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
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(2)
|
where 
and the quantity we are minimizing is
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(3)
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Finding the derivatives gives
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(4)
|
 |  | ![d/(dx)[(1+y^'^2)^(-1/2)y^'],](/images/equations/Point-PointDistance2-Dimensional/Inline11.svg) |
(5)
|
so the Euler-Lagrange differential equation becomes
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(6)
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Integrating and rearranging,
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(7)
|
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(8)
|
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(9)
|
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(10)
|
The solution is therefore
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(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)
|
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(13)
|
Solving for 
and 
gives
 |  |  |
(14)
|
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(15)
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so the distance is
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(16)
|
 |  |  |
(17)
|
 |  |  |
(18)
|
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(19)
|
as expected.
For two points with exact trilinear coordinates 
and 
,
the distance between them is
 |  | ![(sqrt(-abc[a(beta-beta^')(gamma-gamma^')+b(gamma-gamma^')(alpha-alpha^')+c(alpha-alpha^')(beta-beta^')]))/(2Delta)](/images/equations/Point-PointDistance2-Dimensional/Inline44.svg) |
(20)
|
 |  | ![(sqrt(abc[acosA(alpha-alpha^')^2+bcosB(beta-beta^')^2+ccosC(gamma-gamma^')^2]))/(2Delta),](/images/equations/Point-PointDistance2-Dimensional/Inline47.svg) |
(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.
