geometry questions, please...

[Nikolas]

and here comes the time that I ask for help! 


Grab the full size pic HERE

The above is a plan I want to use in one of my pieces, but I'm unable to do anything else than draw it by hand (in an awful way).

It is a single line which starts from the left, goes to the right with three waves, each one smaller than the other, then goes around a couple of times when it happens to coincide with part of the route already taken (the bolder parts) and then repeats adlib. A kind of fractal if you will. Please note that the arrow heads (good willing will make them arrow heads ) point the direction of the line, and the numbers are there just to help and give direction, what comes first, etc... Oh, also: the image is not complete, since I simply run out of paper! But I hope the missing parts are clear.

My questions and problems:

1. Is it possible for the line to coincide (the bold parts)? I mean in order for this to happen the wavy parts and the circly parts need to have identical angles in that particular part of the line, no?
2. If it's possible, would it be possible to have a specific size for the wavy lines, so that the bold parts would fit an exact number or times between then? (I'm pretty sure that this doesn't make sense, so I'll try in a different way: If this single line was a single thread, and we were to color the bold parts, would it be possible to accuretly predict and make the plan as such as the colored/bold parts would fit an exact number of times between each other?)

Once we have these solved

3. Would someone be so kind to draw this on a computer?

Lastly

4. Could it be that this is a known geometrical shape and I'm busting myself and the AGS forums for nothing?

Thank you!

[Evil]

Fibonacci Sequence?

[Jim Reed]

I'm not a mathematician, but it look like you could fake it by drawing half circles or even quarter circles. You would just need a function that can draw you a part of the circle in any given direction with a given radius. It's hard to discern from your drawing if those paired wawes have the same radus or each wawe has it's own. Hence, use half circles for the first case and quarter circles for the second case.
Hope it helps. If you get stuck, just break the problem in smaller steps, so you can understand them. =)

[Bulbapuck]

Mathematically, I believe the answer to (1) is no. I'm pretty sure it can collide in a maximum of 3 points. However, if you had proper mathematical formulae for how you want the line to move you could get so close that the human eye can't see a difference. And also if you account for the thickness of the line you could make an excellent approximation. But you need a proper mathematical discription of the line (and even then it's over my head )

How do you plan to use this for one of your pieces? I'm intrigued now

[Alun]

1. Is it possible for the line to coincide (the bold parts)? I mean in order for this to happen the wavy parts and the circly parts need to have identical angles in that particular part of the line, no?

By "coincide", do you mean match over an extended length?  If so, it's certainly possible, but the derivatives of the lines would probably have to be discontinuous -- in less technical terms, they'd have sharp angles in them.  It would probably be possible to make those sharp angles large enough (that is, close enough to 180 degrees) that they wouldn't be noticeable, though.  (You're going to have a slight discontinuity in the derivative where the waves meet the spiral anyway -- you could arrange it so the first derivative is continuous, but you'd still have to have a discontinuity in the second derivative, or at least somewhere down the line.  But that's probably not worth worrying about.)

However, it may be enough for you to have them tangent at a point... to have them touch at one point, with the same slope there, but not quite touch elsewhere... but they'd probably be close enough.  That's certainly possible, and I'd recommend that rather than trying to have them exactly coincide over extended distances.

2. If it's possible, would it be possible to have a specific size for the wavy lines, so that the bold parts would fit an exact number or times between then? (I'm pretty sure that this doesn't make sense, so I'll try in a different way: If this single line was a single thread, and we were to color the bold parts, would it be possible to accuretly predict and make the plan as such as the colored/bold parts would fit an exact number of times between each other?)

Possible, yes, but it depends on exactly what you want.  The most common kind of "wavy line" is a sine wave, but the "waves" on a sine wave are evenly spaced, which you say they're not supposed to be here.  Aside from that, though, what you've drawn (the "wavy" part of it) does look like a sine wave, and I think a variation on the sine wave would do nicely -- you'd just take the sine of some function of x instead of a sine of x itself.  (Instead of a sine wave, you could, as Jim Reed suggested, fake it by drawing half-circles... but what you've drawn looks closer to a sine wave, and anyway a sine wave is much easier to deal with in graphing programs.  (And with the half-circles you run into again a discontinuity in the second derivative, but nobody's likely to notice that.))

What exact function of x you'd use, though, depends on what kind of spiral you want.  There are many kinds of spiral... one of the most common is the Archimedean spiral, but that has its arms a constant distance apart, which you don't want if the "waves" aren't supposed to be evenly spaced.  So you'd have to decide what kind of spiral you wanted to use... though if you weren't sure, I'd recommend using a circle involute; goes along well with sine waves, and seems to more or less match what you've drawn.

3. Would someone be so kind to draw this on a computer?

I could, but it would take me a little while.  I don't really have a dedicated graphing program on my computer, though I could whip something up using OpenOffice if I had to, and it would take me a bit of work to figure out the exact function to take the sine of to get the "wavy part" to match the distance between the arms of the spiral.  (I could fudge it in Adobe Illustrator, but I'd rather try to actually graph it mathematically rather than fudging it like that.)  Maybe I'll try to get to a it a little later today, if it turns out that what I've described above is what you're looking for.

You know what?  On second thought, I think maybe I'll go ahead and fudge it.  If that turns out not to be good enough, maybe I'll try a more technical version later, but fudging it will be a lot quicker, and if that's good enough, it's good enough.  

4. Could it be that this is a known geometrical shape and I'm busting myself and the AGS forums for nothing?

I'm sure it's not a single known shape, but you could produce it by combining several shapes, as I mentioned above.

[EDIT:

Okay, here's a rough fudged version of the shape, if I understood correctly what you're looking for:



Certainly it's not perfect, but if this is along the lines of what you want I can do it more carefully (and maybe actually plot it mathematically instead of fudging it like this)...  (I can also easily make it larger and/or add more spirals to it; I did this in vector graphics, so resizing is easy.)]

[Nikolas]

Evil: Nope, not the Fibonnacci series. Plus I've used that one previously, so I tend not to use again stuff I've used before! LOL

Jim Reed: Almost there, but in my mind, I'm hoping to have the circlces and the wavy lines moving smoothly...

Bulbapuck: I do think you are right, if the wavy lines were sine waves, so there would actually be maximum 3 points, but while I keep mentioning accurate, etc, I'm not after the mathematical clarity here, exactly, but rather what Alun did!

I can't right now speak of my idea, yet. But hopefully once done I will be able to spill the beans! Sorry...

And here we come to Alun:

1. Yes, over an extended length, and this length needs to be a somewhat exact fraction of the rest of the circle line. This is why I did the first 'sine' wave larger and then kept going smaller, to keep the fraction the same.

I would imagine that I would either have to break the 'fluidity' of the circles, or the waves, and I do think that breaking the waves seems more proper in my weird head.

2. Again yes and spot on. The equal distance circles spiral is not what I'm after, cause it won't work. (the fraction idea I mentioned above). The circle involute seems proper, so the goal here is to have the length of the spirals be divided somehow in a 'tidy' way by the parts which match the waves. Makes sense? :S:S

3. THANKS ! ^_^ For drawing purposes it's cool enough (in fact the original and one repeat would be fine). The only comment I would make is that I would hope to have the last circle of each bunch to go through the previous bunch. (after 22 in my drawing. There are two waves and the third time the circle passes through is from the previous waves and circles. JESUS! Is not enough that I'm temparing with things that I can't do, now I can't even explain them properly)

4. Great for now! I prefer to claim authorship of this shape! LOL

Your image: his is almost what I want with the two exceptions:
a. Two waves after the completion of the first bunch, not three, so that the third time the circle passes, it is forced to pass from the initial starting point.
b. If you can get me the matching lengths of the waves and the circles to have the same relationship for ALL the times this happens (so the same fraction, and if possible a somewhat normal fraction (cause 5.3/9.7 would be rather difficult to use!) that would be simply awesome!

Huge thanks guys! Really, you are the best!

[SSH]

I know what it is, its your design for a new clef! The Siderble Clef!

[Alun]

Your image: his is almost what I want with the two exceptions:
a. Two waves after the completion of the first bunch, not three, so that the third time the circle passes, it is forced to pass from the initial starting point.

Ah, OK; I'd noticed there were only two waves in the second spiral in your drawing, but I wasn't sure whether or not that was intentional.

b. If you can get me the matching lengths of the waves and the circles to have the same relationship for ALL the times this happens (so the same fraction, and if possible a somewhat normal fraction (cause 5.3/9.7 would be rather difficult to use!) that would be simply awesome!

They already were, pretty much, because of the way I put it together.  What that fraction is, however, is hard to say -- technically, while it looks like they're overlapping over a significant length, they're really only tangent, as I mentioned in my first post.  However, the fact that the line has a finite thickness, that tangency does become a real overlap in the drawing (as I think Bulbapuck was getting at in his post).  To figure out exactly how much of the circle overlaps, though... well, honestly, there's no easy way to do it short of measuring the distances.

Hm... actually, on second thought, I take that back, slightly.  Not the part about there being no easy way to measure the exact overlap -- the part about the lengths having the same relationship.  The problem is that as the curves scale up, the lines stay the same width, which means that since the overlap is largely due to the line width it's actually slightly less as the curves get bigger (i.e. at the outside of the spiral).  It may be close enough anyway, but if you want a more precise matching of the ratios, there are two ways to do it: either cheat it by adjusting the curves slightly, or make the line itself get thicker near the outside of the spirals.

But anyway, that aside, here's the revised drawing; hope this is closer to what you're looking for:

[Nikolas]

The image is simply perfact!  :o What I was looking for (cause in fact I was hoping that someone would bring me something which would look 'imperfect' if you get what I mean). What I can't tell is if the waves change size as the move to the right (the ones of almost equal size). I had figured out that the circles get much larger each time, so the waves also need to get much larger over time, which should, theoriticaly, also work better with the angles (bigger both, less angular they get).

The only thing remaining would be to find out the fractions now...

But yes, I hardly want to tax you in any way. I will probably print of this image (can I get a bigger size, please? (like an A4 size)) and try to count 'by hand' the fraction, or something... unless you're more eager to do more work, without knowing why, or having any reward apart from my gratitude!

[Alun]

What I can't tell is if the waves change size as the move to the right (the ones of almost equal size).

They do; each wave is 90% the size of the previous one.  (Well, that's counting the up and down waves; counting just the up waves, each wave is 81% the size of the previous one.)  If that's too subtle, I may be able to make a version with a more noticeable decrease in size of the waves.

I will probably print of this image (can I get a bigger size, please? (like an A4 size))

Sure; like I said, I did it in vector graphics, so I can easily render it any size you want.  It may not be best to post such a big file to the board, though; should I e-mail it as an attachment to the e-mail address listed in your user profile?

[monkey0506]

Geometry questions? Sure, I'll help you out with that:

Square?
Circle?
Triangle?
Rhombus?
Trapezoid?
Parallelogram?
Octagon?
Hexagon?
Dotetracontagon?

Need more?

[Nikolas]

What I can't tell is if the waves change size as the move to the right (the ones of almost equal size).

They do; each wave is 90% the size of the previous one.  (Well, that's counting the up and down waves; counting just the up waves, each wave is 81% the size of the previous one.)  If that's too subtle, I may be able to make a version with a more noticeable decrease in size of the waves.

Ah... right. I couldn't tell.

Tell you what. Once I finish chasing around my tax issues today morning (it's morning in Europe), I'll print off the current one and see what I can make of with the fractions I was talking about. If it's easy to change size of the waves, it might actually be easy to approximate having the same relationship in fractions for each cicle.

I will probably print of this image (can I get a bigger size, please? (like an A4 size))

Sure; like I said, I did it in vector graphics, so I can easily render it any size you want.  It may not be best to post such a big file to the board, though; should I e-mail it as an attachment to the e-mail address listed in your user profile?
[/quote]Better use this: nikolasideris *AT* hotmail.com (not sure if it's the same in my profile, and don't have the time to look right now.

[Alun]

OK; just sent you an A4-sized copy of the image.  Hopefully it'll help.