The physics of a catenary curve

That's a grand title for a simple question.



I clearly remember a physics teacher telling me that a rope/ string/ chain hung between two horizontal points will always...........always....... hang in a catenary curve, (due to gravity) no matter what the horizontal forces are on the suspension points, and no matter how long or short the rope is. Is this actually true?



I wonder about a string of very short length being pulled incredibly taut by some extremely strong horizontally-acting forces. There would seem to me to come a time when gravity acting on the string to drag it into a catenary curve is just not a factor. I mean, if an incredibly strong string fixed to a hook in a vertical wall hung an elephant via a pulley just away from the wall, the horizontal element being, say, 2 inches long.....then I just cannot picture how that string is anything but straight. I am sure there is a special form of catenary curve which is straight, but that isn't my point. Was my physics teacher right to say the string is always curved? Is there some maths on this?

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