Timeline for Bending Beltrami Pseudosphere
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Jul 8, 2016 at 17:25 | comment | added | Narasimham | The sketch is schematic for discussion only,not properly sketched. I suspect the surface gets cut into two cuspidal edges.The surface is not axi-symmetric, does not behave as negative axi-symmetric surface with meridians rotated about $y$ axis. | |
| Jul 8, 2016 at 13:58 | comment | added | Futurologist | It seems to me that you shouldn't have points like S and maybe R (if I'm seeing the picture correctly) because the curvature of the surface at those points will be positive after adding the transverse vertical circles... | |
| Jul 7, 2016 at 18:24 | comment | added | Narasimham | Yes, symmetric deformation. As it is a continuum/ simply connected surface the outer and inner section curves are geometrically related on the surface . But specification of one of them already is a good part of sought answer it appears to me ...We can take a mechanics of materials analogy...On a thin walled shell a certain loading can be applied deforming the shell so that the straight neutral axis of the unloaded shell is bent to a circle making the deformation definition complete as well as problem statement. Please suggest what, for example, may be further needed for problem definition. | |
| Jul 7, 2016 at 15:06 | comment | added | Robert Bryant | Better posed, but not well-posed. Basically, all you are requiring is that the the bent surface should intersect the plane $x=0$ at right angles in two curves that are asymptotic (somehow) to the circle $y^2+z^2=b^2$ in the $x=0$ plane. There is still a lot of choice for these two curves, but the two curves are not independent. Basically, once you choose the intersection curve outside the circle, the intersection curve inside the circle with be uniquely determined, if it exists at all. To determine a formula, you would need to at least specify the outer curve completely. | |
| Jul 7, 2016 at 14:33 | comment | added | Narasimham | Is the problem now better posed? | |
| Jul 7, 2016 at 14:26 | history | edited | Narasimham | CC BY-SA 3.0 |
geometric mode of bending sketched and explained
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| Jul 6, 2016 at 15:10 | comment | added | Narasimham | Thanks for giving attention for the question. My question certainly needs a revision, and sorry about that ... the axis of symmetry $ z=b$ is not on the surface but in the air, unconnected I shall collect my thoughts and pose it properly. Regards | |
| Jul 6, 2016 at 13:41 | comment | added | Robert Bryant | Even with this final remark added (clarifying that you don't want to maintain the entire tubular structure, just the patch $0<t<2\pi$), you haven't described exactly what you want the surface to be. The 'axis of symmetry' is not part of the original surface, so you are only roughly describing where you want the 'bent' surface to go, so there probably no uniqueness and hence no way to determine an explicit parametrization. Presumably, you want the curve $t=\pi/2$ to spiral in to the circle and lie in the plane $x=0$, but you will need to say exactly how you want that to happen. | |
| Jul 6, 2016 at 12:10 | review | Close votes | |||
| Jul 15, 2016 at 3:02 | |||||
| Jul 6, 2016 at 11:45 | history | edited | Narasimham | CC BY-SA 3.0 |
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| Jul 6, 2016 at 11:35 | comment | added | Robert Bryant | I don't think such a thing is possible if the surface is to be at least $C^2$. While you haven't described exactly what you want, it appears that such a surface would have to have a point where $z$ reaches a local minimum (down near the bottom of the circle you have drawn). However, a surface of negative curvature cannot have a point at which $z$ is a local minimum since a negative curvature surface, being locally saddle-shaped, must always cut across its tangent planes. | |
| Jul 6, 2016 at 9:01 | history | edited | Narasimham | CC BY-SA 3.0 |
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| Jul 6, 2016 at 8:55 | history | edited | Narasimham | CC BY-SA 3.0 |
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| Jul 6, 2016 at 8:28 | history | edited | Narasimham | CC BY-SA 3.0 |
spelling minor
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| Jul 6, 2016 at 7:32 | history | asked | Narasimham | CC BY-SA 3.0 |