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Point-Point Distance--3-Dimensional


In Euclidean space R^3, 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 R^3 is given by

 ds=sqrt(dx^2+dy^2+dz^2),
(1)

so the arc length between the points x_1 and x_2 is

 L=intds=int_(x_1)^(x_2)sqrt(1+y^'^2+z^('2))dx
(2)

and the quantity we are minimizing is

 f=sqrt(1+y^'^2+z^('2)).
(3)

Finding the derivatives gives

(partialf)/(partialy)=0
(4)
(partialf)/(partialz)=0
(5)

and

(partialf)/(partialy^')=(y^')/(sqrt(1+y^('2)+z^('2)))
(6)
(partialf)/(partialz^')=(z^')/(sqrt(1+y^('2)+z^('2))),
(7)

so the Euler-Lagrange differential equations become

d/(dx)((y^')/(sqrt(1+y^'^2+z^('2))))=0
(8)
d/(dx)((z^')/(sqrt(1+y^'^2+z^('2))))=0.
(9)

These give

 (y^')/(sqrt(1+y^'^2+z^('2)))=c_1
(10)
 (z^')/(sqrt(1+y^'^2+z^('2)))=c_2.
(11)

Taking the ratio,

 z^'=(c_2)/(c_1)y^'
(12)
 (y^')/(sqrt(1+y^('2)+((c_2)/(c_1))^2y^('2)))=c_1
(13)
 y^('2)=c_1^2[1+y^('2)+((c_2)/(c_1))^2y^('2)]=c_1^2+y^('2)(c_1^2+c_2^2),
(14)

which gives

 y^('2)=(c_1^2)/(1-c_1^2-c_2^2)=a_1^2
(15)
 z^('2)=((c_2)/(c_1))^2y^('2)=(c_2^2)/(1-c_1^2-c_2^2)=b_1^2.
(16)

Therefore, y^'=a_1 and z^'=b_1, so the solution is

 [x; y; z]=[x; a_1x+a_0; b_1x+b_0],
(17)

which is the parametric representation of a straight line with parameter x in [x_1,x_2]. Verifying the arc length gives

 L=sqrt(1+a_1^2+b_1^2)(x_2-x_1)
(18)

where

 [y_1; y_2]=[x_1 1; x_2 1][a_1; a_0]
(19)
 [z_1; z_2]=[x_1 1; x_2 1][b_1; b_0].
(20)

See also

Calculus of Variations, Circle Triangle Picking, Great Circle, Line Line Picking, Point-Point Distance--2-Dimensional, Point-Quadratic Distance, Sphere Point Picking

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 930-931, 1985.

Cite this as:

Weisstein, Eric W. "Point-Point Distance--3-Dimensional." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Point-PointDistance3-Dimensional.html

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