Tuesday 15 March 2011

Catenary - shape of a chain hanging from two points

Problem: Given a chain (string) or length L and it is hanged somewhere by fixed its both endpoints, but the euclidean distance between endpoints in 3D is less than L. And the chain have μ mass per unit length. Determine the shape of the resulting curve.


Commentary: Shape formed is known as Catenary in literature. Solution of this problem is precisely the shape of garlands hanging in weddings or temples. Mathematical treatment of this problem leads to some very interesting insight into the physical characteristics of these types of curves.


Analysis: A nearly complete analysis of this problem is given at http://en.wikipedia.org/wiki/Catenary#Alternative_analysis

Coding: Even though the equations are given on wikipedia page, it is unclear how to code to get the shape of curve given two endpoints and length of the string. But this problem was thoroughly discussed in NSDE lecture 04 (12-01-2011) by Prof. Atanu Mohanty. He gave some hints regarding how to implement it in C.
If you are interested in seeing the code it is available here.

Result:

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