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 cross product with this relation:

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

This can be rewriten:

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

The previous equation can be simplified, and we can prove that the point $ P_4 $ belongs to the plane if :

$$ \det ( \vec{P_1P_2} , \: \vec{P_1P_3} , \:\vec{P_1P_4} ) = 0 $$

Proof (references here or here ):

$$ \begin{align} (\vec{u} \times \vec{v}) \times (\vec{u} \times \vec{w}) &= ( (\vec{u} \times \vec{v}) \cdot \vec{w}) \vec{v} - ((\vec{u} \times \vec{v}) \cdot \vec{v}) \vec{w} \\ (\vec{u} \times \vec{v}) \times (\vec{u} \times \vec{w}) &= ( (\vec{u} \times \vec{v}) \cdot \vec{w}) \vec{v} \\ (\vec{u} \times \vec{v}) \times (\vec{u} \times \vec{w}) &= \det ( \vec{u} , \vec{v} , \vec{w} ) \vec{v} \end{align} $$

Conclusion

The four points are coplanar if, and only if:

$$ \det( \: \vec{P_1P_2} \: , \: \vec{P_1P_3} \: , \: \vec{P_1P_4} \: ) = 0 $$

Matlab source code

close all;
clear all;

%% 3 points for the plan
P1=[0;0;0];
P2=[1;0;1];
P3=[0;1;1];

%% Point to check
% Belong
P4=[2;2;4];
% Do not belong
%P4=[2;2;-2];

%% Display points and plane
plot3(0,0,0);
hold on;
grid on;
axis on;
patch ( [P1(1), P2(1), P4(1), P3(1)], [P1(2), P2(2), P4(2), P3(2)], [P1(3), P2(3), P4(3), P3(3)], 'red' );
plot3 (P1(1), P1(2), P1(3), '.', 'MarkerSize', 50);
plot3 (P2(1), P2(2), P2(3), '.', 'MarkerSize', 50);
plot3 (P3(1), P3(2), P3(3), '.', 'MarkerSize', 50);
plot3 (P4(1), P4(2), P4(3), '.', 'MarkerSize', 50);

%% Method 1: cross product
C = cross (cross(P2-P1, P3-P1), cross(P2-P1, P4-P1));
if (any(C)) 
    disp('P4 does not belong to the plane (cross product)');
else
    disp('P4 belongs to the plane (cross product)');
end;

%% Method 2: determinant 
D = det( [P2-P1, P3-P1, P4-P1]);
if (any(D)) 
    disp('P4 does not belong to the plane (determinant)');
else
    disp('P4 belongs to the plane (determinant)');
end;

How to check if four points are coplanar?

Introduction

Proof

Conclusion

Matlab source code

See also