Direction that is perpendicular to a collection of 3D points?

Question

I have a collection of 3D points which form an imperfect circle and are stored in the order in which they appear in the circle. I'm positioning an object in the centre of the ring by calculating the mean position of all of the points, which works fine. Now, what I want is for the object in the centre to be facing up/down relative to the rest of the points (i.e perpendicular to the ring).

I've included an image to help clarify what I mean. Does anyone know of an algorithm that would be suitable for this?

Answer 1

You have to compute the plane that your points form and get its normal.

If the points are perfectly coplanar, just get three of them, a, b, and c, and compute two vectors. The normal vector n is the cross product of them:

v1 = b - a;
v2 = c - a;
n = v1 x v2;

If the points are not perfectly coplanar, you can get the plane that best fits the points and then, its normal. You can get the plane by solving a linear equation system of the form Ax=0. Since the general equation of a plane is Ax + By + Cz + D = 0, you get one equation per 3D point, obtaining this system:

| x1 y1 z1 1 |   | A |   |  0  |
| x2 y2 z2 1 | x | B | = |  0  |
| x3 y3 z3 1 |   | C |   |  0  |
|    ...     |   | D |   | ... |
| xn yn zn 1 |           |  0  |

The normal vector is (A, B, C).