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Posted 29 Aug 2008

Curve Fitting using Lagrange Interpolation

, 12 Sep 2008
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Fitting a set of points P to a curve using Lagrange Interpolation Polynomial.


In this article, I will explain curve fitting using the Lagrange interpolation polynomial. Curve fitting is used in a wide spectrum in engineering applications such as cars and air crafts surface design. The main problem is, given a set of points in the plan, we want to fit them in a smooth curve that passes through these points. The order of the curve f(x) depends on the number of points given. For example, if we have two points, the function will be of the first order, and the curve will be the line that passes through these two points, while if you have three points, the function will be of the second order f(x) = x2 . Let's first explain the Lagrange polynomial, then we will proceed to the algorithm and the implementation. In this article, I am using C# for coding.


Lagrange Polynomial

An interpolation on two points, (x0, y0) and (x1, y1), results in a linear equation or a straight line. The standard form of a linear equation is given by y = mx + c, where m is the gradient of the line and c is the y-intercept.

m = y1 − y0 / x1 − x0
c = y0 − mx0

which results in:

y = (y1 − y0 / x1 − x0) * x + (x1 y0 − x0 y1 )/ (x1 − x0)

The linear equation is rewritten so that the two interpolated points, (x0, y0) and (x1, y1), are directly represented.

With this in mind, the linear equation is rewritten as:

P1(x) = a0(x − x1) + a1(x − x0)

where a0 and a1 are constants . The points x0 and x1 in the factors of the above equation are called the centers. Applying the equation at (x0, y0), we obtain:

y0 = a0(x0 − x1) + a1(x0 − x0)


a0 = y0/ x0−x1 

At (x1, y1), we get:

y1 = a0(x1 − x1) + a1(x1 − x0), or a1 = y1/ x1−x0

Therefore, the linear equation becomes:

P1(x) = y0 (x− x1) /(x0 − x1) + y1 (x− x0) / (x1 − x0)

The quadratic form of the Lagrange polynomial interpolates three points, (x0, y0), (x1, y1), and (x2, y2). The polynomial has the form:

P2(x) = a0(x− x1)(x− x2) + a1(x− x0) (x− x2) + a2(x− x0)(x− x1)

with centers at x0, x1, and x2. At (x0, y0):

y0 = a0(x0 − x1)(x0 − x2) + a1(x0 − x0)(x0 − x2) + a2(x0 − x0)(x0 − x1),


a0 = y0/ (x0 − x1)(x0 − x2)
a1 = y1 / (x1 − x0)(x1 − x2)
a2 = y2/ (x2 − x0)(x2 − x1)
P2(x) = y0 (x− x1)(x− x2) / (x0 − x1)(x0 − x2) + y1 (x − x0)(x − x2)/ 
       (x1 − x0)(x1 − x2) + y2 (x− x0)(x− x1)/ (x2 − x0)(x2 − x1)

In general, a Lagrange polynomial of degree n is a polynomial that is produced from an interpolation over a set of points, (xi , yi ) for i = 0, 1, . . ., n, as follows:

Pn(x) = y0L0(x) + y1L1(x) + ··· + ynLn(x)

Using the Code

The Algorithm

Given the interpolating points (xi , yi ) for i =0, 1, . . . ,n;

for i = 0 to n
  //the cumulative multiplication from k = 1 (and k not equal i) to n 
  Evaluate Li (x) = ∏nk=1,k != i (x−xk ) / (xi−xk ) ;

Evaluate Pn(x) = y0L0(x) + y1L1(x) + ··· + ynLn(x);

Download the source code from the top of this page to view the code.

Points of Interest

Numerical computing is a very interesting field in software development. This article is related to that field ... And, I will post more articles soon about Computational Geometry ...such as Convex Hull algorithm and Triangulation.


This article, along with any associated source code and files, is licensed under The Code Project Open License (CPOL)


About the Author

Fady Aladdin
Software Developer
Egypt Egypt
No Biography provided

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Comments and Discussions

GeneralI don't mean to quibble but this looks like interpolation Pin
MicroImaging11-Sep-08 11:15
memberMicroImaging11-Sep-08 11:15 
GeneralRe: I don't mean to quibble but this looks like interpolation Pin
axelriet12-Sep-08 5:30
memberaxelriet12-Sep-08 5:30 
GeneralRe: I don't mean to quibble but this looks like interpolation Pin
MicroImaging12-Sep-08 9:20
memberMicroImaging12-Sep-08 9:20 
If you read through this book at

You will see that data interpolation and data fitting are differentiated, although the title is somewhat ambiguous. The author clearly talks about fitting using norms, especially around page 44 where the chapter title "Curve Fitting: An Introduction" talks specifically as i did abuot the L2, L1 and LInfinity norms.
Again I encourage you to use or learn, or whatever you wish, to differentiate interpolation versus fitting.

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