Leaning cards problem I came up with and can’t solve!

Question

I came up with this problem about a year ago and remain unable to solve it! Hopefully someone could help...

So I had a pack of playing cards (as so many things start with!), and was stacking them in quite an unusual way. I had one perpendicular to my table up against a wall, and then had another resting against it at an angle. Because of the friction of the card, I could place them on top of each other in such a way as to get a nice curve.

Curiosity piqued, I then tried to mathematically define this problem and solve it. Here’s what I came up with.

1. Let a plane (with x- and y-axis) be such that 1cm is equivalent to moving along an axis by 1 (thus 3cm -> x from 0 to 3) - useful for maintaining the link to the real world.
2. Have some line, length ‘h’, situated on the positive y-axis (perpendicular to the x-axis). The bottom will have the coordinates (0,0) and the top will have coordinates (0,h).
3. Travel down some distance ‘d’ along this line. Mark off the point (0,h-d).
4. From this point, draw another line with length h at an angle such that the bottom end of this line is touching the x axis (with the two lines created thus far and the x-axis, there is a right-angled triangle; side lengths should be h-d for the adjacent and h for the hypotenuse).
5. As in step 3, travel the distance ‘d’ along this line (starting at the top), and mark off the point once again.
6. As in step 5, draw a line coming off this line so it just touches the x-axis.
7. Repeat steps 5-6 until bored (or, rather, to infinity)

What happens is that a very nice curve is generated. I suspect that this curve is the same no matter the value of d, just more precise (sort of like having a shape with a large number of sides approximating a circle). Thus, as d tends to 0, I suspect the line becomes more and more precise, tending to some curved line.

What is this line, written algebraically? It feels like the sort of thing that will have some elegant solution but I can’t see it. Please find attached two images with I hope should somewhat visually explain the problem. This has been bugging me for a year now and I’d really like a solution! I’m totally stumped. The easier you can make your solution, should you have one, the better.

Thanks in advance.

A visual version of the problem:

The curve produced

Answer 1

This is called a tractrix, or HundKurv in German. Imagine a dog on a leash.