What kind of hyperbola is this?

C

cassacosmin

New member
Joined
Apr 27, 2019
Messages
6
I know that the simplest rectangular hyperbola is described by the reciprocal function y=1/x. It has two orthogonal vertical asymptotes at y=0=x and two horizontal asymptotes at -∞ and +∞ .

Now, I have another kind of hyperbola which has two opposite vertical asymptotes at x=-1 and x=+1, and an inflection point at y=0=x. This curve is constrained between -1 and +1.
What kind of function would describe best this curve?
It should be some variation on the reciprocal function, because its curvature is similar to that of the rectangular hyperbola.
It is similar to a logit function, but the curvature is different.
Thanks.
 
cassacosmin said:
I know that the simplest rectangular hyperbola is described by the reciprocal function y=1/x. It has two orthogonal vertical asymptotes at y=0=x and two horizontal asymptotes at -∞ and +∞ .
Click to expand...
I would say that it has one vertical asymptote, x=0), and one horizontal asymptote, y=0. These are not both vertical. And we don't talk about an asymptote "at -∞" or "at +∞". The curve approaches the one asymptote y=0 when x approaches -∞ and +∞.
cassacosmin said:
Now, I have another kind of hyperbola which has two opposite vertical asymptotes at x=-1 and x=+1, and an inflection point at y=0=x. This curve is constrained between -1 and +1.
What kind of function would describe best this curve?
It should be some variation on the reciprocal function, because its curvature is similar to that of the rectangular hyperbola.
It is similar to a logit function, but the curvature is different.
Click to expand...
Since you are evidently misunderstanding the terminology of asymptotes, I don't know what you mean by this. The lines x=-1 and x=+1 are vertical, but they are not "opposite". More important, you are clearly not talking about a hyperbola, which is a very specific type of curve, and can't have two vertical asymptotes. Probably this is some other rational equation. (Hyperbolas have no inflection points.)

I think it may be related to this:

[MATH]y = \frac{x}{(x+1)(x-1)}[/MATH]FMH115732.png

But, since you mention looking something like the logistic function (I think), I suspect that you are thinking of the inverse of the middle section of this, and that the asymptotes you picture are really horizontal, not vertical. It will probably not be a rational function at all (and therefore will be unrelated to the hyperbola)

Can you tell us what you want this function for? Context may help us understand you. In particular, how is the curve you want different from the logistic function?
 
Thank you for your corrections, excuse my ignorance in mathematics.
I'm tentatively trying to model some cycle in sociology.
Actually, the curve that I'm trying to describe is more complicated than that.
The sigmoidal curves follow one another, and their lenghts is halved as it approaches x=0.
This is a very crude drawing of what I have in mind.
function.png
 
Okay, my guess was closer than I thought. I assumed your "logit" was a misspelling of "logistic", and that you were really talking about horizontal asymptotes. Now I see that the logit function is in fact the inverse of the logistic function (the latter is what "sigmoid" properly refers to). This is, of course, very different from what you initially described.

I don't have time at the moment to dig deep, so I'll mostly ask additional questions. I will say that my first thought is to use something related to the tangent function, since we need a sort of "pseudo-periodic" function with infinitely many asymptotes.

Here are some questions:
  1. How can a function in sociology have vertical asymptotes, meaning that it is undefined at infinitely many places, and very large in even more? I don't need all the background, but possibly a little further context would help me be more sure of your requirements. Maybe, for example, you don't really need the asymptotes as much as you need the "pseudo-periodicity".
  2. How did you draw your graph? It looks as if it were the graph of some actual function you made; how would what you really want be different from this?
  3. You say you are modeling something. Do you have a sample of the actual data you are modeling, to show what it really needs to look like?
  4. I could come up with many different functions with the correct asymptotes and signs; do the actual values between the asymptotes matter? For instance, the tangent looks superficially similar to the logit function, but is not at all the same function, and gives very different specific values. Would that make a difference to you?
 
Last edited:
It appears that you must specify a function that is defined "piecewise."

It might be something like this:

[MATH]f(x) = g(x,\ n) \text { if } 2^n < x < 2^{(n+1)} \text {, where } n \not \in \mathbb Z \text { and } n > 0, and[/MATH]
[MATH]f(x) = h(x,\ n) \text { if } 2^n < x < 2^{(n+1)} \text {, where } n\not \in \mathbb Z \text { and } n < -\ 1.[/MATH]
The exact form of the functions g and h will depend on either some theoretical logic or some observational data.
 
Dr Peterson, ok, my model is even more complicated than that. It behaves like a fractaL, in that every curve can be further subdivided into other self-similar curves.
I'm trying to model the cycles of expansion and contraction of cultural, demographic and economical aspects of human history. Including the concept of a technological singularity. Thta's why I need vertical asymptotes...
Here's an example of these self-similar cycles.
The dates are astronomical years, going from 2985.74 bce to 2027.50 ce (the purported singularity).

2985.74 bce.png

774.19 ce.png

1714.17 ce.png

1949.17 ce.png
 
Dr.Peterson said:
Okay, my guess was closer than I thought. I assumed your "logit" was a misspelling of "logistic", and that you were really talking about horizontal asymptotes. Now I see that the logit function is in fact the inverse of the logistic function (the latter is what "sigmoid" properly refers to). This is, of course, very different from what you initially described.

I don't have time at the moment to dig deep, so I'll mostly ask additional questions. I will say that my first thought is to use something related to the tangent function, since we need a sort of "pseudo-periodic" function with infinitely many asymptotes.

Here are some questions:
  1. How can a function in sociology have vertical asymptotes, meaning that it is undefined at infinitely many places, and very large in even more? I don't need all the background, but possibly a little further context would help me be more sure of your requirements. Maybe, for example, you don't really need the asymptotes as much as you need the "pseudo-periodicity".
  2. How did you draw your graph? It looks as if it were the graph of some actual function you made; how would what you really want be different from this?
  3. You say you are modeling something. Do you have a sample of the actual data you are modeling, to show what it really needs to look like?
  4. I could come up with many different functions with the correct asymptotes and signs; do the actual values between the asymptotes matter? For instance, the tangent looks superficially similar to the logit function, but is not at all the same function, and gives very different specific values. Would that make a difference to you?
Click to expand...
Did I shock you?
 
What's the point of defining a bunch of asymptotes, just to mine a tunnel through them?
 
Last edited:
Dr.Peterson said:
I just don't see value in responding, when the problem keeps changing and there's no data to support any particular answer. I'll let others do so if they want to.
Click to expand...
It doesn't keep changing. I just started by simplifying it for people to be able to answer, because I thought that the "fractal" part would be too difficult to figure out.
 
A "simplified" problem is not the same problem; an answer to a non-fractal problem will not be an answer to a fractal problem. It's like trying to simplify a Rubik's cube, and expecting that a solution will help in solving the real thing. It won't.

But really, I don't think what you've described is (literally) fractal at all, but is just vaguely reminiscent of fractals in your mind; and I doubt that it has a real answer. Very likely, what you want can't be stated clearly enough to be solved.

But, again, if someone else sees something in there worth working on, let them. I just am not interested.
 
You must log in or register to reply here.
Top