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All the equations I've seen assume I know the sag, height, force, or the weight of the catenary I want to plot, but I don't. All I know is the total length ($L$) of the wire and the position of the two points it's attached to ($x_1$, $y_1$, $x_2$, $y_2$, respectively).
If I give it values so that $L = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)$, I'd expect to get a straight line, since $L$ is equal to the distance between the two points. If $L$ was larger, however, I'd expect it to sag like a catenary normally does. $L$ will never be less than the distance between the two points.
How can I plot a curve knowing only this? Is it even possible?