calculate the normal of a 3D plane

Hi! I have a circular plane, whose coordinates of the points of the circumference are:

P = importdata("node_plane.mat");
plot3(P(:,1),P(:,2),P(:,3))

I would like to know if it is correct to calculate the normal of this plane in the following way:

N = cross(P(1,:) - P(2,:), P(3,:) - P(2,:));
N = N/norm(N);

Answer 1

[Torsten]

I would like to know if it is correct to calculate the normal of this plane in the following way:

N = cross(P(1,:) - P(2,:), P(3,:) - P(2,:));
N = N/norm(N);

If you are sure that all points lie in exactly one plane: yes, you can choose any three points and do as you did above.

If you first need to fit a plane to your data points: no.

[Alberto Acri]

Thanks! To be sure that all points (or possibly which points) lie exactly on a plane how can one know?

[Torsten]

Insert all points P in the expression

N*(P-P(1,:))'

and hope that the result is 0 in all cases.

Since it is not: the best fit plane is N'*(P-C)' = 0 with N and C as below.

P = importdata("node_plane.mat");
n = size(P,1);
C = sum(P,1)/n

C = 1x3

-28.9943 -3.2320 44.2401

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

Pn = P-C;
[U, S, V] = svd(Pn);
N = V(:,3)

N = 3x1

-0.0439 -0.1877 0.9813

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

sum(abs(N'*Pn'))   %Sum of distances between plane and points given in node_plane.mat

ans = 4.3799e-12

[Alberto Acri]

Okay, I am not very familiar with it. So the best normal that can be determined with the 'node_plane' points is the one indicated in your code with N? Correct?

[Torsten]

Yes, N is the normal and C is the centroid of your points, and the best-fit plane is N'*(x-C)' = 0.