A plane in 3D is determined by three non-collinear points, or a point and a normal, or a point and two directions within the plane.

General form: ax + by + cz = d, where (a,b,c) is the normal direction. This has 3 effective parameters (ratios a:b:c and the value d/sqrt(a^{2+b}^{2+c}^2)).

Normal form: lx + my + nz = p, where (l,m,n) is the unit outward normal and p >= 0 is the distance from origin.

Intercept form: x/a + y/b + z/c = 1, where a, b, c are the x, y, z intercepts respectively.

Point-normal form: a(x-x1) + b(y-y1) + c(z-z1) = 0, the plane through (x1,y1,z1) with normal (a,b,c).

Three-point form: The plane through P1, P2, P3 can be found using the determinant |x-x1 y-y1 z-z1; x2-x1 y2-y1 z2-z1; x3-x1 y3-y1 z3-z1| = 0.

Family of planes through the intersection of P1=0 and P2=0 is P1 + lambda*P2 = 0. This one-parameter family covers all planes through the line of intersection (except P2 itself, obtained as lambda -> infinity).

Plane containing a line $\frac{x-x1}{a}$=$\frac{y-y1}{b}$=$\frac{z-z1}{c}$: requires (1) the point (x1,y1,z1) on the plane, and (2) the direction (a,b,c) perpendicular to the normal, i.e., aA+bB+c*C=0 where (A,B,C) is the plane normal.

Angle bisector planes of P1=0 and P2=0: $\frac{a1x+b1y+c1z-d1}{sqrt}$(a1^{2+b1}^{2+c1}^2) = +/- $\frac{a2x+b2y+c2z-d2}{sqrt}$(a2^{2+b2}^{2+c2}^2).