[RajatPawar]

I can't honestly think of something else, but what I came up with needs the angle. It's probably useless because it needs the angle which is hard to calculate. You could do computer integration, like getting the co - ords for each point as angle tends to 0.
The circle (semi) you make with the line, suppose has a radius R, you consider a dθ angle and then, R*dθ = the curve's length. But it'd be so small, tending to a point. But I don't know how we can use this to get the co - ordinates. Unless you consider the hinge as the origin (X - Y axis) and then calcuate the distance from there using the distance formula/slope method/perpendicular distance.
All that blab..