The Line Element is
![\begin{displaymath}
ds=\sqrt{dx^2+dy^2+dz^2},
\end{displaymath}](p2_901.gif) |
(1) |
so the Arc Length between the points
and
is
![\begin{displaymath}
L = \int ds = \int_{x_1}^{x_2} \sqrt{1+{y'}^2+z'^2}\,dx
\end{displaymath}](p2_902.gif) |
(2) |
and the quantity we are minimizing is
![\begin{displaymath}
f=\sqrt{1+{y'}^2+z'^2}.
\end{displaymath}](p2_903.gif) |
(3) |
Finding the derivatives gives
and
so the Euler-Lagrange Differential Equations become
These give
![\begin{displaymath}
{y'\over\sqrt{1+{y'}^2+z'^2}}=c_1
\end{displaymath}](p2_912.gif) |
(10) |
![\begin{displaymath}
{z'\over\sqrt{1+{y'}^2+z'^2}}=c_2.
\end{displaymath}](p2_913.gif) |
(11) |
Taking the ratio,
![\begin{displaymath}
z'={c_2\over c_1}y'
\end{displaymath}](p2_914.gif) |
(12) |
![\begin{displaymath}
{y'\over\sqrt{1+y'^2+\left({c_2\over c_1}\right)^2y'^2}} = c_1
\end{displaymath}](p2_915.gif) |
(13) |
![\begin{displaymath}
y'^2={c_1}^2\left[{1+y'^2+\left({{c_2\over c_1}}\right)^2 y'^2}\right]= {c_1}^2+y'^2({c_1}^2+{c_2}^2),
\end{displaymath}](p2_916.gif) |
(14) |
which gives
![\begin{displaymath}
y'^2 = {{c_1}^2\over 1-{c_1}^2-{c_2}^2} \equiv {a_1}^2
\end{displaymath}](p2_917.gif) |
(15) |
![\begin{displaymath}
z'^2 = \left({c_2\over c_1}\right)^2 y'^2 = {{c_2}^2\over 1-{c_1}^2-{c_2}^2} \equiv {b_1}^2.
\end{displaymath}](p2_918.gif) |
(16) |
Therefore,
and
, so the solution is
![\begin{displaymath}
\left[{\matrix{x\cr y\cr z\cr}}\right]=\left[{\matrix{x\cr a_1x+a_0\cr b_1x+b_0\cr}}\right],
\end{displaymath}](p2_921.gif) |
(17) |
which is the parametric representation of a straight line with parameter
. Verifying the Arc Length gives
![\begin{displaymath}
L=\sqrt{1+{a_1}^2+{b_1}^2}\,(x_2-x_1)
\end{displaymath}](p2_923.gif) |
(18) |
where
![\begin{displaymath}
\left[{\matrix{y_1\cr y_2\cr}}\right] = \left[{\matrix{x_1 & 1\cr x_2 & 1\cr}}\right]\left[{\matrix{a_1\cr a_0\cr}}\right]
\end{displaymath}](p2_924.gif) |
(19) |
![\begin{displaymath}
\left[{\matrix{z_1\cr z_2\cr}}\right] = \left[{\matrix{x_1 & 1\cr x_2 & 1\cr}}\right]\left[{\matrix{b_1\cr b_0\cr}}\right].
\end{displaymath}](p2_925.gif) |
(20) |
See also Point-Point Distance--1-D, Point-Point Distance--2-D,
Point-Quadratic Distance
© 1996-9 Eric W. Weisstein
1999-05-25