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Fitting a Circle to Three Given Points

Date: 02/19/2006 at 00:00:06
From: Ritsuko
Subject: curve fitting and equation of a circle

I want to find an equation of a circle passing through three points,
(x1,y1), (x2,y2), and (x3,y3) by solving a system of linear equations.
I cannot figure out what equation (in what form?) must be used for a 
circle.  For a polynomial, I use P(x) = ax + bx + cx^2 when three 
points are given.  I think I could use the circle's general equation
x^2 + y^2 = r^2, but I don't know where to put coefficients a, b and c
and can't make a system of equations.

Date: 02/20/2006 at 09:14:56
From: Doctor George
Subject: Re: curve fitting and equation of a circle

Hi Ritsuko,

Thanks for writing to Doctor Math.  You are on the right track.

A circle is a special case of a general quadratic.  We can write it
like this:

    x^2 + y^2 + Dx + Ey + F = 0

If you substitute your three points into (x,y) you will have three
linear equations with three unknowns (D, E, and F).  Solve for those
three unknowns.  

Then you will need to complete the square on x and y to find the 
center and the radius.

Does that make sense?  Write again if you need more help.

- Doctor George, The Math Forum 
Associated Topics:
High School Conic Sections/Circles

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