How to find the coeffecients(a,b,c,d) of a planar polygon

Question

Given the normal of the plane,Centroid of the plane,the bais vectors,and some(>4) points on the plane , i want to find out the co-effecients(a,b,c,d) of the planar polygon. Is there a better way than substitute points in the plane equation ax + by + cz + d = 0.

Thanks, Harsha.

Answer 1

The equation will be

n_x x + n_y y + n_z z + d = 0

where N[n_x, n_y, n_z] is the normal vector. Then you can substitute any point B(b_x, b_y, b_z) known to be on the plane to solve for d,

d = -( n_x b_x + n_y b_y + n_z b_z )

Why does this work? Let P(x,y,z) be an arbitrary point in the plane. Then the vector P-B must be parallel to the plane and perpendicular to its normal. The dot product of perpendiculars is zero. Consequently,

N dot (P -  B) = (N dot P - N dot B) 
    = n_x x + n_y y + n_z z - (n_x b_x + n_y b_y + n_z b_z) = 0

In the last line you can recognize

a = n_x   b = n_y    c = n_z   d = -(n_x b_x + n_y b_y + n_z b_z)

as already stated.