Problem #130 Ant On Cylinders

Transcription

1 Problem #130 Ant On Cyliners The Distance The Ant Travels Along The Surface John Snyer November, 009 Problem Consier the soli boune by the three right circular cyliners x y (greenish-yellow), x z (re), an y z (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 istance the ant can travel between the two points is given by the following expression. a 8, where The value of the constant is given by the solution to the following transcenental equation. Π 4 sin 1

2 AntOnCyliners130.nb Analysis We begin with a visualization showing the soli forme by the 3 intersecting cyliners. The ant starts his journey at the yellow point shown in the center of the re patch at the point a, a, 0 an ens his journey at the yellow point shown at the center of the blue patch at the point 0, a, a. Blocka 1, p1, p, p1 ContourPlot3Dx y, x z, y z, x,,, y,,, z,,, ContourStyle Re, Green, Blue, Mesh False, MaxRecursion 4, PlotPoints, RegionFunction Functionx, y, z, x y.0005 a && x z.0005 a && y z.0005 a ; p Graphics3DPointSize0.05, Yellow, Spherea, a, 0, 0.04, Sphere0, a, a, 0.04; cyliners Showp1, p, ViewPoint.03986,.67915, , AxesLabel "x", "y", "z", ImageSize Full

3 AntOnCyliners130.nb 3 A geoesic is the curve giving the shortest istance between two points on a surface. From the calculus of variations it is well know that a helix efines a geoesic on the surface of a right circular cyliner. In the present case the ant must travel along two such geoesic curves; each curve has one of its en points at a yellow ot an the other at a common point on the bounary between the re an blue patches efine by the plane x z. In the case of the re patch the parametric equation of the helix can be written as follows. x, y, z a cost, a sint, t Π 4 c Here t is a parameter potentially ranging over the interval 0 t Π n c is a yet to be etermine constant. The corresponing formula for the helix on the blue patch woul be as follows. x, y, z t Π 4 c, a sint, a cost Consiering the helix on the re patch we notice that we must have x z when the helix reaches the bounary plane between the re an blue patches. Calling the x coorinate of this point, we can solve for the value of constant c an the value of the parameter t at the bounary. Reuce Cost, t Π 4 c, 0, Π 4 t Π, t, c, Reals, Backsubstitution True a 0 && 0 a && t ArcCos 4 && c Π 4 ArcCos So the equation of the helix on the re patch is given by the following expression where the parameter t runs over the interval Π 4 t cos 1. x, y, z a cost, a sint, t Π 4 4 cos 1 4 Π By symmetry the istance from the bounary to the yellow point is the same within both the re an blue patches. It is sufficient, therefore, to fin the arc length istance along the geoesic on the surface of the re patch an then ouble the result to fin the total minimum istance along which the ant must travel. The element of arc length s is foun using the well know formula for the this quantity in its parametric form. s SqrtPlus D, t & a Cost, a Sint, t Π ArcCos Π FullSimplify, 0 a && Π 4 t Π & 8 Π 4 ArcCos This can be symbolically integrate to fin the arc length istance of the geoesic along the re patch from the yellow point to the bounary plane between the re an blue patches.

4 4 AntOnCyliners130.nb Assuming0, arc Integrates, t, Π 4, ArcCos FullSimplify 8 Π 4 ArcCos Π 4 ArcCos To simplify things we'll square this arc length an work with it in that form. The square of the minimum arc length istance along the geoesic on the surface of the re patch is then as follows. arc arc FullSimplify, 0 & a Π ArcCos 8 ArcSin Let's plot the square of the istance the ant travels along the re patch for various values of the parameter using the sample value a 1. Blocka 1, Plotarc,, 0, a, AxesOrigin 0, 0.5, PlotStyle Thick, AspectRatio Automatic Recall that the parameter is the value of the x coorinate of the geoesic curve when it reaches the plane bounary x z between the re an blue patches. We see from this plot that there is clearly a value of the parameter which will minimize the istance the ant must travel. We can fin this value of to any egree of precision by taking the erivative of the geoesic arc length, setting it equal to zero, an solving for. In oing this we'll set a 1 an then ajust for alternative values of a later on. Blocka 1, FullSimplifyDarc, 0, 0 a 4 ArcSin Π So the equation which must be solve to fin the value of is as follows. Π 4 sin 1 From this equation we see that the following relationships must hol.

5 AntOnCyliners130.nb 5 sin Π cos Π We can now solve this equation to fin the numerical value of. Blocka 1, ToRulesReuceDarc, 0, 0 a,, Reals Root ArcCos 1 ArcSin &, To sixty ecimal places the value of is as then as follows. N, The reaer may woner what is the physical significance of the constant? In fact,, where is the istance between the centers of two units isks such that the isks overlap by half of each's area. The parameter is iscusse here on MathWorl an is liste by Sloane as sequence A Since the istance the ant must crawl from one yellow point to another is proportional to the value of the parameter a, we can write the shortest istance the ant must travel between the two yellow points in the following form. This result being erive as follows. a 8, where antgeoesic Sqrtarc. a 1, ArcCos ArcSin a Π, 1 4 Π Simplify, 0 1 & We can now illustrate the minimum istance the ant travels between the two yellow points for some selecte values of the parameter a. Block , TableFormTablea, NantGeoesic, 7, a, 10, TableHeaings None, "a", "istance" a istance

6 6 AntOnCyliners130.nb Finally, we can now a the geoesic along which the ant travels to our prior visualization. Blockx , a 1, p1, p, p1 ParametricPlot3D1.01 Cost, 1.01 Sint, 1.01 t Π 4 4 x Π 4 ArcCos x, t, Π 4, ArcCos x, PlotStyle Thick, Cyan; p ParametricPlot3D1.01 t Π 4 4 x Π 4 ArcCos x, 1.01 Sint, 1.01 Cost, x t, ArcCos, Π, PlotStyle Thick, Cyan; 4 Showcyliners, p1, p, ViewPoint.03986,.67915, , AxesLabel "x", "y", "z", ImageSize Full