2-Dimensional Point-Line Distance

The point-line distance in two dimensions is the shortest distance from a point to a line. The equation of a line in slope-intercept form is given by

so the line has slope . Now consider the distance from a point to the line. Points on the line have the vector coordinates

Therefore, the vector

is parallel to the line, and the vector

is perpendicular to it. Now, a vector from the point to the line is given by

Projecting onto ,

If the normal vector points to the positive side of the line, dropping the absolute value in the last expression gives the signed distance.

If the line is specified by two points and , then a vector perpendicular to the line is given by

Let be a vector from the point to the first point on the line

then the distance from to the line is again given by projecting onto , giving

As it must, this formula corresponds to the distance in the three-dimensional case

with all vectors having zero -components, and can be written in the slightly more concise form

where denotes a determinant.

The distance between a point with exact trilinear coordinates and a line is

(Kimberling 1998, p. 31).