Thirty-two millimetres apart over a hundred metres
A chain hanging between two posts settles into a catenary, because
every centimetre of it weighs the same. A suspension bridge cable settles into a
parabola, because what it carries is the road underneath — and the road weighs
the same per metre across, not per metre of cable. Different loads,
different curves, and they are famously easy to confuse because over an ordinary span
they are 32 millimetres apart: a hand's width, at one place, and closer
everywhere else.
The two curves, and the space between them
Same posts, same sag, drawn on top of each other. The red band is the distance between them, drawn a hundred times its true size — at these settings its widest point is the width of your palm.
Why they are so nearly the same
Write the catenary as a series and the parabola is its first term. The next term is a fourth power, and a fourth power of a small number is a very small number — which is the whole of the reason a chain and a cable look alike, and the reason the two curves separate the moment you let the sag get deep.
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Bridges, and one that is not one
A suspension bridge hangs a roadway from its cables, so the cables are parabolic — near enough, and less so the more of the load is the cable's own weight. A chain between two posts has no road under it at all.
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What this page checks about itself
The catenary constant is found by bisection to a part in 1e-12, the
arc length by the closed form 2a·sinh(L/2a) and again by summing the
curve in ten thousand pieces, and the series claim by fitting a slope to the
measured differences.