Distance from a point to a line

In Euclidean geometry, the distance from a point to a line is the shortest distance from a given point to any point on an infinite 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 formula for calculating it can be derived and expressed in several ways.
Knowing the 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 , where , and are real constants with and not both zero, the distance from the line to a point is
The point on this line which is closest to has coordinates:
Horizontal and vertical lines
In the general equation of a line, , and cannot both be zero unless is also zero, in which case the equation does not define a line. If and , the line is horizontal and has equation . The distance from to this line is measured along a vertical line segment of length in accordance with the formula. Similarly, for vertical lines ("b" = 0) the distance between the same point and the line is , as measured along a horizontal line segment.
Line defined by two points.
If the line passes through two points and then the distance of from the line is:
The denominator of this expression is the distance between and . The numerator is twice the area of the triangle with its vertices at the three points, , and . See: . The expression is equivalent to , which can be obtained by rearranging the standard formula for the area of a triangle: , where is the length of a side, and is the perpendicular height from the opposite vertex.
Proofs.
An algebraic proof.
This proof is valid only if the line is neither vertical nor horizontal, that is, we assume that neither nor in the equation of the line is zero.
The line with equation has slope , so any line perpendicular to it will have slope (the negative reciprocal). Let be the point of intersection of the line and the line perpendicular to it which passes through the point (, ). The line through these two points is perpendicular to the original line, so 
Thus,
formula_5
and by squaring this equation we obtain:
Now consider,
using the above squared equation. But we also have,
since is on .
Thus,
and we obtain the length of the line segment determined by these two points,
A geometric proof.
This proof is valid only if the line is not horizontal or vertical.
Drop a perpendicular from the point "P" with coordinates ("x"0, "y"0) to the line with equation "Ax" + "By" + "C" = 0. Label the foot of the perpendicular "R". Draw the vertical line through "P" and label its intersection with the given line "S". At any point "T" on the line, draw a right triangle "TVU" whose sides are horizontal and vertical line segments with hypotenuse "TU" on the given line and horizontal side of length |"B"| (see diagram). The vertical side of ∆"TVU" will have length |"A"| since the line has slope -"A"/"B".
∆"PRS" and ∆"TVU" are similar triangles, since they are both right triangles and ∠"PSR" ≅ ∠"TUV" since they are corresponding angles of a transversal to the parallel lines "PS" and "UV" (both are vertical lines). Corresponding sides of these triangles are in the same ratio, so:
If point "S" has coordinates ("x"0,"m") then |"PS"| = |"y"0 - "m"| and the distance from "P" to the line is:
Since "S" is on the line, we can find the value of m,
and finally obtain:
A variation of this proof is to place V at P and compute the area of the triangle ∆"UVT" two ways to obtain that formula_15
where D is the altitude of ∆"UVT" drawn to the hypotenuse of ∆"UVT" from "P". The distance formula can then used to express formula_16, formula_17, and formula_18in terms of the coordinates of P and the coefficients of the equation of the line to get the indicated formula.
A vector projection proof.
Let "P" be the point with coordinates ("x"0, "y"0) and let the given line have equation "ax" + "by" + "c" = 0. Also, let "Q" = ("x"1, "y"1) be any point on this line and n the vector ("a", "b") starting at point "Q". The vector n is perpendicular to the line, and the distance "d" from point "P" to the line is equal to the length of the orthogonal projection of formula_19 on n. The length of this projection is given by:
Now,
thus
Since "Q" is a point on the line, formula_25, and so,
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 is not vertical or horizontal.
The point P is given with coordinates (formula_27).
The equation of a line is given by formula_28. The equation of the normal of that line which passes through the point P is given formula_29.
The point at which these two lines intersect is the closest point on the original line to the point P. Hence:
We can solve this equation for "x",
The "y" coordinate of the point of intersection can be found by substituting this value of "x" into the equation of the original line,
Using the equation for finding the distance between 2 points, formula_33, we can deduce that the formula to find the shortest distance between a line and a point is the following:
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.
Vector formulation.
The equation of a line can be given in vector form:
Here is a point on the line, and is a unit vector in the direction of the line. Then as scalar "t" varies, gives the locus of the line.
The distance of an arbitrary point to this line is given by
This formula can be derived as follows: formula_37 is a vector from to the point . Then formula_38 is the projected length onto the line and so
is a vector that is the projection of formula_37 onto the line and represents the point on the line closest to formula_41. Thus
is the component of formula_37 perpendicular to the line. The distance from the point to the line is then just the norm of that vector. This more general formula is not restricted to two dimensions.
Another vector formulation.
If the vector space is orthonormal and if the line goes through point and has a direction vector , the distance between point and the line is
Note that cross products only exist in dimensions 3 and 7.