Distance from a point to a line
geometry problem
The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment that joins the point to the line and is perpendicular to the line. The formula 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
Line defined by an equation
In the case of a line in the plane given by the equation
a
x
+
b
y
+
c
=
0
{\displaystyle ax+by+c=0}
where
a
{\displaystyle a}
,
b
{\displaystyle b}
and
c
{\displaystyle c}
are real constants with
a
{\displaystyle a}
and
b
{\displaystyle b}
not both zero, the distance from the line to a point
(
x
0
,
y
0
)
{\displaystyle (x_{0},y_{0})}
is
distance
(
a
x
+
b
y
+
c
=
0
,
(
x
0
,
y
0
)
)
=
|
a
x
0
+
b
y
0
+
c
|
a
2
+
b
2
.
{\displaystyle \operatorname {distance} (ax+by+c=0,(x_{0},y_{0}))={\frac {|ax_{0}+by_{0}+c|}{\sqrt {a^{2}+b^{2}}}}.}
The point on this line that is closest to
(
x
0
,
y
0
)
{\displaystyle (x_{0},y_{0})}
has coordinates:
x
=
b
(
b
x
0
−
a
y
0
)
−
a
c
a
2
+
b
2
and
y
=
a
(
−
b
x
0
+
a
y
0
)
−
b
c
a
2
+
b
2
{\displaystyle 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}}}}
or equivalently
x
=
x
0
−
a
a
x
0
+
b
y
0
+
c
a
2
+
b
2
{\displaystyle x=x_{0}-a{\frac {ax_{0}+by_{0}+c}{a^{2}+b^{2}}}}
and
y
=
y
0
−
b
a
x
0
+
b
y
0
+
c
a
2
+
b
2
{\displaystyle y=y_{0}-b{\frac {ax_{0}+by_{0}+c}{a^{2}+b^{2}}}}
Horizontal and vertical lines
In the general equation of a line,
a
x
+
b
y
+
c
=
0
{\displaystyle ax+by+c=0}
,
a
{\displaystyle a}
and
b
{\displaystyle b}
cannot both be zero unless
c
{\displaystyle c}
is also zero; in that case, the equation does not define a line. If
a
=
0
{\displaystyle a=0}
and
b
≠
0
{\displaystyle b\neq 0}
, the line is horizontal and has equation
y
=
−
c
/
b
{\displaystyle y=-c/b}
. The distance from
(
x
0
,
y
0
)
{\displaystyle (x_{0},y_{0})}
to this line is measured along a vertical line segment of length
|
y
0
−
(
−
c
/
b
)
|
=
|
b
y
0
+
c
|
/
|
b
|
{\displaystyle |y_{0}-(-c/b)|=|by_{0}+c|/|b|}
in accordance with the formula. Similarly, for vertical lines
(
b
=
0
)
{\displaystyle (b=0)}
the distance between the same point and the line is
|
a
x
0
+
c
|
/
|
a
|
{\displaystyle |ax_{0}+c|/|a|}
, as measured along a horizontal line segment.
Begin with the source’s own compact description: “Distance from a point to a line” is geometry problem. The dossier treats that line as a proposition to test through Distance, point and line, not as a finished interpretation.
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