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 Circles around Circles (Posted on 2007-12-23)
Three circles (A, B, and C) are tangentially connected:

Properties:
2) Circle C is of the exact size that if it rolls around A, or it rolls around B, it will touch A and B on the opposite side at identical points E and F.

What is the radius of Circle C if:
1) C > B
2) B > C > A
3) A > C (C is a maximum)

 No Solution Yet Submitted by Leming Rating: 3.7500 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re: solution part 1 -- now parts 2 and 3 Comment 3 of 3 |
(In reply to solution part 1 by Charlie)

I think the only modification needed for multiple rotations of C is to change 2*pi*x - 2*C*x to k*2*pi*x - 2*C*x, as the two portions skipped over remain without multiplication.

Only

needed = 2 * pi * x - 2 * c * x

need be changed, to, say

needed = 2 * 2 * pi * x - 2 * c * x

and the initial estimate of x changed.

This results in

` 1.276999999999969  .0001  1.276999999999969  .00001  1.27704999999997   .000001  1.277054999999969  .0000001  1.27705589999997   .00000001  1.27705596999997   .000000001  1.27705597699997   .0000000001  1.27705597729997   .00000000001  1.27705597735997   .000000000001  1.277055977364971  .0000000000001  1.27705597736567   .00000000000001  1.27705597736575   .000000000000001  1.277055977365758  .0000000000000001 `

indicating a radius of  1.277055977365758 relative to the radius of A for part 2.

With k = 3, we get

` .85                .001  .858               .0001  .8587999999999999  .00001  .8588599999999996  .000001  .8588619999999997  .0000001  .8588622999999995  .00000001  .8588623099999996  .000000001  .8588623099999996  .0000000001  .8588623103999996  .00000000001  .8588623104199996  .000000000001  .8588623104209996  .0000000000001  .8588623104213997  .00000000000001  .8588623104214597  .000000000000001  .8588623104214617  .0000000000000001  .8588623104214621  1D-17 `

Indicating a radius of .8588623104214621 for C. This is the largest that is less than 1 (i.e., less than the radius of circle A).

For curiosity:

With k=4,

` .6                 .01  .64                .001  .648               .0001  .6488999999999999  .00001  .6489199999999998  .000001  .648927            .0000001  .6489276999999997  .00000001  .6489277900000001  .000000001  .648927794         .0000000001  .6489277945        .00000000001  .64892779453       .000000000001 .6489277945349999  .0000000000001  .6489277945358002  .00000000000001  .6489277945358501  .000000000000001  .6489277945358561  .0000000000000001  .6489277945358563  1D-17 `

indicating a radius of .6489277945358563 for circle C.

then

k=5

.5  .01
.52  .001
.522  .0001
.5224  .00001
.5224199999999999  .000001
.522424  .0000001
.5224248999999995  .00000001
.5224249499999998  .000000001
.5224249519999997  .0000000001
.5224249524999998  .00000000001
.5224249525199998  .000000000001
.5224249525259996  .0000000000001
.5224249525261997  .00000000000001
.5224249525262896  .000000000000001
.5224249525262986  .0000000000000001
.5224249525262988  1D-17

k=6

.4  .01
.43  .001
.4370000000000001  .0001
.4377  .00001
.43773  .000001
.437732  .0000001
.4377325  .00000001
.4377325899999999  .000000001
.437732592  .0000000001
.4377325925  .00000000001
.43773259252  .000000000001
.437732592521  .0000000000001
.4377325925216998  .00000000000001
.4377325925217498  .000000000000001
.4377325925217498  .0000000000000001
.4377325925217499  1D-17

k=7

.3  .01
.3700000000000001  .001
.3760000000000001  .0001
.3769  .00001
.3769900000000001  .000001
.376993  .0000001
.3769939  .00000001
.37699391  .000000001
.3769939150000002  .0000000001
.3769939152000002  .00000000001
.3769939152100002  .000000000001
.3769939152150001  .0000000000001
.3769939152155  .00000000000001
.3769939152155499  .000000000000001
.3769939152155589  .0000000000000001
.3769939152155598  1D-17

k=8

.3  .01
.3300000000000001  .001
.3310000000000001  .0001
.3312  .00001
.3312600000000001  .000001
.3312659999999999  .0000001
.3312669  .00000001
.33126693  .000000001
.3312669380000002  .0000000001
.3312669387000002  .00000000001
.3312669387300002  .000000000001
.3312669387370001  .0000000000001
.3312669387375  .00000000000001
.3312669387375799  .000000000000001
.3312669387375869  .0000000000000001
.3312669387375877  1D-17

k=9

.2  .01
.2900000000000001  .001
.2950000000000001  .0001
.2955  .00001
.2955700000000001  .000001
.295575  .0000001
.295575  .00000001
.2955750699999999  .000000001
.2955750760000001  .0000000001
.2955750761000001  .00000000001
.2955750761200001  .000000000001
.295575076124  .0000000000001
.2955750761243999  .00000000000001
.2955750761244399  .000000000000001
.2955750761244409  .0000000000000001
.295575076124441  1D-17

k=10

.2  .01
.2600000000000001  .001
.2660000000000001  .0001
.2669  .00001
.26692  .000001
.2669259999999998  .0000001
.2669265999999998  .00000001
.2669266599999998  .000000001
.2669266619999999  .0000000001
.2669266624999999  .00000000001
.2669266625899999  .000000000001
.2669266625929999  .0000000000001
.2669266625930998  .00000000000001
.2669266625931198  .000000000000001
.2669266625931208  .0000000000000001
.2669266625931216  1D-17

and, skipping, to k = 20

.1  .01
.13  .001
.136  .0001
.1364  .00001
.13648  .000001
.1364870000000001  .0000001
.1364870000000001  .00000001
.13648701  .000000001
.136487012  .0000000001
.1364870124000001  .00000000001
.1364870124300001  .000000000001
.1364870124390001  .0000000000001
.1364870124392001  .00000000000001
.1364870124392501  .000000000000001
.1364870124392591  .0000000000000001
.1364870124392594  1D-17

 Posted by Charlie on 2007-12-27 21:01:17

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