Math Relative position of planes Point and triangle

# Point in a triangle or parallelogram

You can also check whether a point is in the triangle or in the parallelogram. However, this only works with the parametric form.

i

### Method

1. Set up the parametric equation of the plane
2. Point on the plane: calculate $r$ and $s$
3. Check conditions for $r$ and $s$

The parametric equation is set up using the two vectors which make of the triangle or parallelogram.

With triangle ABC:

$\text{E: } \vec{x} = \vec{OA} + r \cdot \vec{AB}$ $+ s \cdot \vec{AC}$

And with the ABCD parallelogram:

$\text{E: } \vec{x} = \vec{OA} + r \cdot \vec{AB}$ $+ s \cdot \vec{AD}$
!

### Remember

The conditions must apply to the parallelogram:
1. $0\le r\le1$
2. $0\le s\le1$

There is another condition for the triangle:
1. $0\le r+s \le1$

### Example

Is the point $P(-0.5|1|1)$ in the triangle ABC with $A(0|1|0)$, $B(0|0|2)$ and $C(-2|2|2)$?

1. #### Set up parametric equation

A parametric equation is set up with the 3 points of the triangle.

$\text{E: } \vec{x} = \vec{OA} + r \cdot \vec{AB}$ $+ s \cdot \vec{AC}$

$\vec{x}=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + r \cdot \begin{pmatrix} 0 \\ -1 \\ 2 \end{pmatrix}$ $+ s \cdot \begin{pmatrix} -2 \\ 1 \\ 2 \end{pmatrix}$

2. #### Insert point

The position vector of $P$ is used for $\vec{x}$ in $E$.

$\begin{pmatrix} -0.5 \\ 1 \\ 1 \end{pmatrix}$ $=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + r \cdot \begin{pmatrix} 0 \\ -1 \\ 2 \end{pmatrix}$ $+ s \cdot \begin{pmatrix} -2 \\ 1 \\ 2 \end{pmatrix}$

Now we set up an equation system and solve it. Every line is an equation.
1. $-0.5=-2s$
2. $1=1-r+s$
3. $1=2r+2s$

From I. one obtains $s=\frac14$, which is used in II..

$1=1-r+\frac14\quad|-1$
$0=-r+\frac14\quad|+r$
$r=\frac14$

$r$ and $s$ are inserted in III..

$1=2\cdot\frac14+2\cdot\frac14$
$1=1$

=> Point lies on the plane

3. #### Check conditions

To find out whether the point lies in a triangle, the conditions for $r$ and $s$ must be checked.

1. $0\le \frac14\le1$
2. $0\le \frac14\le1$
3. $0\le \frac14+\frac14 \le1$

All conditions apply.
=> P lies in the triangle ABC.