Value between two curves
Value between two curves
(OP)
Hi there,
I have a graph with multiple curves of which I know their respective polynomials. The graph shows on the x-axis the angle of attack of a wing whereas the y-axis gives the lift coefficient for several aspect ratios (hence multiple curves)
How can I find the value in between two "adjacent curves" for a given value of x? I thought of linear interpolation but cannot apply it here because x does not change.
Many thanks
I have a graph with multiple curves of which I know their respective polynomials. The graph shows on the x-axis the angle of attack of a wing whereas the y-axis gives the lift coefficient for several aspect ratios (hence multiple curves)
How can I find the value in between two "adjacent curves" for a given value of x? I thought of linear interpolation but cannot apply it here because x does not change.
Many thanks





RE: Value between two curves
Fe
RE: Value between two curves
say you have y1(x) and y2(x), which are values from each of the shown curves. You want to find y`(x), which is a value at an intermediate point, between the curves y1 and y2, at the given value of x.
Assuming, as an example, that the curves y1(x) and y2(x) represent lift for angles-of-attack A1 and A2 respectively, and you want lift at A`, use linear interpolation as follows:
A` = y2(x) + (y1(x)-y2(x))*A`/(A1-A2)
alternatively,
(A'-A1)/(A2-A1) = (y`(x)-y1(x))/(y2(x)-y1(x))
RE: Value between two curves
i guess we're missing something ...
RE: Value between two curves
The graph I'm referring to is just one of a series of graphs which I have digitised and obtained polynomial expressions for the curves that fit the data. I had originally implemented a "mean" as suggested by FeX32. My only worry is that some curves cross each other
example for a certain value of x (AoA) the curve representing the lift coeff. for aspect-ratio 1.5 is below the one for aspect-ratio 3.5. As x increases the situation is reversed.
Hope this clarifies a bit.
RE: Value between two curves
i'd've thought linear interpolation on each of the curves would get you there, or "goal seek" in excel (y=f(x) ... find "x" for a given "y").
RE: Value between two curves
if you've got polynominal expressions for teh two curves, you can average the coefficients to obtain the mean curve, so long as you're using the same powers for both curves.
RE: Value between two curves
RE: Value between two curves
Lift = f(AoA, AR)
Pick fixed a value of AoA. Then we have
Lift = f(AR) and apply simple linear interpolation based on aspect ratio AR
You want to interpolate to find lift for some AR between your fixed values AR1 and AR2
Lift = Lift1 + (AR - AR1) * slope where
Slope = (Lift2-Lift1)/(AR2-AR1)
Again all the above would be computed at the same value of AoA.
(it doesn't matter whether Lift1 or Lift2 is higher, the algebra will give the correct result)
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RE: Value between two curves
I'll implement the solutions posted and take it from there
Cheers
RE: Value between two curves
It would seem to me that given several curves with Xp,Yp and aspect ratio ratio, Z, one could write the polynomial
Y= AmnX^nZ^m, where n and m range from 0 to N and M
from which you get the best fit
Standard techniques are available to get the best RMS fit. to wit
Error squared =sum(Y-Yp)^2 summed over p
To get the coefficients, take the partial derivative with respect o Amn to get a set of MN+ N +M linear equations in Amn which solution (resulting in minimizing the error of the fit)
yields the coefficients, Amn