Intersection Between a Plane and a Segment

$\begin{tikzpicture} % grid \draw[help lines] (-4,-3) grid (5,4); % origin %\draw[red, line width=.1mm] (-0.1,-0.1) -- (0.1,0.1) % (0.1,-0.1) -- (-0.1,0.1); %\coordinate[label={[red]above:$O$}] (O) at (0,0); %\draw[black,line width=5mm] (-3, {3 * sqrt(3) - sqrt(3)}) -- ((-3,0); %\node [square,rotate={30},minimum size=10mm] at (-3, {3 * sqrt(3) - sqrt(3)}) [draw] (d2) [orange,fill,text=white] {$d_{2}$}; %\draw[orange,line width=.1mm] (-3, {3 * sqrt(3) - sqrt(3)}) -- (-2,0); %\draw[orange,line width=.1mm] (-3, {3 * sqrt(3) - sqrt(3)}) -- (0,2); % coordinates \coordinate[label={[black]left:$A$}] (A) at (-1,1.5); \coordinate[label={[black]left:$B$}] (B) at (-2,-1); \coordinate[label={[black]right:$C$}] (C) at (2,-1); \coordinate[label={[black]right:$D$}] (D) at (3,1.5); \coordinate[label={[black]left:$P_{0}$}] (P0) at (-.5,-2); \coordinate[label={[black]right:$P_{1}$}] (P1) at (1,2.5); \coordinate (R) at (-.15,-1); \coordinate[label={[black]right:$I$}] (I) at (.35,.5); \coordinate (N) at (.35, 3); % plane \draw[black, line width=.1mm] (A) -- (B) -- (C) -- (D) -- cycle; % mark points \drawpoint{P0}{.5mm}{black}; \drawpoint{I}{.5mm}{black}; \drawpoint{P1}{.5mm}{black}; % line \draw[black, line width=.1mm] (P0) -- (R); \draw[dotted, black, line width=.1mm] (R) -- (I); \draw[black, line width=.1mm] (I) -- (P1); % normal \draw[->, black, line width=.1mm] (I) -- (N); % brace normal \drawbrace{I}{N}{2mm}{cyan}{$\vec{n}$}{-6mm}{2mm}{} \end{tikzpicture}$

The segment may $P_{0}, P_{1}$ intersect the plane formed by $ABCD$ with the normal $\vec{n}$ in any (or all) generic point $I$ on the segment.

The line equation in parametric form can be expressed as:

$\begin{eqnarray*} P &=& P_{0} + t(P_{1} - P_{0}) \end{eqnarray*}$

where $P$ is the intersection between the segment $P_{0}, P_{1}$ and the plane $ABCD$ and values of $t$ will yield points on that segment.

The equation of the plane can be expressed as:

$\begin{eqnarray*} \vec{n} \cdot P &=& \vec{n} \cdot P_{2} \end{eqnarray*}$

where $P_{2}$ can be any known point in the plane $ABCD$ .

We plug the line equation into the equation of the plane:

$\begin{eqnarray*} \vec{n} \cdot (P_{0} + t(P_{1} - P_{0})) &=& \vec{n} \cdot P_{2} \end{eqnarray*}$

and distribute the normal vector $\vec{n}$ :

$\begin{eqnarray*} \vec{n} \cdot P_{0} + \vec{n}\cdot t(P_{1} - P_{0}) &=& \vec{n} \cdot P_{2} \end{eqnarray*}$

then collect for $t$ :

$\begin{eqnarray*} \vec{n} \cdot t(P_{1} - P_{0}) &=& \vec{n} \cdot P_{2} - \vec{n} \cdot P_{0} \end{eqnarray*}$

then group on the right-hand side of the equation:

$\begin{eqnarray*} \vec{n} \cdot t(P_{1} - P_{0}) &=& \vec{n} \cdot (P_{2} - \cdot P_{0}) \end{eqnarray*}$

and collect again for $t$ we obtain the equation for $t$ :

$\begin{eqnarray*} t &=& \frac{\vec{n} \cdot (P_{2} - \cdot P_{0})}{\vec{n} \cdot (P_{1} - P_{0})} \end{eqnarray*}$

Now, based on the equation, we can make the following judgments:

We know from the vector properties that if $\vec{n} \cdot (P_{1} - P_{0}) = 0$ then the vector $\vec{n}$ and the segment $P_{0}, P_{1}$ are perpendicular meaning that the segment $P_{0}, P_{1}$ is either parallel to the plane formed by $ABCD$ or that the segment $P_{0}, P_{1}$ is contained entirely in the plane described by $ABCD$ .
- If any point on the segment $P_{0}, P_{1}$ is contained in the plane described by $ABCD$ then the entire segment is contained within the plane.
- If any point on the segment $P_{0}, P_{1}$ is contained in the plane described by $ABCD$ then the segment is parallel with the plane described by $ABCD$ .
If $0 \le t \le 1$ then the segment $P_{0}, P_{1}$ interesects the plane.
- If $t = 0$ then the intersection falls on the first end-point.
- If $t = 1$ then the intersection falls on the second end-point.
If $t < 0$ or $t > 1$ then the segment $P_{0}, P_{1}$ does not intersect the plane.
- If $t > 1$ then intersection occurs beyond the second end-point.
- If $t < 0$ then intersection occurs before the first end-point.