Finding the equation of a curve

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In summary, to find the equation of a curve given an arbitrary number of points, you first choose a functional form to fit the points. This can be done by producing a scatterplot and finding a functional form that looks close to the data. Possible options include using a log function for data that decreases for small positive values and a spline function for a smoother fit. Alternatively, you can use a least squares method, which may not pass through all the points but will be close to them. There are an infinite number of curves that can fit the data, so the user must make a decision on the best fit. Existing programs, such as MS Excel, can also be used for this purpose. To find the equation of a specific curve, such
  • #1
Alex
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How would I go about finding the equation of a curve given an arbitrary number of points? Please don't give me a full explanation if you don't want to, just the name of a process by which I could do this would be fine.
 
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  • #2
You first have to choose a functional form to which to fit your points. In order to choose wisely, you should produce a scatterplot and try to find a functional form that looks close. For instance if in your data table the value of the dependent variable plunges downward for very small positive values of the dependent variable, and the function increases monotonically, then you might try to fit a log function to the data.

So howsabout you post your data set?
 
  • #3
Given any finite set of points, there exist an infinite number of curves that will pass through those points so you have to decide what conditions you want to put on the curve you are looking for. Google on "curve fitting" and you will see some options.
If you are looking for a function of the form y= f(x), then putting the x and y values of n points into that equation will give you n equations which you could solve for n unknowns. In particular, a polynomial of degree n-1 will have n coefficients so given n points, there always exists a unique polynomial of degree n-1 passing through those points. Those tend to be very "wavy" so many applications use a "spline" function instead- a function that is "piece-wise" polynomial. Google on "spline functions-" in particular you might look at
http://www.cse.unsw.edu.au/~lambert/splines/


On the other hand, the best choice may not be a curve that actualy passes through the points but one that is "close" in some sense. For that, you might use a "least squares" method. Google on "least squares". Mathworld has this:
http://mathworld.wolfram.com/LeastSquaresFitting.html
 
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  • #4
Tom Mattson said:
You first have to choose a functional form to which to fit your points. In order to choose wisely, you should produce a scatterplot and try to find a functional form that looks close. For instance if in your data table the value of the dependent variable plunges downward for very small positive values of the dependent variable, and the function increases monotonically, then you might try to fit a log function to the data.
So howsabout you post your data set?

Well that's the thing - I don't have a data set. I'm trying to write a program that will return an equation to a set of data that I input. I suppose I could write one to graph the data and estimate the family of that function based on that - as you mentioned - but I thought there might be an easier way.
 
  • #5
HallsofIvy said:
Given any finite set of points, there exist an infinite number of curves that will pass through those points so you have to decide what conditions you want to put on the curve you are looking for. Google on "curve fitting" and you will see some options.
If you are looking for a function of the form y= f(x), then putting the x and y values of n points into that equation will give you n equations which you could solve for n unknowns. In particular, a polynomial of degree n-1 will have n coefficients so given n points, there always exists a unique polynomial of degree n-1 passing through those points. Those tend to be very "wavy" so many applications use a "spline" function instead- a function that is "piece-wise" polynomial. Google on "spline functions-" in particular you might look at
http://www.cse.unsw.edu.au/~lambert/splines/
On the other hand, the best choice may not be a curve that actualy passes through the points but one that is "close" in some sense. For that, you might use a "least squares" method. Google on "least squares". Mathworld has this:
http://mathworld.wolfram.com/LeastSquaresFitting.html


I checked the least squares method before I posted here but I wasnt sure if it would be quite right. I just skimmed over it, but I got the impression that the least squares method gave a "line of best fit" result. I need a function that will pass through every one of my points.

I'll check out spline functions. Thanks
 
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  • #6
Alex said:
Well that's the thing - I don't have a data set. I'm trying to write a program that will return an equation to a set of data that I input.

You should probably look at an already existing program that can do this. By far the most ubiquitous one is MS Excel. It will fit your data to a curve, but the user has to choose what kind of regression he wants before anything can happen. As HallsofIvy said, there are an infinite number of curves that will fit the data, so the user must make a decision.
 
  • #7
Tom Mattson said:
You should probably look at an already existing program that can do this. By far the most ubiquitous one is MS Excel. It will fit your data to a curve, but the user has to choose what kind of regression he wants before anything can happen. As HallsofIvy said, there are an infinite number of curves that will fit the data, so the user must make a decision.

I've tried to search for an already existing program to do this but I've found nothing that is open-source or easily incorporatable in a c program.

Maybe you could help me with an example? Ie: How would I go about finding the equation of a sin curve for the points [(1,7), (2,21), (3, -14),]? Also how would I choose a regression for that curve? I'm sure that I did this in calculus back in high school but I can't really remember the process.
 
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  • #8

1. What is the equation of a curve?

The equation of a curve is a mathematical expression that represents the relationship between the x and y coordinates of points on a curve. It can be written in various forms, such as y = f(x), where f(x) is a function of x, or in parametric form as x = f(t) and y = g(t).

2. How do you find the equation of a curve?

To find the equation of a curve, you need to have at least three points on the curve. You can then use these points to solve for the coefficients in a polynomial equation, or use techniques such as calculus to find the equation in terms of derivatives and integrals.

3. What is the importance of finding the equation of a curve?

Finding the equation of a curve is important in many areas of science and mathematics. It allows you to describe and analyze the behavior of a curve, make predictions about its future values, and compare it to other curves. It also provides a way to model real-world phenomena and make scientific discoveries.

4. What are some common methods for finding the equation of a curve?

Some common methods for finding the equation of a curve include using regression analysis, which involves fitting a curve to a set of data points, and using calculus techniques such as differentiation and integration. Other methods may depend on the specific characteristics of the curve, such as using parametric equations for curves that cannot be expressed in terms of x and y.

5. Can the equation of a curve change?

Yes, the equation of a curve can change depending on the properties of the curve and the conditions under which it is being observed. For example, the equation of a parabola may change if the curve is translated or rotated. In addition, as more data points are collected, the equation of a curve may need to be adjusted to better fit the data.

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