Creating a rectangular patch using random numbers

Hi I am currently producing four random points in a 3D space. With these points I would like to create a rectangular patch which i can then use to find the vectors between them, then the normal and finally the plane equation. However my problem with this is that I need the patch to be a quadrilateral with parallel lines. When using three points it is fine as all three points are within the same plane however when using four points. One of my points is always off. How can I fix this.

An example of what I mean is provided below.

Answer 1

[Les Beckham]

The chances of four random points in 3d being coplanar are virtually zero. The chances of them forming a parallelogram are infinitesimal. The first 3 points can be random, but if you want a parallelogram, you will have to calculate the fourth point from the first three points

p = rand(3,3); % create three random points in 3d
p(:,4) = p(:,3) + (p(:,1) - p(:,2)); % calculate the 4th point to form a parallelogram
p(:,5) = p(:,1); % duplicate the first point to form a closed shape

plot3(p(1,:), p(2,:), p(3,:), 'o-')
patch(p(1,:), p(2,:), p(3,:), 'g', 'FaceAlpha', 0.3)

text(p(1,1), p(2,1), p(3,1), 'p1') % label the points
text(p(1,2), p(2,2), p(3,2), 'p2')
text(p(1,3), p(2,3), p(3,3), 'p3')
text(p(1,4), p(2,4), p(3,4), 'p4')
grid on

[Tishan Anantharajah]

Ok thank you this is what I was thinking as well but wanted confirmation.

[Les Beckham]

You are quite welcome.

Answer 2

[chicken vector]

You can't really fix this.

A plane is univoquely defined by 3 points in space, so, most-probably, a fourth point won't lie on that plane and no bi-dimensional figure can match your requirements.

If you really want to pull-off some graphics you can plot every possible triangular patch as follows:

nPoints = 4;
dims = 3;
patchColor = [.8 .8 .8];

points = rand(dims,nPoints);
polyData = struct;

figure;
hold on;
for p = 1 : nPoints
    scatter3(points(1,1),points(2,1),points(3,1),50,'k','filled');
    polyData(p).patch = patch(points(1,1:3),points(2,1:3),points(3,1:3),patchColor,'FaceAlpha',0.3);
    points = circshift(points,1,2);
end
hold off;
view([1,1,1])

polyData(1).patch

ans = 
  Patch with properties:

    FaceColor: [0.8000 0.8000 0.8000]
    FaceAlpha: 0.3000
    EdgeColor: [0 0 0]
    LineStyle: '-'
        Faces: [1 2 3]
     Vertices: [3×3 double]

  Show all properties

This way you can access patch properties.

Notice that if you are looking ofr the vector connecting two points, you can just do the difference.

point1 = [3 4]';
point2 = [10 0]';
vector = point2 - point1

vector = 2×1

7 -4

versor = vector/norm(vector)

versor = 2×1

0.8682 -0.4961