Author Topic: Geometric Proof  (Read 356 times)

olddog

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Geometric Proof
« on: August 06, 2021, 04:43:51 PM »
Bob;

There has been a debate over at AGN.  I've got this geometric proof I'd like to run past you.  I've done my best with it and I think it is correct.  I've handed a copy of it to Yrrah as he is familiar with the subject and I've handed a copy to my son who is also a computer scientist (and has a younger fresher mind).

I'd appreciate your thinking on it if you have time.  It might turn out to be fairly important to the pellet manufacturers, or not.  Still I'd appreciate your thoughts.

Image file is attached.  I did manage to make one error that I have found (and not fixed).  I don't think it invalidates the proof.
Here is the post over at AGN:
https://www.airgunnation.com/topic/d430l-177-hn-ftt-25meters-yrrah-roll-anecdotal-1/#post-1060966

Thanks
Mike
« Last Edit: August 06, 2021, 04:50:45 PM by olddog »


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rsterne

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Re: Geometric Proof
« Reply #1 on: August 06, 2021, 05:42:40 PM »
Your geometric proof looks good to me, I can see no errors in it.... Any change of R1, R2 or H1 should produce a change in H2....

Bob
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olddog

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Re: Geometric Proof
« Reply #2 on: August 06, 2021, 08:32:21 PM »
Your geometric proof looks good to me, I can see no errors in it.... Any change of R1, R2 or H1 should produce a change in H2....

Bob
Thank you for your time, sir.
Mike
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olddog

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Re: Geometric Proof
« Reply #3 on: August 07, 2021, 07:36:11 PM »
Bob,

Here is an exception to the proof.

They do not have to overlap completely.  They only need to overlap on the shared point where they are released.  Clearly this does not mean they will share the same "contact surface" entirely.  Said another way, two cones which overlap will roll on the same radius where they overlap.

If, for example, we were rolling two pellets which met that criteria and there was some point where they were gated the only way they could both pass a perfectly fitting gate would be for one to be completely overlapped by the other... but you get the idea.

Thanks again.
Mike
« Last Edit: August 08, 2021, 09:07:43 AM by olddog »
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rsterne

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Re: Geometric Proof
« Reply #4 on: August 08, 2021, 09:27:54 AM »
There are many pellets which would roll about the same vertex of the cone.... You could have a .177 cal, a .22 cal, and a .25 cal which would all roll about the same vertex point, but the radius would be different, and a function of the caliber....

In a similar manner, you could have vastly different pellets have the same starting point and ending point because they would follow a different route (radius) to get from where they started to a fixed end point, along different radii....



However, if they start sitting with one edge parallel (eg. against a ruler), they can only end up at the same point if "similar" tapers.... That is why it is critical for this test to work properly to use a fixed point on the pellet (the back of the skirt is ideal) indexed to the same point each time.... If in your "exception" example the skirts were both at the "end" of the cone, they would still follow the same path.... They should do that if the difference in diameters, in relation to the length (ie the taper), and the skirt diameter, was the same....



Bob
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olddog

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Re: Geometric Proof
« Reply #5 on: August 08, 2021, 06:48:21 PM »
Bob;

Thanks again for taking the time to look this over.  You are a treasure, sir.

There are many pellets which would roll about the same vertex of the cone.... You could have a .177 cal, a .22 cal, and a .25 cal which would all roll about the same vertex point, but the radius would be different, and a function of the caliber....
Yes, understood.

One cone which rolls about a different vertex (not the same vertex) could roll on a chord from the start point to the end point.  But because we actually tie it to a point on a line which it must touch at two places that is prevented?  It has to start on the same line, with the same angle. See attached.  I think that is the point you make below?

In a similar manner, you could have vastly different pellets have the same starting point and ending point because they would follow a different route (radius) to get from where they started to a fixed end point, along different radii....



However, if they start sitting with one edge parallel (eg. against a ruler), they can only end up at the same point if "similar" tapers....

Also not sure I understand this.   Similar tapers are defined along the entire length of the cone defined in the first proof in the thread.  Any section of a cone which has a similar taper is by definition part of the same extended cone. They can not roll to the same place because the vertex upon which they roll moves with the starting point and they exist at a different radius from the vertex.  The cone is also required to touch the starting line at r1 and r2 as defined in the first post.  So one pellet from one portion of the cone can not roll on the same radius as one from a different portion of the cone when you establish the starting point and require them to be indexed to that point.  The circle which a section of a cone describes when rolled is defined by the distance from the vertex whence it was extracted.  Even if it has the same slope it's rolling radius is different. Proof 1.

That is why it is critical for this test to work properly to use a fixed point on the pellet (the back of the skirt is ideal) indexed to the same point each time.... If in your "exception" example the skirts were both at the "end" of the cone, they would still follow the same path.... They should do that if the difference in diameters, in relation to the length (ie the taper), and the skirt diameter, was the same....



I can see that it is critical for the pellets to be indexed upon some feature, the waist, or the base and further for them to be indexed on a line at the starting point, which should be parallel to the slope of the cone for the test to work.  As far as I can tell there is only one way for a perfect machine to fail to distinguish between two cones.  Perfect is clearly not achievable.  Good enough certainly seems to be.

The example is the special case which only exists if the two "pellets" actually overlap in the cone which is probably pretty common in actuality. When they overlap in the cone the slope/taper (the tangent of the angle at the base of the cone) is the same for both sections.

In the previous examples we were discussing a a circle described by a line about the vertex, we are now talking about an area bounded by two concentric circles defined by r1 and r2 in proof 1.  The distance between those concentric circles (h1 in proof 1) is the width of the gate they must pass.  If one is fully contained within the other they can both pass through the gate only if the gate will pass the larger.  If they only partially overlap only one can pass through the gate. If the gate is arranged so that one will pass the other will not pass.  That presumes a perfect gate.

There is only one exception I can see.  The special case I pointed out in proof 2.

Again thanks for your advice.  I greatly appreciate it.
I am also attaching a picture @Yrrah gave me of his machine. He stacked those by rolling them.
Mike
« Last Edit: August 08, 2021, 07:18:54 PM by olddog »
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Alan

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Re: Geometric Proof
« Reply #6 on: August 09, 2021, 04:39:39 AM »
Maybe I'm reading something into this that isn't there, but the gist I get, is that indexing pellets improves accuracy?
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Re: Geometric Proof
« Reply #7 on: August 09, 2021, 08:09:25 AM »
Seems about right on all points.... Not a perfect system, but certainly good enough....

Bob
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olddog

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Re: Geometric Proof
« Reply #8 on: August 09, 2021, 09:08:39 AM »
Maybe I'm reading something into this that isn't there, but the gist I get, is that indexing pellets improves accuracy?
Yes, sir. That is the point Bob made.  The starting point and the angle of the lay at the start is very important.  The angle of the lay is very sensitive and that is why you need a strait edge and must align the two points on the pellet against that edge.  Any error in that lay is significant because it accumulates with each turn of the pellet.  If you can see even a sliver of light between the pellet and the edge it is too much.  An error in placement of the base is linear and translates one for one at the far end of the roll.  An error in the lay against the line is more significant.

The standard deviation of variation on his machine is under one percent.of the diameter of the roll.

Harry uses a feather to start the roll on his machine.  That is how he manages to get pellets to stack like in the above image and how he gets that less than 1% standard deviation.

We are working up specifications for a rolling table at this point.  I think they are about finalized.

« Last Edit: August 10, 2021, 04:37:44 PM by olddog »
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olddog

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Re: Geometric Proof
« Reply #9 on: August 09, 2021, 09:11:31 AM »
Seems about right on all points.... Not a perfect system, but certainly good enough....

Bob
Bob;
Thank you again, sir.  This is tedious stuff and your thinking is invaluable.
Here is something I got from Harry @yrrah last night.  You might find it interesting.

with respect;
Mike
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olddog

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Re: Geometric Proof
« Reply #10 on: August 12, 2021, 09:21:19 PM »
Bob;

I've released the 3d STL files for the table I designed. People can get them here:
https://www.airgunnation.com/topic/pellet-rolling-table-3d-print-files/

Thanks for your time, sir.  Much appreciated.

/r
Mike
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olddog

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Re: Geometric Proof
« Reply #11 on: August 19, 2021, 11:04:04 AM »
Your geometric proof looks good to me, I can see no errors in it.... Any change of R1, R2 or H1 should produce a change in H2....

Bob
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olddog

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Re: Geometric Proof
« Reply #12 on: August 19, 2021, 11:33:39 AM »
Seems about right on all points.... Not a perfect system, but certainly good enough....

Bob
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