KAPOW!!! Spontaneous brain explosion.Cat curves are mathematically much messier than parabolas, and yet there is very little quantitative difference between them. I thought that if I considered the question using parabolas and found an answer quite a bit different than 5/6, that would be strong evidence that the answer to the question is "no".
So. y = a*x^2 + b is a parabola. The derivative is
dy/dx = 2*a*x. The question asks about the curve where the angle of the hammock at the ridge-line is 30 degrees. In geek-speak that is PI/6.
x of interest here is half the ridge-line length, because the parabola's minimum is at the center of the ridge-line. So fixing x at whatever, say 108/2, we can set up
dy/dx = 2*a*x = PI/6
and solve for a
a = PI/(12*x)
and this characterizes the parabola of interest.
Now there is some formula for the arc-length of a parabola over a range but it too is nasty. I've got code I've cooked up for bridge hammock designs that numerically computes this quantity, and so here I can empirically compute the ratio of ridge-line to hammock-length for a variety of hammock lengths to see what I can see.
What I see (accepting of course the possibility of some blunder-headed mistake along the way), is a ratio that over a wide range of ridge-line lengths is 0.957 with any differences in digits beyond that.
In particular, no-where close to 0.833333
suggesting to me that wherever 83% comes from, it ain't from a cat curve's slope when that slope is 30 degrees.
here ended the first lesson