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I know two points, say A and B, that are the endpoints of a curve (a regular function) over a certain range. I also have a constraint on the integral over this range, say P. Given only this information, it could specify an infinite number of possible curves, but each endpoint also has an "influence" value that governs how much that endpoint should impact the shape of the curve.

For the given curve below, the endpoints are at A=[1.7, 100] with an influence of 0.74, B=[2.1, 550] with an influence of 0.3, and P=145, but this is just an example:

curve to find

How could I find the curve itself over this range? It seems the root of the problem is what exactly is meant by the "influence" parameter. I know it lies in the range [0,1] and that at 0 the endpoint has no effect and at 1 the endpoint has maximal effect on the resulting curve.

What could the influence parameter represent in terms of the generation of the curve?

As you can see, because the first point has a higher influence value, it remains closer to that value over the course of the curve. Also, because the area under the curve is constrained, the curve has to jump above the last point to ensure there is enough area.

Part of what I am wondering is how to even define this problem mathematically. I know the influence values constrain the space of possible curves in some way, but I don't know how to capture this effect in mathematical terms.

Again, all I know is the endpoints, their influence values and the area under the curve. How can I generate the given curve from this information?

I'm not sure where to start solving this problem! Is there a general method for this class of problems? If someone can point me in the right direction I could read up on a method for this, but I'm not even sure where to begin. Thanks!

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What do you mean by "influence"? Note that the solution is never unique, so it's better to specify the class of functions you're interested in. – TZakrevskiy Jul 21 '13 at 8:59
@TZakrevskiy Well, the meaning of the "influence" parameter is what I am trying to figure out. I do know it is in the range [0,1] and that at 0, it has no effect and that at 1 it has maximal effect. The curve itself appears to be a polynomial of at least degree 4? Does that narrow it down at all? – prismofeverything Jul 21 '13 at 9:42

One idea is to try a generalized sort of interpolation, namely, suppose we get rid of the influence thing, then one can solve the problem by finding a polynomial $p$ such that $$p(A)=p_A, \\ p(B)=p_B, \\ \int_A^B p(x)dx = P.$$ This can be solved with a second degree polynomial, but it seems to me that you can fix the influence issue by adding extra variables to the problem, that is, by raising the degree of the resulting polynomial, though I myself cannot think of a way of reflecting this into equations. Nonetheless, it does not look like the solution is unique.

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Alright, forgetting the "influence" for a moment, how would I solve for the given constraints? – prismofeverything Jul 26 '13 at 17:32
Well, suppose the curve is a second-degree polynomial, then it's in the form $p(x)=ax^2+bx+c$. Then you impose the three conditions above and you get a linear system in $a$, $b$ and $c$ which has a unique solution. – busman Jul 26 '13 at 22:32

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