Deriving the equation of points for exact fitting and shape analysis

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Discussion Overview

The discussion revolves around deriving equations for closed curves, specifically focusing on exact fitting through a set of known points and exploring methods for shape analysis of similar curves. It includes theoretical and practical approaches to curve fitting, including polynomial equations and spline methods.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that to find an exact polynomial equation passing through a set of points, the degree of the polynomial should be one less than the number of points.
  • Another participant clarifies that while there is a unique polynomial of degree n-1 for n points, there are infinitely many smooth curves that can pass through those points.
  • A suggestion is made to use the Lagrange Polynomial as a method for deriving the polynomial equation.
  • Participants mention alternative methods such as spline fitting and Bezier curves, noting their common use in generating smooth curves for applications like airfoil profiles.
  • A reference to a cubic spline fitting method for closed figures is provided, indicating a resource for further exploration.

Areas of Agreement / Disagreement

Participants generally agree on the existence of a unique polynomial for a given set of points, but there is no consensus on the best method for achieving the desired curve fitting or shape analysis, as multiple approaches are proposed.

Contextual Notes

Some limitations include the dependency on the definitions of the curves and the mathematical complexity involved in coding spline methods or Bezier curves. The discussion does not resolve which method is most efficient or mathematically correct.

giusyvenezia
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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
 
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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.
 
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.
 

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