248
4 Problems with Variable Boundaries
Substituting y
= c 1 into the above two expressions, and simplify, we get
1 + ϕ
(x 0 )c 1 = 0
(4.1.36)
1 + ψ
(x 1 )c 1 = 0
(4.1.37)
Equations (4.1.36) and (4.1.37) show that the extremal curve and the two known
curves are orthogonal respectively, namely the distance between the two disjoint
curves is the length of their common perpendicular, the transversality condition
(4.1.34) and the transversality condition (4.1.35) are reduced to the orthogonal
condition. So the system of equations with four unknowns x 0 , x 1 , c 1 and c 2 can
be obtained
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
c 1 x 0 + c 2 = ϕ(x 0 )
c 1 x 1 + c 2 = ψ(x 1 )
1 + ϕ
(x 0 )c 1 = 0
1 + ψ
(x 1 )c 1 = 0
(4.1.38)
Solving the system of Eqs. (4.1.38), the four unknowns can be found out, then
the concrete form of the extremal curve can be determined, and can calculate the
minimum distance between the two known curves namely the minimum of the
functional (4.1.31) can be calculated.
It is also can be seen from Eqs. (4.1.36) and (4.1.37) that
ϕ
(x 0 ) = ψ
(x 1 )
(4.1.39)
This shows that the slopes of the tangent lines for the known curves are equal at
two intersection points, namely two tangent lines are parallel to each other.
If to find the shortest distance of a known point to a known curve, then there are
only three unknowns. At the moment, if the known point is the left endpoint A(x 0 , y 0 )
of the extremal function for the functional, then the system of the equations can be
written as
⎧
⎨
⎩
c 1 x 0 + c 2 = y 0
c 1 x 1 + c 2 = ψ(x 1 )
1 + ψ
(x 1 )c 1 = 0
(4.1.40)
If the known point is the right endpoint B(x 1 , y 1 ) of the extremal function for the
functional, then the system of the equations can be written as
⎧
⎨
⎩
c 1 x 0 + c 2 = ϕ(x 0 )
c 1 x 1 + c 2 = y 1
1 + ϕ
(x 0 )c 1 = 0
(4.1.41)
4 Problems with Variable Boundaries
Substituting y
= c 1 into the above two expressions, and simplify, we get
1 + ϕ
(x 0 )c 1 = 0
(4.1.36)
1 + ψ
(x 1 )c 1 = 0
(4.1.37)
Equations (4.1.36) and (4.1.37) show that the extremal curve and the two known
curves are orthogonal respectively, namely the distance between the two disjoint
curves is the length of their common perpendicular, the transversality condition
(4.1.34) and the transversality condition (4.1.35) are reduced to the orthogonal
condition. So the system of equations with four unknowns x 0 , x 1 , c 1 and c 2 can
be obtained
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
c 1 x 0 + c 2 = ϕ(x 0 )
c 1 x 1 + c 2 = ψ(x 1 )
1 + ϕ
(x 0 )c 1 = 0
1 + ψ
(x 1 )c 1 = 0
(4.1.38)
Solving the system of Eqs. (4.1.38), the four unknowns can be found out, then
the concrete form of the extremal curve can be determined, and can calculate the
minimum distance between the two known curves namely the minimum of the
functional (4.1.31) can be calculated.
It is also can be seen from Eqs. (4.1.36) and (4.1.37) that
ϕ
(x 0 ) = ψ
(x 1 )
(4.1.39)
This shows that the slopes of the tangent lines for the known curves are equal at
two intersection points, namely two tangent lines are parallel to each other.
If to find the shortest distance of a known point to a known curve, then there are
only three unknowns. At the moment, if the known point is the left endpoint A(x 0 , y 0 )
of the extremal function for the functional, then the system of the equations can be
written as
⎧
⎨
⎩
c 1 x 0 + c 2 = y 0
c 1 x 1 + c 2 = ψ(x 1 )
1 + ψ
(x 1 )c 1 = 0
(4.1.40)
If the known point is the right endpoint B(x 1 , y 1 ) of the extremal function for the
functional, then the system of the equations can be written as
⎧
⎨
⎩
c 1 x 0 + c 2 = ϕ(x 0 )
c 1 x 1 + c 2 = y 1
1 + ϕ
(x 0 )c 1 = 0
(4.1.41)