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Joined: January 17, 2013 Posts: 373 Submissions: 5 Location: Probably in the garage...

AR and what it does to rings 

Posted on Sat Apr 12, 2014 9:16 pm  Last edited by Levi on Wed Nov 11, 2015 4:19 am; edited 1 time in total 
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While playing around in Blender the other day I started thinking about an easy way to display the effects that AR has on rings, so I created five rings with an AR of 1 through 5 respectively, the minor radius or "gauge" of 0.25 remained the same only the major radius or "ID" was increased. I then added as many of each ring size into one of the same sized ring as I could while keeping them at 90 degrees from the linking ring.
Here is the result. Even though they may look smaller all of the rings have the same WD or gauge.
UPDATE: The above image has some issues as outlined below, here is a more accurate one.
FYI: If you want to try it at home, you need to bump the major and minor segments of the torus to at least 96/48 otherwise the torus is too square which causes the rings to sit too close to each other, providing room for extra rings that won't fit in real life.
My question is not directly about calculating AR or how the angles of the rings will affect the ring count, just the simple idea that increasing the AR by one point lets you add 3 more rings than you could at the previous AR?
My not so mathematically inclined mind knows that Pi shows up somewhere in all of this and that the ring count increase is probably 3.14, which means by an AR of 8 you would be able to add one additional ring for a total of 22 rings instead of 21.
AR1=1
AR2=3
AR3=6
AR4=9
AR5=12
AR6=15
AR7=18
AR8=22
Given the typical AR range for maille purposes is between 2 and 7, it seems to me that AR progression at the same gauge between AR2 and 7 can be described as linear at three additional rings per increment?
Are these results accurate enough and reasonable for learning purposes to start explaining the concept of AR?
Mostly Harmless 


LordPyridine
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Joined: March 21, 2013 Posts: 127 Submissions: 3 Location: Don\'t worry about it...



Posted on Sat Apr 12, 2014 10:01 pm 
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*facepalms for not trying this first*
As far as accuracy goes, it should work fine as long as the rings are as round as possible, meaning saw cut. A 1 AR ring may be difficult to make conventionally, perhaps have one created from a nonmetallic material? The picture alone would be very useful in teaching AR, and the ring within AR x formula would've certainly helped when I first learned.
These are fleeting moments within we live.
In tomorrow I die.
Unto today I live forever. 


Joined: February 8, 2013 Posts: 737 Submissions: 61 Location: Australia



Posted on Sat Apr 12, 2014 10:11 pm 
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That is a great illustration of AR!



Joined: January 17, 2013 Posts: 373 Submissions: 5 Location: Probably in the garage...



Posted on Sun Apr 13, 2014 1:41 am 
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Thanks both of you. I thought it was an easy way to convey the base idea, I hope it helps others.
It's a fairly simple build process in Blender to reproduce what I've done, I'll try to get the process documented in the next few days and post it.
Mostly Harmless 


LordPyridine
[ Big Voice ]
Joined: March 21, 2013 Posts: 127 Submissions: 3 Location: Don\'t worry about it...



Posted on Sun Apr 13, 2014 3:10 am 
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Before you go on a Blender binge, have you tried animating them with hardbody? Assuming your computer doesn't burst into flames, it could help determine the looseness of the maximumamountofringswithinaringfigure. That's a mouthful, how about ring cramination limit?
These are fleeting moments within we live.
In tomorrow I die.
Unto today I live forever. 


Joined: January 17, 2013 Posts: 373 Submissions: 5 Location: Probably in the garage...



Posted on Sun Apr 13, 2014 3:20 pm 
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LordPyridine wrote:  have you tried animating them with hardbody? 
By that I'm guessing you mean, did i apply a rigid body to the rings and run physics simulations on them, for which the answer is yes. I let the physics engine settle most of the rings into place, took a few nudges here and there but for the most part I just let "gravity" do it's thing. The AR 2 cluster is so tight the rings jiggle around in a counterclockwise direction stuck in a seemingly perpetual tug of war. Gravity pulls on one ring which bumps the next that moves the third to the point where gravity starts pulling it and the cycle continues. As for performance the impact is minimal there are only 40 objects in the scene and once a cluster has settled into place, remove the "Dynamic" check mark for those rings to stop them from wiggling around while you work on the next cluster.
In Blender, as I mentioned about increasing the major and minor segments to make the ring more round. If left at the default the "cramination limit" goes up because the rings are not really round, they are 12 sided which means they squeeze together closer than real round rings would. Even at 48 sided they still catch on each other a little but it's close enough for most purposes. To get really good results the segments would have to be doubled or tripled, so 96 or 128. At that point performance could become an issue, I'll play around and let you know.
To determine the true "craminiation limit" multiply the AR by 3.14 then subtract 3.14 from that total. The whole number is the amount of rings you can fit, everything after the decimal point is "slack", a fraction of a whole ring. Here is a breakdown of AR 2.0 to AR 3.0 ring count to slack space. The math seems to work out but I could be missing something...
AR 2.0 is 2.0 x 3.14 = 6.28  3.14 = 3.14 , slack .14 * .3 = 0.042 * 2 = 0.084
Total rings: 3 Total slack: 0.084 of a ring
AR 2.1 is 2.1 x 3.14 = 6.594  3.14 = 3.454 , slack .45 * .3 = 0.135 * 2 = 0.27
Total rings: 3 Total slack: 0.27 of a ring
AR 2.2 is 2.2 x 3.14 = 6.908  3.14 = 3.768 , slack .76 * .3 = 0.228 * 2 = 0.456
Total rings: 3 Total slack: 0.456 of a ring
AR 2.3 is 2.3 x 3.14 = 7.22  3.14 = 4.08 , slack .08 * .4 = 0.032 * 2 = 0.064
Total rings: 4 Total slack: 0.064 of a ring
AR 2.4 is 2.4 x 3.14 = 7.536  3.14 = 4.396 , slack .39 * .4 = 0.156 * 2 = 0.312
Total rings: 4 Total slack: 0.312 of a ring
AR 2.5 is 2.5 x 3.14 = 7.85  3.14 = 4.71 , slack .71 * .4 = 0.284 * 2 = 0.568
Total rings: 4 Total slack: 0.568 of a ring
AR 2.6 is 2.6 x 3.14 = 8.16  3.14 = 5.02 , slack .02 * .5 = 0.01 * 2 = 0.02
Total rings: 5 Total slack: 0.02 of a ring
AR 2.7 is 2.7 x 3.14 = 8.478  3.14 = 5.338 , slack .33 * .5 = 0.165 * 2 = 0.33
Total rings: 5 Total slack: 0.33 of a ring
AR 2.8 is 2.8 x 3.14 = 8.79  3.14 = 5.65 , slack .65 * .5 = 0.325 * 2 = 0.65
Total rings: 5 Total slack: 0.65 of a ring
AR 2.9 is 2.9 x 3.14 = 9.10  3.14 = 5.96 , slack .96 * .5 = 0.48 * 2 = 0.96
Total rings: 5 Total slack: 0.96 of a ring
AR 3.0 is 3.0 x 3.14 = 9.42  3.14 = 6.28 , slack .28 * .6 = 0.168 * 2 = 0.336
Total rings: 6 Total slack: 0.336 of a ring
Let me know if that makes sense.
Mostly Harmless 


Joined: March 3, 2002 Posts: 987 Submissions: 244



Posted on Tue Apr 15, 2014 4:48 am 
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Your program is lying to you. Maximum amount of rings at AR 2 is exactly 2 by definition. Do the math by hand and you'll see that it takes an AR of minimum 2.155 to accommodate 3 rings.
Here's a list of minimum AR's for vertical rings.
1 ring=1.00 AR
2 rings=2.00 AR
3 rings=2.16 AR
4 rings=2.41 AR
5 rings=2.71 AR
6 rings=3.00 AR
7 rings=3.31 AR
8 rings=3.63 AR
9 rings=3.93 AR
10 rings=4.24 AR
11 rings=4.56 AR
12 rings=4.87 AR
This metric is really only useful for Japanese weaves so there's not much point in going above 12 rings.



Joined: August 05, 2010 Posts: 601 Submissions: 28 Location: Bar Harbor, ME, USA



Posted on Tue Apr 15, 2014 3:30 pm 
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Im finding it difficult ending with one equation for all ARs...
But here is the best I got. This is valid for when N is greater than 4 (at an AR of about 2.107) because it was generated geometrically with angle quadrant assumptions.
N <= Pi/sin(1/(AR1)), where N is the number of rings.
Interestingly, the Pi*(AR1) approximates this closely (small underestimate). Neither can/should be used below an AR of 2.1 to estimate. Also as I've seen with math behind orbital weaves, despite getting an exact mathematical result... rings made in real life are not a circle, which will cause significant variations in how many you can physically fit.
while(!project.isFinished())
project.addRing();
// Maille Code V2.0 T7.1 R5.6 Eo.n Fper MFe.s Wsm Caws G0.81.6 I2.48.0 Pn Dcdejst Xw1 S07 


Joined: January 17, 2013 Posts: 373 Submissions: 5 Location: Probably in the garage...



Posted on Tue Apr 15, 2014 5:32 pm 
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Thanks lorenzo, Blender isn't lying it's just trying to work with what I've given it, which as it turns out are not quite specific enough numbers. My "done by hand" math above is flawed which I presumed it to be on one level or another. Thank you, TCG for providing a better equation.
Once I redid the four rings with an AR of 2.16 (0.790.25/.25 in Blender) they do simulate much better, the AR 2.0 rings did have some very minute overlap of the meshes upon closer inspection, which as indicated, in the real world would stop it from working. It also explains the perpetual tug of war my previous AR 2 rings where performing along with slower simulations.
Now given I'm trying to keep this as simple as possible for someone who's learning AR, and while that .155 does make all the difference it's the sort of detail that makes eyes glaze over and minds wander. As you learn AR you'll realize just how stupidly specific you have to be in certain situations but that's something to be covered later in the "advanced topics" sections.
Where do you start? You link one ring into another.
One of the next questions seems to be. How many more rings can I fit in this other ring?
Telling someone that each time you increase the AR by 1 you can add roughly 3 more rings into that new ring, really seems to help people grasp the idea. Followed by showing them a few strips of E4in1 each with the same gauge but differing ARs really helps solidify the concept. Other complexities can be further defined from there.
Oh and LordPyridine, I realize now what you're implying about the "cramination limit", I'm expecting the math on that one would be evil. The partial spaces between rings that the other rings would fit into along with the induced angles would throw all sorts of things into the math.
Mostly Harmless 


Joined: March 3, 2002 Posts: 987 Submissions: 244



Posted on Tue Apr 15, 2014 7:11 pm 
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I haven't used blender in probably about 15 years but most engineering CAD programs have an intereference check tool. It's pretty useful for these types of things.
If you want to understand that correct math behind AR it's also relatively easy. I assume that you can draw regular polygons in blender? Any Ngon with sides=wire diameter and N=number of rings is a perfect analog. Draw the minor diameter of your rings centered on the points of the Ngon and the ID tangent to the outside of those minor diameters.



Joined: January 17, 2013 Posts: 373 Submissions: 5 Location: Probably in the garage...



Posted on Sat Apr 19, 2014 1:24 am 
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If in my first post I cited that the silver rings had an AR of 2.16 vs 2.0 would you be able to confirm or deny it visually?
Probably not, the math is important but it's a finer detail that can be covered once the user/student understands the base idea. To that end lorenzo, your list of vertical rings/AR mins is great sort of closing slide to a presentation on the topic.
My goal here is to simplify the understanding of AR via visual queues for someone first learning about mailling or to provide others with an easy way to teach the concept. Being able to eyeball a ring and know roughly what AR it is, is a very helpful thing.
Many people when starting out don't know what "the rings aspect ratio" implies, they may know what a ring is, how aspect works and what a ratio is but putting it all together is often a challenge. For some the math clears things up, for many it just muddies the waters.
I'm trying to find/build visual examples along with simple rules that exemplify the concept without having to create spreadsheets or by applying math with decimal places. Of course those spreadsheets and math become required items over time if you keep at it but in the beginning as always the simpler the better.
Blender certainly isn't SolidWorks but it does the trick for the price. From a quick search it seems like Blender is lacking such a notification check at the moment which doesn't really surprise me given engineering CAD is not a primary focus for Blender but then again I haven't searched very hard.
If someone wanted to replicate my examples for say teaching purposes, a different layout, other colours or to change the AR of the rings. The process is fairly simple and just about anyone can get/install Blender even if the results are not technically perfect with my current noob Blender setup.
While the collision detection may have issues if not feed the proper math, I'm fairly certain the tori are being sized correctly. ID/WD=AR, replicated using the major radius minus the minor radius to get the ID then divide the ID by the minor radius or 0.790.25/.25 for a ring with an AR of 2.16. In the example below I used copper rings with an AR of 4, set the MajR=1.25, minR=0.25 then added a cylinder with a radius of 0.75 to fill the center. Lorenzo, I presume this is sort of what you implied above about placing the minor diameter to points on the Ngon.
After further reducing the collision margins to 0.0001 from 0.001 the rings sit/relax and move in a much more true to life fashion. Sadly, the above images where generated prior to adjusting the margins so the spacing is a bit off.
Here is a second visual AR helper I quickly whipped up. Ring groups from top to bottom are (if memory serves...) 2.16, 3.4, 4.4, 5.2 all chosen for no particular reason. I suppose they should be 2.16, 3.16, 4.16 and 5.16 to keep things as linear as possible but I was tired. I'll do up a better one later with colored rings and an AR listing. The copper rings simply provide a visual marker between clusters.
lorenzo wrote:  If you want to understand that correct math behind AR it's also relatively easy. 
I'm all for relatively easy math but by that I take it you mean something more than ID/WD=AR or you use a different formula from the one suggested by TCG to calculate the vertical rings per AR?
Mostly Harmless 


Joined: March 3, 2002 Posts: 987 Submissions: 244



Posted on Wed Apr 23, 2014 7:52 am 
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I could tell pretty much instantly that your AR was off on the silver rings. The interference was very noticeable, for me at least. The new renders are much better, I don't notice any obvious abnormalities at first glance. It looks like there might be some interference on the bottom left side of the 4.4 AR rings but that would be consistent with trying to cram 11 rings into that AR so it's safe to say that you're making the rings correctly sized.
What I was trying to point out about the real math behind AR is that it's based on geometry, not algebra. Rather than trying to fit the numbers into crude approximations of a formula it's more intuitive and accurate to actually see that the rings with AR of 3 will form a regular hexagon. If it looks like a sloppy pentagon then you know that the AR is only around 2.82.9, a reasonably tight square is about 2.5, etc. With a bit of practice anyone can distinguish polygons up to a 12 sided figure(~AR5) just by visualizing a clock face.



Joined: January 17, 2013 Posts: 373 Submissions: 5 Location: Probably in the garage...



Posted on Wed Apr 23, 2014 2:16 pm 
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Ah...Now I get what you meant, thanks for the clarification. The clock face is an easy way to think about it.
Mostly Harmless 


Joined: January 17, 2013 Posts: 373 Submissions: 5 Location: Probably in the garage...



Posted on Wed Nov 11, 2015 4:11 am 
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I'm slowly working down my todo list, so I remade the AR chain image with some more details and a slightly better render. The chains are sitting where "gravity" left them after a few "minutes" in the physics engine. They are also at almost 90 degrees off from each other, seemed a little too blueprint ish when squared.
AR Chain v2
I have a few other blends to finish first but I'll also redo the first layout with properly sized rings and linear AR values when I have a few minutes. Following that I'd also like to recreate it using triangles, squares and pentagons keeping with lorenzo's geometry explanation but that will take a bit longer.
Hmm, seems by taking one item off my todo list, I've added two more to the list...
/shrugs, sounds about right.
Mostly Harmless 


Joined: August 05, 2010 Posts: 601 Submissions: 28 Location: Bar Harbor, ME, USA



Posted on Wed Nov 11, 2015 12:05 pm 
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Levi wrote:  Hmm, seems by taking one item off my todo list, I've added two more to the list...
/shrugs, sounds about right. 
Yup, sounds right. *Glances at his own todo list in horror.*
while(!project.isFinished())
project.addRing();
// Maille Code V2.0 T7.1 R5.6 Eo.n Fper MFe.s Wsm Caws G0.81.6 I2.48.0 Pn Dcdejst Xw1 S07 

