distance from a point to a plane

Suppose we wish to find the nearest point on a plane to the point ($X\_0,\; Y\_0,\; Z\_0$), where the plane is given by $aX\; +\; bY\; +\; cZ\; =\; D$. We define $x\; =\; X\; -\; X\_0$, $y\; =\; Y\; -\; Y\_0$, $z\; =\; Z\; -\; Z\_0$, and $d\; =\; D\; -\; aX\_0\; -\; bY\_0\; -\; cZ\_0$, to obtain $ax\; +\; by\; +\; cz\; =\; d$ as the plane expressed in terms of the transformed variables. Now the problem has become one of finding the nearest point on this plane to the origin, and its distance from the origin. The point on the plane in terms of the original coordinates can be found from this point using the above relationships between $x$ and $X$, between $y$ and $Y$, and between $z$ and $Z$; the distance in terms of the original coordinates is the same as the distance in terms of the revised coordinates.

The formula for the closest point to the origin may be expressed more succinctly using notation from linear algebra. The expression $ax+by+cz$ in the definition of a plane is a dot product $(a,b,c)\backslash cdot(x,y,z)$, and the expression $a^2+b^2+c^2$ appearing in the solution is the squared norm $|(a,b,c)|^2$. Thus, if $\backslash mathbf\{v\}=(a,b,c)$ is a given vector, the plane may be described as the set of vectors $\backslash mathbf\{w\}$ for which $\backslash mathbf\{v\}\backslash cdot\backslash mathbf\{w\}=d$ and the closest point on this plane to the origin is the vector

:$\backslash mathbf\{p\}=\backslash frac\{\backslash mathbf\{v\}d\}$

In either the coordinate or vector formulations, one may verify that the given point lies on the given plane by plugging the point into the equation of the plane.

To see that it is the closest point to the origin on the plane, observe that $\backslash mathbf\{p\}$ is a scalar multiple of the vector $\backslash mathbf\{v\}$ defining the plane, and is therefore orthogonal to the plane.

Thus, if $\backslash mathbf\{q\}$ is any point on the plane other than $\backslash mathbf\{p\}$ itself, then the line segments from the origin to $\backslash mathbf\{p\}$ and from $\backslash mathbf\{p\}$ to $\backslash mathbf\{q\}$ form a right triangle, and by the Pythagorean theorem the distance from the origin to $q$ is

:$\backslash sqrt\{|\backslash mathbf\{p\}|^2+|\backslash mathbf\{p\}-\backslash mathbf\{q\}|^2\}$.

Since $|\backslash mathbf\{p\}-\backslash mathbf\{q\}|^2$ must be a positive number, this distance is greater than $|\backslash mathbf\{p\}|$, the distance from the origin to $\backslash mathbf\{p\}$.

Alternatively, it is possible to rewrite the equation of the plane using dot products with $\backslash mathbf\{p\}$ in place of the original dot product with $\backslash mathbf\{v\}$ (because these two vectors are scalar multiples of each other) after which the fact that $\backslash mathbf\{p\}$ is the closest point becomes an immediate consequence of the Cauchy–Schwarz inequality.

The vector equation for a hyperplane in $n$-dimensional Euclidean space $\backslash mathbb\{R\}^n$ through a point $\backslash mathbf\{p\}$ with normal vector $\backslash mathbf\{a\}\; \backslash ne\; \backslash mathbf\{0\}$ is $(\backslash mathbf\{x\}-\backslash mathbf\{p\})\backslash cdot\backslash mathbf\{a\}\; =\; 0$ or $\backslash mathbf\{x\}\backslash cdot\backslash mathbf\{a\}=d$ where $d=\backslash mathbf\{p\}\backslash cdot\backslash mathbf\{a\}$.{{cite book|last1=Cheney|first1=Ward|last2=Kincaid|first2=David|title=Linear Algebra: Theory and Applications|date=2010|publisher=Jones & Bartlett Publishers|isbn=9781449613525|pages=450, 451}}

The corresponding Cartesian form is $a\_1x\_1+a\_2x\_2+\backslash cdots+a\_nx\_n=d$ where $d=\backslash mathbf\{p\}\backslash cdot\backslash mathbf\{a\}=a\_1p\_1+a\_2p\_2+\backslash cdots\; a\_np\_n$.

The closest point on this hyperplane to an arbitrary point $\backslash mathbf\{y\}$ is

:$\backslash mathbf\{x\}=\backslash mathbf\{y\}-\backslash left[\backslash dfrac\{(\backslash mathbf\{y\}-\backslash mathbf\{p\})\backslash cdot\backslash mathbf\{a\}\}\{\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{a\}\}\backslash right]\backslash mathbf\{a\}=\backslash mathbf\{y\}-\backslash left[\backslash dfrac\{\backslash mathbf\{y\}\backslash cdot\backslash mathbf\{a\}-d\}\{\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{a\}\}\backslash right]\backslash mathbf\{a\}$

and the distance from $\backslash mathbf\{y\}$ to the hyperplane is

:$\backslash left\backslash |\backslash mathbf\{x\}-\backslash mathbf\{y\}\backslash right\backslash |\; =\; \backslash left\backslash |\backslash left[\backslash dfrac\{(\backslash mathbf\{y\}-\backslash mathbf\{p\})\backslash cdot\backslash mathbf\{a\}\}\{\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{a\}\}\backslash right]\backslash mathbf\{a\}\backslash right\backslash |=\backslash dfrac\{\backslash left|(\backslash mathbf\{y\}-\backslash mathbf\{p\})\backslash cdot\backslash mathbf\{a\}\backslash right|\}\{\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\}=\backslash dfrac\{\backslash left|\backslash mathbf\{y\}\backslash cdot\backslash mathbf\{a\}-d\backslash right|\}\{\backslash left\backslash |\backslash mathbf\{a\}\backslash right\backslash |\}$.

Written in Cartesian form, the closest point is given by $x\_i=y\_i-ka\_i$ for $1\backslash le\; i\backslash le\; n$ where

:$k=\backslash dfrac\{\backslash mathbf\{y\}\backslash cdot\backslash mathbf\{a\}-d\}\{\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{a\}\}=\backslash dfrac\{a\_1y\_1+a\_2y\_2+\backslash cdots\; a\_ny\_n-d\}\{a\_1^2+a\_2^2+\backslash cdots\; a\_n^2\}$,

and the distance from $\backslash mathbf\{y\}$ to the hyperplane is

:$\backslash dfrac\{\backslash left|a\_1y\_1+a\_2y\_2+\backslash cdots\; a\_ny\_n-d\backslash right|\}\{\backslash sqrt\{a\_1^2+a\_2^2+\backslash cdots\; a\_n^2\}\}$.

Thus in $\backslash mathbb\{R\}^3$ the point on a plane $ax+by+cz=d$ closest to an arbitrary point $(x\_1,y\_1,z\_1)$ is $(x,y,z)$ given by

:$\backslash left.\backslash begin\{array\}\{l\}x=x\_1-ka\backslash \backslash y=y\_1-kb\backslash \backslash z=z\_1-kc\backslash end\{array\}\backslash right\backslash \}$

where

:$k=\backslash dfrac\{ax\_1+by\_1+cz\_1-d\}\{a^2+b^2+c^2\}$,

and the distance from the point to the plane is

:$\backslash dfrac\{\backslash left|ax\_1+by\_1+cz\_1-d\backslash right|\}\{\backslash sqrt\{a^2+b^2+c^2\}\}$.

• Distance from a point to a line
• Hesse normal form
• {{section link|Skew lines|Distance}}

{{Reflist}}

{{DEFAULTSORT:Point On Plane Closest To Origin}}

Category:Euclidean geometry

Category:Distance

Category:Planes (geometry)