How to check if four points are coplanar

Introduction

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 \( P_3 \)“.

Proof

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:

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

Conclusion

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

See also


Last update : 11/26/2018