How to compute normal vector to least square plane in PovRay only

Question

Is it possible to calculate normal vector to a plane defined by set of points using PovRay only (proper set has more than 3 points)? At the moment I'm using external program that computes via Jacobi eigenvalues of a least square plane.

Still it would be nice not to have to switch for this step to different program but just to use internal procedures of PovRay.

Kris

Answer 1

Here, I assume that the position of the plane is known and that you want to calculate the normal vector. I do not answer the question how this plane is calculated in the first place.

The normal vector to a plane defined by three points (e.g. A, B, C) in (3D) space can be calculated with the cross product of two vectors a=A-C and b=B-C (in the graph below, point A and B would be at the tips of the arrows a and b, respectively. Point C is at the starting point of the two vectors). I also assume that the points are not in a line.

The resulting vector is perpendicular to a and b, therefore it is normal to the given plane.

To get a vector of length 1, you would have to divide it by its length. Now to the question: In POVRay, I assume you have the coordinates in some variables. Then (omitting any #declare statements) you calculate the normal vector (the inices 1,2,3 correspond to the components in x,y,z direction):

n1 = a2*b3 - a3*b2;

n2 = a3*b1 - a1*b3;

n3 = a1*b2 - a2*b1;

The length L of vector n is L=sqrt(n1*n1 + n2*n2 + n3*n3). Now you divide each component by L and you have the normal vector of unit length.

The direction of the normal vector depends on how the three points are arranged on the plane. Counter-clockwise means that the normal vector is directed upwards (or outwards) from the plane, and vice versa. I used this to check whether a surface is visible or not (i.e. its normal vector points towards the point where the camera is located or away from the camera).