ \delta \end{pmatrix} & = \begin{bmatrix} -\epsilon {\bf 1} & \vec{w}\times \\ \vec{w}^\top & 0 \end{bmatrix} \begin{pmatrix} \vec{l} \\ \vec{m} \end{pmatrix} \end{aligned}$$

where $\vec{w}\times = \begin{pmatrix}x\\y\\z\end{pmatrix} \times = \begin{bmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{bmatrix}$ is the cross product matrix operator in 3×3 form.

• The meet operator $[W\times]$ has dimensions 4×6 to work between lines and points.

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Example

• A plane normal to the $x$ axis located at $x=3$ has coordinates $W=(1,0,0;-3)$
• A line through $y=2$ directed towards $\hat{i}+\hat{k}$ has coordinates $L=[1,0,1;2,0,-2]$
• The meet operator is $$ [W\times] = \left[ \begin{array}{ccc|ccc} 3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 & -1 \\ 0 & 0 & 3 & 0 & 1 & 0 \\ \hline 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$$
• The point where the line meets the plane is $P=[W\times]L$ $$P=\left[ \begin{array}{ccc|ccc} 3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 & -1 \\ 0 & 0 & 3 & 0 & 1 & 0 \\ \hline 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \begin{bmatrix}1\\0\\1\\ \hline 2 \\ 0 \\ -2 \end{bmatrix} =\begin{pmatrix}3\\2\\3\\ \hline 1 \end{pmatrix}$$
• The point is located at $\vec{r} = (3,2,3)$