Point-Plane Distance

LINK: http://mathworld.wolfram.com/Point-PlaneDistance.html
Point-Plane Distance

Given a plane


and a point x_0=(x_0,y_0,z_0), the normal to the plane is given by

 v=[a; b; c],

and a vector from the plane to the point is given by

 w=-[x-x_0; y-y_0; z-z_0].

Projecting w onto v gives the distance D from the point to the plane as

D = |proj_(v)w|
= (|v·w|)/(|v|)
= (|a(x-x_0)+b(y-y_0)+c(z-z_0)|)/(sqrt(a^2+b^2+c^2))
= (|ax+by+cz-ax_0-by_0-cz_0|)/(sqrt(a^2+b^2+c^2))
= (|ax_0+by_0+cz_0+d|)/(sqrt(a^2+b^2+c^2)).

Dropping the absolute value signs gives the signed distance,


which is positive if x_0 is on the same side of the plane as the normal vector v and negative if it is on the opposite side.

This can be expressed particularly conveniently for a plane specified in Hessian normal form by the simple equation


where n^^=v/|v| is the unit normal vector. Therefore, the distance of the plane from the origin is simply given by p (Gellert et al. 1989, p. 541).

Given three points x_i for i=1, 2, 3, compute the unit normal


Then the distance from a point x_0 to the plane containing the three points is given by


where x_i is any of the three points. Expanding out the coordinates shows that


as it must since all points are in the same plane, although this is far from obvious based on the above vector equation.

SEE ALSO: Hessian Normal Form, Plane, Point, Projection Theorem


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


Weisstein, Eric W. "Point-Plane Distance." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Point-PlaneDistance.html

posted on 2009-03-24 17:18 zmj 阅读(725) 评论(0)  编辑 收藏 引用

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