Accurately interpolating more Data points for a bump profile

[Hughezy]

Hi there,

I require some advice on how to interpolate more data points on a bump profile. Basically i have a curve defined be twentyfive (x,y) data points. The bump is meant to accurately represent half of an aerofoil. I would like to accurately interpolate more data points (lets say up to 50) so that i can generate a smoother curve in my computer simulation.
The bump geometry profile it:

x y
-1.5 , 0
-1.4 , 0.002
-1.2 , 0.014
-1 , 0.044
-0.8 ,0.099
-0.6 , 0.178
-0.4 , 0.257
-0.3, 0.2835
-0.2 , 0.31
-0 , 0.331
0.2 , 0.325
0.35 , 0.308
0.4 ,0.3
0.6 ,0.258
0.63 ,0.251
0.8 ,0.207
0.9 ,0.181
1 ,0.155
1.2 ,0.103
1.24 ,0.093
1.4 ,0.058
1.58 ,0.031
1.6 ,0.027
1.8 ,0.006
2 ,0

I do not know what type of function this is wheter its quadratic or cubic etc...

Any asistance would be greatly apreciated.

 

[chiro]

Hey Hughezy and welcome to the forums.

For interpolation, there are quite a number of techniques available.

The simplest is the Lagrange Interpolation formula which generates an n+1th degree polynomial for n data points, but if you want more control over the actual interpolation model you will need something a bit more developed, and for that you should check out BSPLINES.

http://en.wikipedia.org/wiki/Lagrange_polynomial

http://en.wikipedia.org/wiki/B-spline

In terms of calculating these, you can use standard numerical platforms like MATLAB or something like Octave which is free, and then get some coded routines to generate the right data structures with all the information for that model.

If you have MATLAB, then a google search returns this:

http://www.mathworks.com/matlabcentral/fileexchange/27047-b-spline-tools

This kind of stuff is a big topic in 3D animation and games design, so the relevant literature in this area (i.e. computer graphics) should have more information if you wish to dig deeper (as well as certain areas of applied mathematics).

 

[Hughezy]

Hi Chiro,

Thankyou for the reply!
I do indeed have MATLAB although my experience is limited. I have no idea where to start with B-spline tools. Do you know if there is a tutorial avaliable? Or do you yourself have any further advise?

Looking at the langrange polynomial on
http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html

Solving an n+1th polynomial for 25 data points by hand would be a ridiculously long calculation. And I am not sure how accurate it would end up being. I was ignorant as to how difficult this will be, and with my current time constraints I am not sure whether this will be possible.

But again thankyou

 

[chiro]

You can solve the Lagrange polynomial by computer (and you should use a computer for something like this not only because its tedious, but because you can make errors that also compound).

http://www.mathworks.com/matlabcentral/fileexchange/899-lagrange-polynomial-interpolation

The above is a really short function that calculates an interpolated value given a set of data points.

 

[CellCoree]

I would suggest using a polynomial interpolation method to accurately interpolate more data points for your bump profile. This method involves fitting a polynomial function to your existing data points and then using the function to calculate the values for additional data points.

To start, you will need to determine the degree of the polynomial function that best fits your data. This can be done by visually inspecting the data points and trying different degrees until you find the best fit. Alternatively, you can use a mathematical method such as the least squares method to determine the degree.

Once you have determined the degree of the polynomial, you can use a mathematical software or programming language to fit the function to your data points. This will generate a polynomial equation that represents your data.

Next, you can use this equation to calculate the values for additional data points. It is important to note that the accuracy of the interpolated data points will depend on the accuracy of the original data points and the degree of the polynomial used.

It is also important to keep in mind that interpolating more data points does not necessarily guarantee a smoother curve in your computer simulation. Other factors such as the accuracy of the original data, the interpolation method used, and the resolution of your simulation will also affect the smoothness of the curve.

In conclusion, using a polynomial interpolation method can help you accurately interpolate more data points for your bump profile. However, it is important to carefully consider other factors that may affect the smoothness of the curve in your simulation.