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January 9

Function for a catenary given unequal heights

All treatments of catenaries involving unequal heights (h1 and h2) seem to be only interested in arc length. What is the general function for any catenary connecting two unequal heights, y = f(x)? Arc length is a secondary concern. — Preceding unsigned comment added by 98.14.205.209 (talk) 00:55, 9 January 2018 (UTC)[reply]

The general formula for a catenary, allowing for scaling and translation, is

y-k=a\cosh \left({\frac {x-h}{a}}\right).

There are three parameters so requiring that the curve pass through two points does not allow you to solve for all three and get a specific equation. You need some additional piece of information, such as the arc length, to determine the curve. It might be possible to determine the catenary passing through three given points, but I don't know how complicated the expressions get. --RDBury (talk) 02:49, 9 January 2018 (UTC)[reply]

You have one degree of freedom in your problem, so, as RDBury said above, you need additional condition to solve for one free parameter.
Given a horizontal and vertical displacement you have a slope of a line segment, which shall become a chord of a catenary arc. You can have multiple chords with the same slope on a single catenary curve. Each of them can be scaled to your original problem thus giving different catenary arcs through the two given points. See images in Catenary # Mathematical description # Equation section.
So you can choose some additional constraint (say, the arc length, the maximum curvature, the height of minimum point etc.) to make a solution unique (which does not necessarily mean easy). --CiaPan (talk) 10:35, 10 January 2018 (UTC)[reply]