Shortest Distance Between Two Points on the Surface of
a Cone Given a cone with a half angle whose axis of symmetry is the z
axis, using polar cylindrical coordinates, the distance between two points that
are di¤erentially separated is
q
ds = dz 2 + d 2 + 2 d 2 :
z and
Hence
are related via z = = tan ; which implies dz = d = tan
q
q
ds =
cot2 d 2 + d 2 + 2 d 2 = d 2 = sin2
q
1
ds =
d 2 + sin2 2 d 2 :
sin
+
= cot d :
2d 2;
The distance between two points can be represented in two di¤erent ways.
(1) In this method the total distance is given by
L=
Z
1
sin
2
1
q
1 + sin2
2 02 d
;
where 0 = d =d : This integrand, f , is independent of
Lagrange equation is
This implies
2 0
sin
p
2 02
1 + sin2
where
o
0
@f
d @f
sin 2
d
p
=
0 =
@
d @
d
1 + sin2
is a constant. Solving for
0
4 02
=
sin2
2
2
o
This leads to the integral
Z
sin
d =(
= 0:
o;
results in
1 + sin2
2
o
2 02
2
sin
d
=
d
sin
=
2 02
: Hence the Euler-
po
2
o ) sin
=
2
o
=
2 02
;
2
o;
:
o
Z
p
d
2
2
o
:
This integral is easily performed with the substitution
=
o = cos
!
2
2
o
=
2
o
tan2 ; and d =d =
1
o
tan = cos :
The integral now becomes
Z
(
o ) sin
=
o
(
o ) sin
=
cos
(
1
cos )
o
1
tan
d =
cos o tan
o
( o= ) :
Choosing the initial point to lie at
o
Z
d =
= 0; the curve for the shortest distance is
cos ( sin ) =
o:
Some care must be taken here as it is necessary that the range in
or else it is shorter to go the opposite way around the cone.
is less than
(2) In this method the total distance is given by
Z
1
L=
sin
2
1
q
02
+ sin2
2d
;
where 0 = d =d : Since the integrand, f; is independent of the independent
variable the …rst integral of the Euler- Lagrange equation is
0
f
@f
=
@ 0
a constant. The expression becomes
q
0
1
02 + sin2
2
p
sin
sin
Multiplying by
to
1
sin
p
02
+ sin2
2
02
+ sin2
2
2
sin
o;
0
02
+ sin2
2
=
o:
and squaring both sides of the equation leads
02
=
sin
o
q
02
sin2
4
=
2
2
=
2
02
o
2 02
o
2
o
=
o
2
o
sin
p
2
Separating and Integrating this expression yields
Z
d
p
(
)
sin
=
o
o
2
0
+ sin2
+ sin2
=
o
2
o
2;
2
;
d
:
d
:
This is the same integral as that obtained in method (1), hence the same curve,
cos ( sin ) =
2
o: