mathworld.wolfram.com

Point-Line Distance--2-Dimensional -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorld

PointLineDistance2D

The equation of a line ax+by+c=0 in slope-intercept form is given by

y=-a/bx-c/b,

(1)

so the line has slope -a/b. Now consider the distance from a point (x_0,y_0) to the line. Points on the line have the vector coordinates

[x; -a/bx-c/b]=[0; -c/b]-1/b[-b; a]x.

(2)

Therefore, the vector

[-b; a]

(3)

is parallel to the line, and the vector

v=[a; b]

(4)

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

r=[x-x_0; y-y_0].

(5)

Projecting r onto v,

PointLineDistance2DVec

If the line is specified by two points x_1=(x_1,y_1) and x_2=(x_2,y_2), then a vector perpendicular to the line is given by

v=[y_2-y_1; -(x_2-x_1)].

(12)

Let r be a vector from the point x_0=(x_0,y_0) to the first point on the line

r=[x_1-x_0; y_1-y_0],

(13)

then the distance from (x_0,y_0) to the line is again given by projecting r onto v, giving

d=|v^^·r|=(|(x_2-x_1)(y_1-y_0)-(x_1-x_0)(y_2-y_1)|)/(sqrt((x_2-x_1)^2+(y_2-y_1)^2)).

(14)

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

d=(|(x_2-x_1)x(x_1-x_0)|)/(|x_2-x_1|)

(15)

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

d=(|det(x_2-x_1 x_1-x_0)|)/(|x_2-x_1|),

(16)

where det(A) denotes a determinant.

The distance between a point with exact trilinear coordinates (alpha^',beta^',gamma^') and a line lalpha+mbeta+ngamma=0 is

d=(|lalpha^'+mbeta^'+ngamma^'|)/(sqrt(l^2+m^2+n^2-2mncosA-2nlcosB-2lmcosC))

(17)

(Kimberling 1998, p. 31).

See also

Collinear, Point-Line Distance--3-Dimensional

Explore with Wolfram|Alpha

References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Cite this as:

Weisstein, Eric W. "Point-Line Distance--2-Dimensional." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html

Subject classifications