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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

 ax+by+cz+d=0
(1)

by defining the components of the unit normal vector n^^=(n_x,n_y,n_z),

n_x=a/(sqrt(a^2+b^2+c^2))
(2)
n_y=b/(sqrt(a^2+b^2+c^2))
(3)
n_z=c/(sqrt(a^2+b^2+c^2))
(4)

and the constant

 p=d/(sqrt(a^2+b^2+c^2)).
(5)

Then the Hessian normal form of the plane is

 n^^·x=-p,
(6)

and p is the distance of the plane from the origin (Gellert et al. 1989, pp. 540-541). Here, the sign of p determines the side of the plane on which the origin is located. If p>0, it is in the half-space determined by the direction of n^^, and if p<0, it is in the other half-space.

The point-plane distance from a point x_0 to a plane (6) is given by the simple equation

 D=n^^·x_0+p
(7)

(Gellert et al. 1989, p. 541). If the point x_0 is in the half-space determined by the direction of n^^, then D>0; if it is in the other half-space, then D<0.


See also

Plane, Point-Plane Distance

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References

Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, pp. 539-543, 1989.

Referenced on Wolfram|Alpha

Hessian Normal Form

Cite this as:

Weisstein, Eric W. "Hessian Normal Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HessianNormalForm.html

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