Here's about what I think the curve looks like:
In working through a theory of mine, I have come across a curve I cannot identify. Along with $x$ and $y$, this curve needs an additional input to complete the curve. Things I know about this curve:
The curve is identical when reflected around the line $y=x$.
The curve always contains the points $(0,10), (10,0), (5,a),$ and $(a,5)$.
The area under the curve from $0$ to $10$ is always $10a$.
At $a=5$, the curve matches perfectly with the line $10-x$.
I am nearly certain that the curve is hyperbolical in nature.
Knowing this, what is the equation of this curve?
$\endgroup$ 101 Answer
$\begingroup$In fact, the constraints are such that you will not obtain a smooth curve in general. Let us consider, for the sake of simplicity the case $a=1$ for which the objective is to have a total area equal to $10$.
If we consider the figure below, you see that $90\%$ of the area is already taken by the rectangular shapes. It remains a tiny one-unit area to be "distributed" under the 3 arcs of the curve. It means that each arc must stick to the borders, generating uneasthetic spikes, rather far from the shape(s) you desire.
As $a$ increases, the spikes will be attenuated till $a=5$. A symmetrical issue will take place beyond $a=5$.
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