It is especially convenient to specify planes in so-called Hessian normal form. This is obtained from the general equation of a plane
(1)
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by defining the components of the unit normal vector ,
(2)
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(3)
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(4)
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and the constant
(5)
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Then the Hessian normal form of the plane is
(6)
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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)
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(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 .