Distance from a point to a line: Difference between revisions

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InThe [[Euclidean'''distance''' geometry]], the(or '''perpendicular distance''') '''from a point to a line''' is the shortest [[Euclidean distance|distance]] from a givenfixed [[Point (geometry)|point]] to any point on ana fixed infinite [[Line (mathematics)|straight line]]. It is thein [[perpendicularEuclidean distancegeometry]]. ofIt the point to the line,is the length of the [[line segment]] which joins the point to nearestthe pointline onand is [[perpendicular]] to the line. The [[algebraic expression]]formula for calculating it can be derived and expressed in several ways.

{{Short description|Problem in coordinate geometry}}

In [[Euclidean geometry]], the '''distance from a point to a line''' is the shortest [[Euclidean distance|distance]] from a given [[Point (geometry)|point]] to any point on an infinite [[Line (mathematics)|straight line]]. It is the [[perpendicular distance]] of the point to the line, the length of the [[line segment]] which joins the point to nearest point on the line. The [[algebraic expression]] for calculating it can be derived and expressed in several ways.

Knowing the shortest distance from a point to a line can be useful in various situations—for example, finding the shortest distance to reach a road, quantifying the scatter on a graph, etc. In [[Deming regression]], a type of linear [[curve fitting]], if the dependent and independent variables have equal variance this results in [[orthogonal regression]] in which the degree of imperfection of the fit is measured for each data point as the perpendicular distance of the point from the regression line.

==Cartesian coordinates==

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[[File:PointToLineDistance v2.png|thumb|''This '''picture''' is a stub, help improve it.''. It tries to explain the roles of the values in the formula. Note: values are NOT to scale. Cyan line represents the formula to the left.]]

== =Line defined by an equation ===

[[File:PointToLineDistance.png|PointToLineDistance]] -->

In the case of a line in the plane given by the equation {{mathnowrap|1=''ax'' + ''by'' + ''c'' = 0,}}, where {{mvar|''a}}'', {{mvar|''b}}'' and {{mvar|''c}}'' are [[real number|real]] constants with {{mvar|''a}}'' and {{mvar|''b}}'' not both zero, the distance from the line to a point {{math|(''x''₀, ''y''₀)}} is<ref>{{harvnb|Larson|Hostetler|2007|loc=p. 452}}</ref><ref>{{harvnb|Spain|2007}}</ref>{{rp|p.14}}

== Line defined by an equation ==

In the case of a line in the plane given by the equation {{math|1=''ax'' + ''by'' + ''c'' = 0}}, where {{mvar|a}}, {{mvar|b}} and {{mvar|c}} are [[real number|real]] constants with {{mvar|a}} and {{mvar|b}} not both zero, the distance from the line to a point {{math|(''x''₀, ''y''₀)}} is<ref>{{harvnb|Larson|Hostetler|2007|loc=p. 452}}</ref><ref>{{harvnb|Spain|2007}}</ref>{{rp|p.14}}

:<math>\operatorname{distance}(ax+by+c=0, (x_0, y_0)) = \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}. </math>

The point on this line which is closest to {{math|(''x''₀, ''y''₀)}} has coordinates:<ref name="Larson 2007 loc=p. 522">{{harvnb|Larson|Hostetler|2007|loc=p. 522}}</ref>

:<math>x = \frac{b(bx_0 - ay_0)-ac}{a^2 + b^2} \text{ and } y = \frac{a(-bx_0 + ay_0) - bc}{a^2+b^2}.</math>

'''Horizontal and vertical lines'''

In the general equation of a line, {{math|1=''ax'' + ''by'' + ''c'' = 0}}, {{mvar|''a}}'' and {{mvar|''b}}'' cannot both be zero unless {{mvar|''c}}'' is also zero, in which case the equation does not define a line. If {{math|1=''a'' &nbsp;= &nbsp;0}} and {{math|''b'' &nbsp;{{math|≠ 0}}&nbsp;0, the line is horizontal and has equation {{math|1=''y'' = −{{sfrac|-''c''|/''b''}}}}. The distance from {{math|(''x''₀, ''y''₀)}} to this line is measured along a vertical line segment of length {{math|1={{!}}''y''₀ −- (−{{sfrac|-''c''|/''b''}}){{!}}| = {{sfrac|{{!}}''by''₀ + ''c''{{!}}|{{!}} / |''b''{{!}}}}}}| in accordance with the formula. Similarly, for vertical lines (''b'' = 0) the distance between the same point and the line is {{math|1={{sfrac|{{!}}''ax''₀ + ''c''{{!}}|{{!}} / |''a''{{!}}}} }}|, as measured along a horizontal line segment.

== Line defined by two points ==

If the line passes through two points {{math|1=''P''₁ = (''x''₁, ''y''₁)}} and {{math|1=''P''₂ = (''x₂'', ''y₂'')}} then the distance of {{math|(''x''₀, ''y''₀)}} from the line is:<ref name=GEO />

:<math>\operatorname{distance}(P_1, P_2, (x_0, y_0)) = \frac{|(x_2-x_1)(y_0-y_1)-(x_0-x_1)(y_2-y_1)|}{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}. </math>

== =Line defined by two points ===

The denominator of this expression is the distance between {{math|''P''₁}} and {{math|''P''₂}}. The numerator is twice the area of the triangle with its vertices at the three points, {{math|(''x''₀, ''y''₀)}}, {{math|''P''₁}} and {{math|''P''₂}}. See: {{slink|Area of a triangle|Using coordinates}}. The expression is equivalent to {{math|1=''h'' = {{sfrac|2''A''|''b''}}}}, which can be obtained by rearranging the standard formula for the area of a triangle: {{math|1=''A'' = {{sfrac|1|2}} ''bh''}}, where {{mvar|b}} is the length of a side, and {{mvar|h}} is the perpendicular height from the opposite vertex.

If the line passes through two points {{math|1=''P''₁ = (''x''₁, ''y''₁)}} and {{math|1=''P''₂ = (''x₂'', ''y₂'')}} then the distance of {{math|(''x''₀, ''y''₀)}} from the line is:<ref name=GEO />

:<math>\operatorname{distance}(P_1, P_2, (x_0, y_0)) = \frac{|(x_2-x_1)(y_0y_2-y_1)x_0-(x_0x_2-x_1)(y_0+x_2 y_1-y_2-y_1) x_1|}{\sqrt{(x_2y_2-x_1y_1)^2+(y_2x_2-y_1x_1)^2}}. </math>

The denominator of this expression is the distance between {{math|''P''₁}}'' and {{math|''P''₂}}''. The numerator is twice the area of the triangle with its vertices at the three points, {{math|(''x''₀, ''y''₀)}}, {{math|''P''₁}}'' and {{math|''P''₂}}''. See: {{slink|Area of a triangle|Using coordinates}}. The expression is equivalent to {{<math|1 display=''"inline">h'' = \frac{2A}{sfrac|2''A''|''b''}}}}</math>, which can be obtained by rearranging the standard formula for the area of a triangle: {{<math|1 display=''"inline">A'' = \frac{1}{sfrac|1|2}} ''bh''}}</math>, where {{mvar|''b}}'' is the length of a side, and {{mvar|''h}}'' is the perpendicular height from the opposite vertex.

== Line defined by point and angle ==

If the line passes through the point {{math|1=''P'' = (''P''_x, ''P''_y)}} with angle {{math|''θ''}}, then the distance of some point {{math|(''x''₀, ''y''₀)}} to the line is

:<math>\operatorname{distance}(P, \theta, (x_0, y_0)) = |\cos(\theta)(P_y-y_0) -\sin(\theta)(P_x-x_0)| </math>

==Proofs==

===An algebraic proof===

This proof is valid only if the line is neither vertical nor horizontal, that is, we assume that neither {{mvar|''a}}'' nor {{mvar|''b}}'' in the equation of the line is zero.

The line with equation {{math|1=''ax'' + ''by'' + ''c'' = 0}} has slope {{math|−-''a''/''b''}}, so any line perpendicular to it will have slope {{math|''b''/''a''}} (the negative reciprocal). Let {{math|(''m'', ''n'')}} be the point of intersection of the line {{math|1=''ax'' + ''by'' + ''c'' = 0}} and the line perpendicular to it which passes through the point ({{math|''x''₀}}, {{math|''y''₀}}). The line through these two points is perpendicular to the original line, so

:<math>\frac{y_0 - n}{x_0 - m}=\frac{b}{a}.</math>

Thus,

Line 43 ⟶ 36:

Now consider,

:<math> (a(x_0 - m) + b(y_0 - n))^2 & = a^2(x_0 - m)^2 + 2ab(y_0 -n)(x_0 - m) + b^2(y_0 - n)^2 \\= (a^2 + b^2)((x_0 - m)^2 + (y_0 - n)^2)</math>

:<math>

\begin{align}

(a(x_0 - m) + b(y_0 - n))^2 & = a^2(x_0 - m)^2 + 2ab(y_0 -n)(x_0 - m) + b^2(y_0 - n)^2 \\

& = \left(a^2 + b^2\right) \left((x_0 - m)^2 + (y_0 - n)^2\right)

\end{align}

</math>

using the above squared equation. But we also have,

:<math> (a(x_0 - m) + b(y_0 - n))^2 = (ax_0 + by_0 - am - bn )^2 = (ax_0 + by_0 + c)^2</math>

since {{math|(''m'', ''n'')}} is on {{math|1=''ax'' + ''by'' + ''c'' = 0}}.

Thus,

:<math>\left(a^2 + b^2\right) \left((x_0 - m)^2 + (y_0 - n)^2\right) = (ax_0 + by_0 + c)^2 </math>

and we obtain the length of the line segment determined by these two points,

:<math>d=\sqrt{(x_0 - m)^2+(y_0 - n)^2} = \frac{|ax_0+ by_0 +c|}{\sqrt{a^2+b^2}}.</math><ref>Between Certainty and Uncertainty: Statistics and Probability in Five Units With Notes on Historical Origins and Illustrative Numerical Examples</ref>

===A geometric proof===

Line 72 ⟶ 60:

and finally obtain:<ref>{{harvnb|Ballantine|Jerbert|1952}}</ref>

:<math> |\overline{PR}| = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}.</math>

A variation of this proof is to place V at P and compute the area of the triangle ∆''UVT'' two ways to obtain that <math>D|\overline{TU}| = |\overline{VU}||\overline{VT}|</math>

where D is the altitude of ∆''UVT'' drawn to the hypoteneuse of ∆''UVT'' from ''P''. The distance formula can then used to express <math>|\overline{TU}|</math>, <math>|\overline{VU}|</math>, and <math>|\overline{VT}|</math>in terms of the coordinates of P and the coefficients of the equation of the line to get the indicated formula.{{citation needed|date=April 2015}}

===A vector projection proof===

Line 84 ⟶ 75:

Since ''Q'' is a point on the line, <math>c = -ax_1 - by_1</math>, and so,<ref>{{harvnb|Anton|1994|loc=pp. 138-9}}</ref>

:<math> d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}.</math>

Although the distance is given as a modulus, the sign can be useful to determine which side of the line the point is on, in a sense determined by the direction of normal vector (a,b)

== Another formula ==

It is possible to produce another expression to find the shortest distance of a point to a line. This derivation also requires that the line beis not vertical or horizontal.

The point P is given with coordinates (<math>x_0, y_0</math>).

Line 103 ⟶ 93:

:<math>d=\sqrt{ \left( {\frac{x_0 + m y_0-mk}{m^2+1}-x_0 } \right) ^2 + \left( {m\frac{x_0+m y_0-mk}{m^2+1}+k-y_0 }\right) ^2 } = \frac{|k + m x_0 - y_0|}\sqrt{1 + m^2} .</math>

Recalling that ''m'' = -''a''/''b'' and ''k'' = - ''c''/''b'' for the line with equation ''ax'' + ''by'' + c = 0, a little algebraic simplification reduces this to the standard expression.<ref name=">{{harvnb|Larson |Hostetler|2007 |loc=p. 522"}}</ref>

==Vector formulation==

Line 112 ⟶ 102:

: <math> \mathbf{x} = \mathbf{a} + t\mathbf{n}</math>

Here {{math|'''a'''}} is the position of a point on the line, and {{math|'''n'''}} is a [[unit vector]] in the direction of the line. Then as scalar ''t'' varies, {{math|'''x'''}} gives the [[locus (mathematics)|locus]] of the line.

The distance of an arbitrary point {{math|'''p'''}} to this line is given by

: <math>\operatorname{distance}(\mathbf{x} = \mathbf{a} + t\mathbf{n}, \mathbf{p}) = \| (\mathbf{pa}-\mathbf{ap}) - ((\mathbf{pa}-\mathbf{ap}) \cdot \mathbf{n})\mathbf{n} \|. </math>

This formula can be derived as follows: <math>\mathbf{pa}-\mathbf{ap}</math> is a vector from {{math|'''ap'''}} to the point {{math|'''pa'''}} on the line. Then <math>(\mathbf{pa} - \mathbf{ap}) \cdot \mathbf{n}</math> is the projected length onto the line and so

:<math>\mathbf{a} + ((\mathbf{pa} - \mathbf{ap}) \cdot \mathbf{n})\mathbf{n}</math>

is a vector that is the [[projection (linear algebra)|projection]] of <math>\mathbf{pa}-\mathbf{ap}</math> onto the line and represents the point on the line closest to <math>\mathbf{p}</math>. Thus

:<math>(\mathbf{pa}-\mathbf{ap}) - ((\mathbf{pa}-\mathbf{ap}) \cdot \mathbf{n})\mathbf{n}</math>

is the component of <math>\mathbf{pa}-\mathbf{ap}</math> perpendicular to the line. The distance from the point to the line is then just the [[norm (mathematics)|norm]] of that vector.<ref name=GEO>{{cite web|last=Sunday|first=Dan|title=Lines and Distance of a Point to a Line|url=httpshttp://geomalgorithms.com/a02-_lines.html|publisher=softSurfer|url-statusaccessdate=dead|archive-url=https://web.archive.org/web/20210507021429/https://geomalgorithms.com/a02-_lines.html|archive-date=2021-05-076 December 2013}}</ref> This more general formula is not restricted to two dimensions.

== Another vector formulation ==

If the vector space is [[orthonormality|orthonormal]] and if the line (''l'' ) goes through point {{math|'''a'''}}A and has a [[Euclidean vector|direction vector]] {{<math|'''n'''}}>\vec u</math>, the distance between point {{math|P and line (''l'p'''}} and the line) is<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Point-Line Distance--3-Dimensional

: <math>d(\mathrm{P}, (l))= \frac{\left\|\overrightarrow{\mathrm{AP}} \times\vec u\right\|}{\|\vec u\|}</math>

|url=https://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html|access-date=2021-06-06|website=mathworld.wolfram.com|language=en}}</ref>

where <math>\overrightarrow{\mathrm{AP}} \times\vec u</math> is the [[cross product]] of the vectors <math>\overrightarrow{\mathrm{AP}}</math> and <math>\vec u</math> and where <math>\|\vec u\|</math> is the vector norm of <math>\vec u</math>.

: <math>\operatorname{distance}(\mathbf{x} = \mathbf{a} + t\mathbf{n}, \mathbf{p}) = \frac{\left\|(\mathbf{p}-\mathbf{a}) \times \mathbf{n}\right\|}{\|\mathbf{n}\|}.</math>

Note that cross products only exist in dimensions 3 and 7.

Line 145 ⟶ 134:

==References==

* {{citation|first=Howard|last=Anton|title=Elementary Linear Algebra|edition=7th|year=1994|publisher=John Wiley & Sons|isbn=0-471-58742-7}}

* {{citation|first1=J.P.|last1=Ballantine|first2=A.R.|last2=Jerbert|year=1952|volume=59|title=Distance from a line or plane to a point|journal=American Mathematical Monthly|issue=4|pages=242–243|doi=10.2307/2306514|jstor=2306514}}

* {{citation|first1=Ron|last1=Larson|first2=Robert|last2=Hostetler|title=Precalculus: A Concise Course|year=2007|publisher=Houghton Mifflin Co.|isbn=978-0-618-62719-67|url-access=registration|url=https://archive.org/details/precalculusconci00lars}}

* {{citation|first=Barry|last=Spain|title=Analytical Conics|year=2007|orig-yearorigyear=1957|publisher=Dover Publications|isbn=978-0-486-45773-47}}

* {{cite web|url=https://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html|title=Point-Line Distance--3-Dimensional|last=Weisstein|first=Eric W.|author-link=Eric W. Weisstein|publisher=[[MathWorld]]}}

==Further reading==

*{{citation|title=Encyclopedia of Distances|first1=Michel Marie|last1=Deza|author1-link=Michel Deza|first2=Elena|last2=Deza|author2-link=Elena Deza|edition=2nd|publisher=Springer|year=2013|isbn=9783642309588|page=86|url=https://books.google.com/books?id=QxX2CX5OVMsC&pg=PA86}}

[[Category:Euclidean geometry]]