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),
and the constant
| [画像: 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 DistanceExplore with Wolfram|Alpha
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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 FormCite this as:
Weisstein, Eric W. "Hessian Normal Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HessianNormalForm.html