Problem #130–Ant On Cylinders
The Distance The Ant Travels Along The Surface
John Snyder
November, 2009
Problem
Consider the solid bounded by the three right circular cylinders x2 ! y2 " 2 a2 (greenish-yellow), x2 ! z2 " 2 a2 (red), and
y2 ! z2 " 2 a2 (blue) shown in the figure below. An ant wants to follow the shortest path along the surface from the point !a, a, 0"
to the point !0, a, a". What is the length of this path?
Solution
! Summary
The shortest distance the ant can travel between the two points is given by the following expression.
ad
8 # 2 d 2 , where d " 0.5713037465453777195781044 ...
The value of the constant d is given by the solution to the following transcendental equation.
2d
2 # d 2 ! Π # 4 sin#1
d
2
2
AntOnCylinders130.nb
! Analysis
We begin with a visualization showing the solid formed by the 3 intersecting cylinders. The ant starts his journey at the yellow
point shown in the center of the red patch at the point !a, a, 0" and ends his journey at the yellow point shown at the center of the
blue patch at the point !0, a, a".
Block!"a ! 1, p1, p2#,
p1 ! ContourPlot3D!$x2 " y2 # 2 a2 , x2 " z2 !! 2 a2 , y2 " z2 # 2 a2 %, &x, $
&y, $
2 a,
2 a', &z, $
2 a,
2 a,
2 a',
2 a', ContourStyle % ""Red#, "Green#, "Blue##,
Mesh % False, MaxRecursion % 4, PlotPoints % 22, RegionFunction %
Function("x, y, z#, x2 " y2 & 2.0005 a2 && x2 " z2 & 2.0005 a2 && y2 " z2 & 2.0005 a2 )*;
p2 ! Graphics3D+"PointSize+0.025,, Yellow, Sphere+"a, a, 0#, 0.04,, Sphere+"0, a, a#, 0.04,#,;
cylinders ! Show+p1, p2, ViewPoint % "2.03986, 2.67915, 0.333396#,
AxesLabel % ""x", "y", "z"#, ImageSize % Full,*
A geodesic is the curve giving the shortest distance between two points on a surface. From the calculus of variations it is well
know that a helix defines a geodesic on the surface of a right circular cylinder. In the present case the ant must travel along two
such geodesic curves; each curve has one of its end points at a yellow dot and the other at a common point on the boundary
between the red and blue patches defined by the plane x " z. In the case of the red patch the parametric equation of the helix can be
written as follows.
AntOnCylinders130.nb
3
A geodesic is the curve giving the shortest distance between two points on a surface. From the calculus of variations it is well
know that a helix defines a geodesic on the surface of a right circular cylinder. In the present case the ant must travel along two
such geodesic curves; each curve has one of its end points at a yellow dot and the other at a common point on the boundary
between the red and blue patches defined by the plane x " z. In the case of the red patch the parametric equation of the helix can be
written as follows.
!x, y, z" " #a
2 cos$t%, a
2 sin$t%, t #
Π
4
c&
Here t is a parameter potentially ranging over the interval 0 % t % Π ' 2 and c is a yet to be determined constant.
The corresponding formula for the helix on the blue patch would be as follows.
!x, y, z" " # t #
Π
4
c, a
2 sin$t%, a
2 cos$t%&
Considering the helix on the red patch we notice that we must have x " z when the helix reaches the boundary plane between the
red and blue patches. Calling the x coordinate of this point d, we can solve for the value of constant c and the value of the parameter t at the boundary.
Reduce!&d !!
2 a Cos+t,, d !! t $
Π
4
c, 0 ( d (
"t, c#, Reals, Backsubstitution % True*
d
a ! 0 && 0 " d " a && t # ArcCos!
2 a, Π - 4 & t & Π - 2',
" && c # $
4d
Π $ 4 ArcCos!
2 a
d
2 a
"
So the equation of the helix on the red patch is given by the following expression where the parameter t runs over the interval
Π ' 4 % t % cos#1 (d ) * 2 a+,.
!x, y, z" " #a
2 cos$t%, a
2 sin$t%, t #
Π
4
4d
4 cos#1 -
d
2 a
.#Π
&
By symmetry the distance from the boundary to the yellow point is the same within both the red and blue patches. It is sufficient,
therefore, to find the arc length distance along the geodesic on the surface of the red patch and then double the result to find the
total minimum distance along which the ant must travel. The element of arc length & s is found using the well know formula for the
this quantity in its parametric form.
ds ! Sqrt!Plus )) D+*, t,2 & -) &a
FullSimplify!*, 0 ( d ( a
2
a2 &
8 d2
#Π $ 4 ArcCos!
d
2 a
2 Cos+t,, a
2 Sin+t,, t $
2 && Π - 4 ( t ( Π - 2* &
Π
4
4d
4 ArcCos!
d
2 a
*$Π
' * --
"$
2
This can be symbolically integrated to find the arc length distance of the geodesic along the red patch from the yellow point to the
boundary plane between the red and blue patches.
4
AntOnCylinders130.nb
Assuming!0 ( d (
2 a, arc ! Integrate!ds, &t, Π - 4, ArcCos!
d
*'* -- FullSimplify*
2 a
2
a2 &
8 d2
#Π $ 4 ArcCos!
d
2 a
"$
2
$
Π
4
& ArcCos!
d
"
2 a
To simplify things we'll square this arc length and work with it in that form. The square of the minimum arc length distance along
the geodesic on the surface of the red patch is then as follows.
arc2 ! arc2 -- FullSimplify!*, 0 ( d (
d2 &
a2 Π2
8
" ArcSin!
d
$ 2 a2 ArcCos!
2 a
2 a* &
d
"
2 a
Let's plot the square of the distance the ant travels along the red patch for various values of the parameter d using the sample value
a " 1.
Block+"a ! 1#,
Plot+arc2, "d, 0, a#, AxesOrigin % "0, 0.5#, PlotStyle % Thick, AspectRatio % Automatic,,
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.2
0.4
0.6
0.8
1.0
Recall that the parameter d is the value of the x coordinate of the geodesic curve when it reaches the plane boundary x " z between
the red and blue patches. We see from this plot that there is clearly a value of the parameter d which will minimize the distance the
ant must travel. We can find this value of d to any degree of precision by taking the derivative of the geodesic arc length, setting it
equal to zero, and solving for d. In doing this we'll set a " 1 and then adjust for alternative values of a later on.
Block+"a ! 1#,
FullSimplify+D+arc2, d, # 0, 0 ( d ( a,,
2d
2 $ d2 & 4 ArcSin!
d
"#Π
2
So the equation which must be solved to find the value of d is as follows.
2d
2 # d 2 ! Π # 4 sin#1
d
2
From this equation we see that the following relationships must hold.
AntOnCylinders130.nb
sin#1
d
!
2
cos#1
d
!
2
1
4
1
4
Π#2d
2 # d2
Π!2d
2 # d2
5
We can now solve this equation to find the numerical value of d.
Block+"a ! 1#,
ToRules+Reduce+"D+arc2, d, # 0, 0 ( d ( a#, d, Reals,,,
%d ' Root!%$ ArcCos!
(1
" & ArcSin!
2
(1
" & (1
2 $ (12 &, 0.57130374654537771958&"&
2
To sixty decimal places the value of d is as then as follows.
N++, 60,
'd ' 0.571303746545377719578104409356755207958952404071611754027852(
The reader may wonder what is the physical significance of the constant d? In fact, d " "/
2 , where " is the distance between
the centers of two units disks such that the disks overlap by half of each's area. The parameter " ' 0.8079455 ... is discussed here
on MathWorld and is listed by Sloane as sequence A133741.
Since the distance the ant must crawl from one yellow point to another is proportional to the value of the parameter a, we can write
the shortest distance the ant must travel between the two yellow points in the following form.
ad
8 # 2 d 2 , where d " 0.5713037465453777195781044 ...
This result being derived as follows.
antGeodesic ! 2 a Sqrt!arc2 -. &a % 1, ArcCos!
d
2 a
ArcSin!
d
2 a
ad
*%
1
4
Π$2d
2 $ d2
*%
1
4
Π"2d
2 $ d2
,
'* -- Simplify+*, 0 ( d ( 1, &
8 $ 2 d2
We can now illustrate the minimum distance the ant travels between the two yellow points for some selected values of the parameter a.
Block+"d ! 0.57130374654537771958#,
TableForm+Table+"a, N+antGeodesic, 7,#, "a, 10#,, TableHeadings % "None, ""a", "distance"##,,
a
1
2
3
4
5
6
7
8
9
10
distance
1.548562
3.097125
4.645687
6.194250
7.742812
9.291374
10.83994
12.38850
13.93706
15.48562
Finally, we can now add the geodesic along which the ant travels to our prior visualization.
6
AntOnCylinders130.nb
Finally, we can now add the geodesic along which the ant travels to our prior visualization.
Block!"x ! 0.5713, a ! 1, p1, p2#,
p1 ! ParametricPlot3D!&1.01
2 a Cos+t,, 1.01
2 a Sin+t,, 1.01 t $
Π
4
4x
$
Π $ 4 ArcCos!
x
2 a
&t,
Π
4
, ArcCos!
',
*', PlotStyle % "Thick, Cyan#*;
x
2 a
p2 ! ParametricPlot3D!&1.01 t $
Π
4
$
4x
Π $ 4 ArcCos!
x
2 a
&t, ArcCos!
*
*
, 1.01
2 a Sin+t,, 1.01
*, ', PlotStyle % "Thick, Cyan#*;
4
2 a
Show+cylinders, p1, p2, ViewPoint % "2.03986, 2.67915, 0.333396#,
x
Π
AxesLabel % ""x", "y", "z"#, ImageSize % Full,*
2 a Cos+t,',