Register to reply

Deriving the equation of points for exact fitting and shape analysis

by giusyvenezia
Tags: analysis, deriving, equation, exact, fitting, points, shape
Share this thread:
Nov29-12, 08:09 AM
P: 1

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,

Phys.Org News Partner Science news on
'Office life' of bacteria may be their weak spot
Lunar explorers will walk at higher speeds than thought
Philips introduces BlueTouch, PulseRelief control for pain relief
Nov29-12, 08:23 AM
Sci Advisor
PF Gold
P: 39,682
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",, is probably the simplest method in concept though not always simplest to compute.
Nov29-12, 10:22 AM
P: 280
If you have matlab, you can also fit a 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.

Nov29-12, 11:14 AM
Sci Advisor
HW Helper
PF Gold
P: 6,746
Deriving the equation of points for exact fitting and shape analysis

From PF's own files:

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

Register to reply

Related Discussions
Fitting a graph to data points. Precalculus Mathematics Homework 2
Fitting a Function given 3 points Precalculus Mathematics Homework 4
Data points and fitting functions Advanced Physics Homework 0
Fitting a quadratic to three points - gradient Advanced Physics Homework 0
What do you do if your exact equation isn't exact? and they give u an Integrating F Calculus & Beyond Homework 2