Hessian Normal Form

It is especially convenient to specify planes in so-called Hessian normal form. This is obtained from the general equation of a plane

(1)

by defining the components of the unit normal vector ,

			(2)
			(3)
			(4)

and the constant

(5)

Then the Hessian normal form of the plane is

(6)

and is the distance of the plane from the origin (Gellert et al. 1989, pp. 540-541). Here, the sign of determines the side of the plane on which the origin is located. If , it is in the half-space determined by the direction of , and if , it is in the other half-space.

The point-plane distance from a point to a plane (6) is given by the simple equation

(7)

(Gellert et al. 1989, p. 541). If the point is in the half-space determined by the direction of , then ; if it is in the other half-space, then .

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