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A heavy string attached at two ends at same horizontal level and when central dip is very small approaches the following curve

The shape of cable is the equivalent to the shape of the bending moment diagram of a simply supported beam subjected to the same loading as the cable.

When a heavy string is attached at two ends at the same horizontal level and when the central dip is very small approaches the following curve shape is a parabola.

If the cable is subjected to Horizontal UDL, then the BMD of the simply supported beam has a parabolic shape and hence, the shape of the cable is also parabolic.