2

I'll explain my idea because it is very likely that this is not the correct name. Suppose you have a surface, tie a string around it and want to move it along one axis (say X axis). What I'm thinking is that if the surface has "constant perimeter" the string would still fit to the surface's shape and move through it. However, if this wasn't the case the string would either not be able to move through the whole surface or just be too loose. A simple example of this is a cylinder where you can wrap a string around it in a circular shape and move it freely through one axis.

I was wondering if the concept of "constant perimeter surface" exists and if there are any resources or you know about more complex examples of it.

pato
  • 21
  • 1
    I think maybe what you are looking for are products of surfaces with constant curvature, for example the cylinder ($S^1 \times \mathbb R$) or the torus ($S^1 \times S^1$). But you'll have to formalise your definition first. – Frido Dec 29 '23 at 09:45
  • 1
    @Frido I'm not talking about those kind of surfaces. I was thinking if it was possible to have this property for a surface that gets thinner but taller along that axis, for example. – pato Dec 29 '23 at 10:36
  • How is the string tied around the surface? Are any points of the string fixed in place? – FabrizzioMuzz Jan 20 '24 at 23:13
  • 1
    @FabrizzioMuzz The points of the string are not fixed in place. The string is like a closed curve with fixed length around the surface that is flexible so that it can change its shape but without modifying the length. And this string can only be moved in one direction. – pato Jan 24 '24 at 07:21
  • @pato Would the surface at https://www.desmos.com/3d/aedaaff506 be of "constant perimeter" when moved in the positive $x$ direction? Press the play button on variable slider $t$ to see this motion. – FabrizzioMuzz Jan 25 '24 at 03:14
  • In other words, would the circle-shaped string whose initial center is the origin slide around the torus section when the surface is moved as shown in the Desmos graph? I think that the answer is no, but I'm just making sure. – FabrizzioMuzz Jan 25 '24 at 03:17
  • 1
    @FabrizzioMuzz I think it would be of "constant perimeter". I didn't consider surfaces like that one I was just thinking of "straight" surfaces. Technically you could slide the string through the surface and it would fit perfectly. The idea behind this concept is that if you tied a string around the surface and pulled up from it the surface would slide out of the string and fall to the ground due to gravity if it has "constant perimeter" or if the perimeter is always smaller than the string's length (supposing that the string does not stay in equilibrium and not move). I hope this helps. – pato Jan 26 '24 at 06:46
  • @pato It does help. Would something like the surface in the photo at https://math.stackexchange.com/questions/1229716/how-to-mathematically-formulate-the-surface-of-a-spring#1324931 be of constant perimeter? I think the answer is no, due to the gravity analogy you mentioned. – FabrizzioMuzz Jan 26 '24 at 14:18
  • 1
    @FabrizzioMuzz I think that surface could classify as constant perimeter because it would slip out of the string. When I first thought about this I considered more "compact" surfaces like a fruit for example. – pato Jan 30 '24 at 10:47
  • Which fruit? How is a fruit shaped? – FabrizzioMuzz Jan 30 '24 at 13:46
  • 1
    @FabrizzioMuzz An apple for example. I'm not thinking about their shape but more about how they are constructed. They don't bend over themselves like the spring surface you sent. Think of simpler solid objects like your phone. You can tie a string around it but it won't move through the whole surface because the perimeter is larger where the camera is located. – pato Jan 31 '24 at 14:04
  • Would a sphere have constant perimeter? – FabrizzioMuzz Jan 31 '24 at 14:28
  • 1
    It would not have constant perimeter because if you were to intersect it with a plane in any direction the resulting circumference will have a different perimeter. – pato Feb 01 '24 at 08:52
  • Are self-intersections allowed? – FabrizzioMuzz Feb 02 '24 at 20:46
  • I don't think self-intersections are allowed for this type of surfaces – pato Feb 06 '24 at 07:36
  • Informally, consider an injective parametrized surface in $\mathbb{R}^3$ that consists of profile Jordan plane curve swept along a trajectory space curve such that the plane in which the profile curve lies at each point of the trajectory curve is normal to the tangent of the trajectory at that same point. I think that allowing the profile to continuously deform in such a way that its total arc length is preserved as the profile is swept along the trajectory should yield a constant perimeter surface. – FabrizzioMuzz Feb 06 '24 at 20:06
  • You can find more details on "swept surfaces" on per page 472 of the first edition of The NURBS Book by Les Piegl. In this case, the matrix $M(v)$ mentioned in the text should only rotate and not scale $\textbf{C}(u)$ such that the normal condition in the previous comment is satisfied. – FabrizzioMuzz Feb 06 '24 at 20:09
  • I'll check it out. Thanks a lot. – pato Feb 07 '24 at 12:31

0 Answers0