Can you parametrize this helix?
Oscarville
x=t
y=sin5t
z=cos5t gives a helix along the x-axis
I'm trying to parametrize a helix, like this, that follows the circumference of a circle. Can anyone help?
gudi
You mean geodesics on a torus? They are formed by bending the above
helix around x-axis.
Narasimham
gudi
Oscarville
The one problem, though, is that the helix grows as it curves around the inner boundaries of the circle.
I think, actually, I'll post the drawing I came into contact with, and see if there's anything anyone cares to do with it. I believe the math is beyond me, however the illustration has some interesting features. So, I'll just put it out there in another post.
Oscarville
Oscarville
Also, the expression of the "spiraling (logarithmic?) curve" is of interest to me. I want to find a connection between the two. I will explain why. But here's what I have.
I call this thing, a "contuatia". It is a circle of undefined radius, divided into twelve segments. The first line begins at the tenth division of the circle and descends and ascends across the circle, intersecting at the first division of the circle, with a value of e.
You might ask, how did I get such a value? Some people are able to estimate the value of certain things, simply by looking at them closely. e appears to my mind, as an absolute value -somewhere off in space. Hence, when I consider the value of e in my mind, I find that it falls exactly where I have indicated on the illustration. The same principle applies for the indication of the other numbers within the drawing as well. I'm simply able to imagine the locations of certain values. The drawing itself, was not a creative escapade. The lines appeared before me, just as the values on them did. Hence I simply drew what was all ready there. Poetic, n'est pas?
However, moving along. The second line leaves the tenth division of the first circle and approaches the apex of the circle, the twelfth division. Where the helix assumes its maximal outer value, there is an intersection between the twelfth division. The value of the helix at this point is 108. Here after it grows in value and intersects with the circle at every 30 degrees, only to terminate at the ninth division of the circle. This line, the second line, is a theoretical model of what may be the nature and formation of numbers, describing their distribution, in relation to the "first" line. I don't know, but...
Some things to consider. Where the first line equals 1, the second line equals 12. This is the primary relationship between the lines. Where the second line equals 36, the first equals 2. Hence a linear relationship is out of the equation. Obviously, from the drawing, that is the case. I'm not a genius, believe me.
Therefore the connection between line 1 and 2 is non-linear and, more importantly, can be understood by a simple description of the parameters which define each line. Notice also that where the first line equals (pi), one can draw a perfect circle, intersecting the value of 360 on the second line, the first division of the circle, and e, where it stands on the second line. If that's not worth calling mom, I don't know what possibly could be.
If the values on these illustrations seem too fascinating to be true, believe me, it is not because I have spent hours concocting this "contuatia", as I call it. As, I've already said, the lines appeared before me in my mind and I simply drew them. The values on each line were found by pure estimation.
Here's the basic value of this drawing. Where 12 and 1 connect on the second and the first line, the "distance" establishing their relationship is more or less linear. Where 36 and 2 connect, the relationship is becoming less linear. Notice, that everything important is in base 12 on the second line, and is something "else" on the first. The number 1, for instance. 2, e, pi, and so on -for the first. On the second, every intersect between the circle's bound and the maximal outer reach of the helix is a base 12 number. The growth is quite rapid. Termination at the ninth division might be the end of numbers as we know it. If you're curious what the spiral at the ninth to the middle is, I think it is a very rapid descent from a very HIGH number to zero, very rapidly. TOO rapidly, one might say.
e and 108 relate by the arc length formed by 30 degrees of the circle. And, finally, the last thing I can say is that where the first line equals (pi), the relationship it shares with the second is perfectly semi-circular. Hence, my interesting drawing.
I am not smart enough to put these lines into mathematical form. But if I could, I would be going nuts over this thing.
The lucky one will be the person who can paramaterize the first line, parameterize the second, describe the "distance" between the two and how it changes such that we get such interesting results between such interesting numbers on two such distinct curves. -finally, continue finding things within the patterned structure of the helix as it intersects the circle every 30 degrees (excepting the second loop, which occurs at 15 degrees).
The value of number line as it terminates at the ninth division represents the highest imaginable number. All other values are phantasms.
Attachments available from http://mathforum.org/kb/servlet/JiveServlet/download/125-2220049-7336587-661792/contuatia%20002.jpg and http://mathforum.org/kb/servlet/JiveServlet/download/125-2220049-7336587-661791/contuatia%20001.jpg
Oscarville
"You're crazy."
"This doesn't mean anything."
"What does this mean?"
"Where did you take this from?"
"Wow."
"Here's the answer."
Anything like that. And I'll be happy. Cheers. Happy Holidays.
Avni Pllana
Hi Oscarville,
a circular spiral (see attachment) can be obtained using following equations:
x = (R+r*cos(w1*t))*cos(w2*t)
y = (R+r*cos(w1*t))*sin(w2*t)
z = r*sin(w1*t)
where w1 = n*w2 .
Best regards,
Avni
Attachment available from http://mathforum.org/kb/servlet/JiveServlet/download/125-2220049-7339195-662599/circular_spiral.png
Oscarville
Here's some things, I've realized, later, in thinking about this whole thing.
So, say: s=r(theta)
As s increases, the initial line emanating from the 11th division of the circle begins. For instance when the circle has gone 15 degrees clockwise, the first line has reached a minimum in its decent at the value 1. And as the circle reaches 30 degrees the value of the spiral is e. At 45 degrees, I believe, the spiral (the first one I mean, not the circular spiral) has a value of pi. And so on. As the circle continues along its path, the logarithmic spiral begins to turn in on itself faster and faster.
Thus, as the circle approaches the ninth division, the spirals are spiraling very fast. And then it all quits at the ninth division. So, although this first spiral isn't a logarithmic spiral, per say (I don't know what it is exactly), it might have an exponential kind of growth, of some kind, to a power of (theta), in some manner.
Just something to ponder, if anyone's interested.
I think this particular rant I'm going on now, better explains what I meant initially by the "relationship" between the circle, its growth, and the creation of the first spiral. The second spiral is also related to the growth of the circle, as well as the first spiral. So with s=r(theta), and the equations for the second spiral, the question of what these drawings mean might be answered, as long as the initial spiral can be put into terms.
If anyone is interested, I've got some more drawings that show how uniform the logarithmic spiral is in relation to the circle. I'll post them when I get a chance. But, in fact, with the circle laying flat on a plane (or at a slight angle) its possible to see how the initial spiral generates itself quite nicely in two dimensions.
Thank you again for the equations, Avni.
gudi
If you wish to give torsion so the tube goes out of its plane, then
add a twist to z with respect to polar angle t.
a = 1; b = .4;
ParametricPlot3D[ { (a + b Cos[ph]) Cos[t], (a + b Cos[ph]) Sin[t], b
Sin[ph] + .5 t }, {t, 0, 4 Pi}, {ph, -Pi, Pi},PlotPoints -> {140, 30}]
Narasimham