Oops, I notice you used "your" abcd, where I use a=sun, d=outer, b/c are planets. The change of variables should be r'_planet = r_ring - r_planet r'_sun = r_sun + r_planet On Mon, Aug 3, 2015 at 1:47 PM, Tom Rokicki <rokicki@gmail.com> wrote:
Right. And if you rewrite things with a small change of variables:
b' = d - b a' = a + b
both of which are invariant under offsets, then you'll see that offsetting just works (since offsets don't change b' or a', and the final result multiplies d by 2pi so that drops out as well.)
On Mon, Aug 3, 2015 at 1:37 PM, William R Somsky <wrsomsky@gmail.com> wrote:
Assuming I've made no typos, the core is the following:
Define Ph[a,c,d,b] = b pi + (a+c) arccos [ ( (a-b)^2 + d^2 - (c+b)^2 ) / ( 2 (a-b) d ) ] + (b+c) arccos [ ( (a-b)^2 + (c+b)^2 - d^2 ) / ( 2 (a-b) (c+b) ) ]
(a is the ring radius, c is the sun radius, b is the planet radius, and d is the offset of the sun from the center of the ring)
Find {a,c,d,b,b,...} w/ d non-negative real, a,c,b... positive integers, a > c+d, a-d-c < 2b < a+d-c such that for all pairs of b1,b2: Ph[a,c,d,b1] - Ph[a,c,d,b2] is a multiple of 2pi
Once you have that, it's just solving: bx^2+by^2 == (a-b)^2 && (bx+d)^2+by^2 == (c+b)^2
On 08/03/15 12:38, Tom Rokicki wrote:
How many bits do you need?
I can give you an equation. It is not very complicated.
I think they are algebraic in some cases but not all.
On Aug 3, 2015 9:33 AM, "Bill Gosper" <billgosper@gmail.com> wrote:
Are these approximate x and y numbers roots of polynomials? If not, thought to be algebraic anyway? If the latter, and available to high precision, I could run them through RootApproximant. --rwg
On Mon, Aug 3, 2015 at 12:02 PM, William R Somsky <wrsomsky@gmail.com> wrote:
WHOOPS! Sorry, went to plot it, and found I'd done the wrong set.
The proper match to 26-9-7-5 is 26-9-5,10:
26 9 5 10 13.702903 0.072038 26 9 5 10 11.174443 0.338222 26 9 5 10 9.588008 0.585295 26 9 5 10 8.527280 0.807389 26 9 5 10 7.805237 0.997014 26 9 5 10 7.330664 0.144455 26 9 5 10 7.064324 0.236996
And 25-10-6-4 is 25-10-4,9:
25 10 4 9 13.702903 0.572038 25 10 4 9 11.174443 0.838222 25 10 4 9 9.588008 0.085295 25 10 4 9 8.527280 0.307389 25 10 4 9 7.805237 0.497014 25 10 4 9 7.330664 0.644455 25 10 4 9 7.064324 0.736996
Also 24-11-3,8:
24 11 3 8 11.174443 0.338222 24 11 3 8 9.588008 0.585295 24 11 3 8 8.527280 0.807389 24 11 3 8 7.805237 0.997014 24 11 3 8 7.330664 0.144455 24 11 3 8 7.064324 0.236996
So, using the 7.805 displacement you get the attached diagram, (also available as https://drive.google.com/open?id=0B2889vNnzpsTWVJzTTZ6TUNod0U) which has concentric gears.
WRSomsky
On 08/03/15 11:23, William R Somsky wrote:
In my tabulations, the set you call 26-9-7-5 is 26-9-5,12 (ring-sun-list-planets) where the planets are in increasing size and all from one side of the centerline. (7 & 12 form a complementary "somsky-set", if you want to call it that, but the 7 is on the opposite side of the 5)
For 26-9-5,12 you get the solutions (ring, sun, planet, planet, offset, sun-phase):
26 9 5 12 16.048313 0.500000 26 9 5 12 13.388239 1.000000 26 9 5 12 11.624141 0.500000 26 9 5 12 10.372542 0.000000 26 9 5 12 9.448488 0.500000 26 9 5 12 8.750000 1.000000 26 9 5 12 8.215897 0.500000 26 9 5 12 7.807356 1.000000 26 9 5 12 7.498839 0.500000 26 9 5 12 7.273254 1.000000 26 9 5 12 7.119219 0.500000 26 9 5 12 7.029480 1.000000 26 9 5 12 7.000000 0.500000
For 25-10-4,11 you get:
25 10 4 11 13.388239 0.500000 25 10 4 11 11.624141 1.000000 25 10 4 11 10.372542 0.500000 25 10 4 11 9.448488 1.000000 25 10 4 11 8.750000 0.500000 25 10 4 11 8.215897 0.000000 25 10 4 11 7.807356 0.500000 25 10 4 11 7.498839 0.000000 25 10 4 11 7.273254 0.500000 25 10 4 11 7.119219 0.000000 25 10 4 11 7.029480 0.500000 25 10 4 11 7.000000 0.000000
If you use the same offset, you get concentric matches.
"But you said that they can always be concentric, barring overlap? What about the 16.04?" you ask. Well, in the 25-10-4,11 case, an offset of 16.04 causes the sun and ring to overlap, and my program (and Tom's I certainly believe) doesn't even consider those cases.
If you used Tom's program, I expect it might be using the greatest possible offset, which would be 16.04 for one case, and 13.38 for the other. (Also, there may be scaling problems, as I don't know if Tom scales his image to the size of the ring gear -- mine normally does -- otherwise a set w/ a ring of 7 would look tiny, while a ring of 57 would go off the image area.)
Bill
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