Deriving the equation of points for exact fitting and shape analysis

[giusyvenezia]

Hello,

I would like to ask you some questions.

1) I've a closed curve (for example an ellipse, which may represent the contour of an object) represented by the set of its (known) points. I need to find the equation of that curve to pass through all and every point (exact fit). I think that to do this I need a polynomial whose grade is equal to the number of points less 1.

Something like this:

a0+a1 x1+a2 x1^2+ ...+ an x1^n = y1
a0+a2 x2+a2 x2^2+ ...+ an x2^n = y2
...
a0+a2 xn+a2 xn^2+ ...+ an xn^n = yn

This argument is right? Do you have suggestions (or anything else relevant) for me in this regard for which is the best way to solve my problem? This equation can be made in parametric form?

2) After I got the exact equation of this curve. Suppose we have a set of curves very similar to each other (represented by their equation), I would like to find the equation that represents the shape which best approaches to all previous curves, a sort of average curve created from those previously acquired.
Do you know if this thing can be done and how? What is the best way (most efficient and / or mathematically more correct) to do this?

Best Regards,

Giusy

 

[HallsofIvy]

It's not clear what you want. Given any finite number of points, there exist an infinite number of smooth curves passing through those points. In particular, given n points, there exist a unique n- 1 degree polynomial giving those points. The "Lagrange Polynomial", http://en.wikipedia.org/wiki/Lagrange_polynomial, is probably the simplest method in concept though not always simplest to compute.

 

[meldraft]

If you have matlab, you can also fit a spline.

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

Bezier curves are also widely used. These are two of the most popular methods for these kinds of problems and are commonly used to generate the equation for a smooth solid body, e.g. an airfoil profile. They are a bit involved to code, but you can still use software to generate the functions for you.

 

[SteamKing]

From PF's own files:
https://www.physicsforums.com/showthread.php?t=178945&highlight=parametric+spline

This shows how one can fit a cubic spline to points describing closed figures.

 

[blue_raver22]

Hello Giusy,

Thank you for your questions. To answer your first question, the approach you have described is correct. To find the equation of a curve that passes through a set of points, you can use polynomial interpolation. This involves finding a polynomial function of degree n-1 (where n is the number of points) that passes through all of the points. This can be done by solving a system of equations, as you described. This equation can also be written in parametric form, where x and y are both functions of a parameter t, such as in the case of an ellipse.

To address your second question, what you are describing is known as shape analysis or shape averaging. This involves finding a common shape or average shape among a set of similar curves. There are various mathematical approaches to solving this problem, including using statistical methods or geometric methods. Some common techniques include principal component analysis (PCA) and dynamic time warping (DTW). These methods involve analyzing the variability and similarity among the curves to find a common shape or average curve. The most appropriate method will depend on the specific characteristics of your data and the desired outcome. Further research in this area may help you determine the best approach for your specific problem.

I hope this helps and wish you the best in your research.

Best regards,