Find center of a plane and project center point perpendicular to plane

For instance, lets say I have 4 data points (each with XYZ coordinates). I want to make a 3 dimensional planar surface out of these set of points and find the center of this plane. Then, subsequently be able to project this center perpendicular to the plane by a variable distance.

Any suggestions?

[Walter Roberson]

Center of the plane is just the centroid of the points.

But you have problems if the points are not coplanar.

Answer 1

[Fabio Freschi]

As already pointed out by Walter Roberson, 4 points cannot be coplanar. Assuming that they are, you may try this

clear variables, close all

% three random points in 3d
P = rand(3,3);

% vectors
V1 = P(2,:)-P(1,:);
V2 = P(3,:)-P(1,:);

% add one point on the plane defined by the previous three points
P = [P; P(1,:)+V1+V2];

% plot points
figure, hold on, axis equal
plot3(P(:,1),P(:,2),P(:,3),'o');
quiver3(P(1,1),P(1,2),P(1,3),V1(1),V1(2),V1(3),'AutoScale','off')
quiver3(P(1,1),P(1,2),P(1,3),V2(1),V2(2),V2(3),'AutoScale','off')

% plane center
B0 = mean(P,1);

% plot
plot3(B0(1),B0(2),B0(3),'*');

% orthogonal vector
W = cross(V2,V1);

% unit vector
U = W/sqrt(sum(W.^2,2));

% project B by the distance k
k = 2;
B = B0+k*U;

% plot
quiver3(B0(1),B0(2),B0(3),k*U(1),k*U(2),k*U(3),'AutoScale','off');
plot3(B(1),B(2),B(3),'s');

Basically two vectors are created from the first three points, assuming the first point as reference. The orthogonal direction is obtained with a cross product of these vectors. Note that you can control the positive side by changing the order of the cross product.

[Hannah Weiss]

Thank you for the help and for the thoughful explainations! I really appreciate it.