The aim of this article is to check if four points are coplanar, i.e. they lie
on the same plane. Let’s consider four points \( P_1 \), \( P_2 \), \( P_3 \)
and \( P_4 \) defined in \( \mathbb{R}^3 \). This question may be reformulated as
*“is the point \(P_4\) belongs to the plane defined by points \( P_1 \), \( P_2 \) and \( P_3 \)“*.

First, let’s compute the normal vector to the plane defined by points \( P_1 \), \( P_2 \) and \( P_3 \):

$$ \vec{n_1}=\vec{P_1P_2} \times \vec{P_1P_3} $$

Let’s now compute the normal vector to the plane defined by points \(P_1\), \(P_2\) and \(P_4\):

$$ \vec{n_2}=\vec{P_1P_2} \times \vec{P_1P_4} $$

If the points lie on the same plane, \( \vec{n_1} \) and \( \vec{n_2} \) are colinear and this can be check thanks to the dot product with this relation:

$$ \vec{n_1} \cdot \vec{n_2} =0 $$

This can be rewriten:

$$ (\vec{P_1P_2} \times \vec{P_1P_3}) \cdot (\vec{P_1P_3} \times \vec{P_1P_4}) = 0 $$

The previous equation can be simplified:

$$ \vec{P_1P_2} \cdot (\vec{P_1P_2} \times \vec{P_1P_4}) =0 $$

The four points are coplanar if, and only if \( \vec{P_1P_2} \cdot (\vec{P_1P_3} \times \vec{P_1P_4}) =0 \).

- Calculating the transformation between two set of points
- Check if a point belongs on a line segment
- Cross product
- Common derivatives rules
- Common derivatives
- Dot product
- Common integrals
- Sines, cosines and tangeantes of common angles
- Singular value decomposition (SVD) of a 2×2 matrix
- Understanding covariance matrices

Last update : 11/26/2018