In the case of a general surface, the distance between two points measured along the surface is known as a geodesic. For example, the shortest distance between two points on a sphere is an arc of a great circle.

In the Euclidean plane , the curve that minimizes the distance between two points is clearly a straight line segment. This can be shown mathematically as follows using calculus of variations and the so-called Euler-Lagrange differential equation. The line element in is given by

(1)

so the arc length between two points and is

(2)

where and the quantity we are minimizing is

(3)

Finding the derivatives gives

			(4)
			(5)

so the Euler-Lagrange differential equation becomes

(6)

Integrating and rearranging,

(7)

(8)

(9)

(10)

The solution is therefore

(11)

which is a straight line. Now verify that the arc length is indeed the straight-line distance between the points. and are determined from

			(12)
			(13)

Solving for and gives

			(14)
			(15)

so the distance is

			(16)
			(17)
			(18)
			(19)

as expected.

For two points with exact trilinear coordinates and , the distance between them is

			(20)
			(21)

where is the area of the triangle (Scott 1894; Carr 1970; Kimberling 1998, p. 31).

The shortest distance between two points on a sphere is the so-called great circle distance.

Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, 1970.

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

Scott, C. A. Projective Methods in Plane Analytical Geometry, 3rd ed. New York: Chelsea, 1894.

CITE THIS AS:

Weisstein, Eric W. "Point-Point Distance--2-Dimensional." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Point-PointDistance2-Dimensional.html