Equation of page of unfolded paper with a crease

[angellus]

Okay, I think I got the right sub-forum here. I have never posted around here, so sorry if it is wrong. I came here for some help with a math problem I need to solve for a GLSL shader for programming (math in the real world!).

I apologize in advance if it seems I did not put a lot of effort into this, but I really do not know where to start. I am a programmer and I know where to look when I need help with programming, but I have not had to do math I could not figure out on my own for programming on my own yet, so I thought I would turn to a forum for help. Thanks ahead of time

Problem:
I need to figure the mathematical equation for a piece of paper that has been folded in half landscape and then unfolded. An example, take a piece of page and fold it like you would a birthday card, make sure it has a good crease in it and then unfold it. When you do this, there is a arch right where the crease if from the fold that causes the paper to go up quickly and then slowly goes back down.

This is the say the equation of the line would be like a upside-down "W" (curvy, not pointed) with the parts after the valleys stretched really far out. If you can understand that at all. The best way to see what I mean is take a piece of notebook paper and folder it.

My main problem is that I have no idea what type of line this would be graphed to. I was thinking something like a logarithmic line, but it does not quite fit well. I have no idea how to Google around for this and I am not at my University so I cannot just find a random math professor and ask them, so I have no idea where to start. I also do not have my graphing calculator with me or know of any computer tools to assist me with this.

 

[Ackbach]

There probably is no one equation that is "correct". However, you could approximate it with a couple of curves: $e^{-|x|}$ would give you a sharp corner right at the crease. If you want a more curved peak, then you could try
$$\text{sech}(x):= \frac{2}{e^{x}+e^{-x}},$$
or even a Gaussian function $e^{-x^{2}}$. Are these sort of what you're after? You can play games with stretching them out or shrinking them in, of course.

[EDIT]: Ah, but you want the central point of the crease to be down, right?

 

[Ackbach]

Here's another option:
$$\frac{1-e^{-x^{2}}}{1+x^{2}}.$$
The plot is here.

 

[angellus]

Awesome. That is what I need. Thanks. Time to go check out the rest of the forums.

 

[CellCoree]

Hello,

Thank you for reaching out for help with your math problem. I can offer some guidance and suggestions to help you find the equation for the page of unfolded paper with a crease.

First, let's define some variables to make it easier to understand and work with the problem. Let's say the width of the paper is w and the length is l. When you fold the paper in half, the crease will be at the midpoint of the length, so the distance from the crease to the edge of the paper will be l/2. When you unfold the paper, the crease will create an arch or curve in the paper, and we can measure the height of this arch as h.

Now, let's think about the shape of the curve created by the crease. It sounds like you are trying to find an equation for a curve that is similar to an upside-down W, with the valleys being stretched out. This could be modeled using a polynomial function, such as a cubic or quartic function. You could also try using a trigonometric function, such as a sine or cosine function, to model the curve.

To determine the exact equation, you will need to gather more information about the curve. One way to do this is to take measurements of the height of the arch at different points along the length of the paper. You could then plot these points on a graph and try to fit a curve to them. Alternatively, you could use a computer program or graphing calculator to help you plot points and find the best fit curve.

I hope this helps point you in the right direction. Good luck with your problem!