Least-Squares Circle Fit Given a finite set of points in R2, say { (x i,yi ...
Least-Squares Circle Fit Given a finite set of points in R2, say { (x i,yi ...
Least-Squares Circle Fit Given a finite set of points in R2, say { (x i,yi ...
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October 24, 2006 10:22 am MDT Page 3 <strong>of</strong> 3<br />
Example : Let’s take a few <strong>po<strong>in</strong>ts</strong> from the parabola y = x 2 and fit a circle to them. Here’s a table<br />
giv<strong>in</strong>g the <strong>po<strong>in</strong>ts</strong> used:<br />
i x i y i u i v i<br />
0 0.000 0.000 -1.500 -3.250<br />
1 0.500 0.250 -1.000 -3.000<br />
2 1.000 1.000 -0.500 -2.250<br />
3 1.500 2.250 0.000 -1.000<br />
4 2.000 4.000 0.500 0.750<br />
5 2.500 6.250 1.000 3.000<br />
6 3.000 9.000 1.500 5.750<br />
Here we have N = 7, x = 1.5, and y = 3.25. Also, S uu = 7, S uv = 21, S vv = 68.25, S uuu = 0,<br />
S vvv = 143.81, S uvv = 31.5, S vuu = 5.25. Thus (us<strong>in</strong>g Eq. 4 and Eq. 5) we have the follow<strong>in</strong>g 2 × 2<br />
l<strong>in</strong>ear system for (u c , v c ):<br />
[ 7 21<br />
21 68.25<br />
] [<br />
uc<br />
v c<br />
]<br />
=<br />
[ ] 15.75<br />
74.531<br />
Solv<strong>in</strong>g this system gives (u c , v c ) = (−13.339, 5.1964), and thus (x c , y c ) = (−11.839, 8.4464). Substitut<strong>in</strong>g<br />
these values <strong>in</strong>to Eq. 6 gives α = 215.69, and hence R = 14.686. A plot <strong>of</strong> this example appears<br />
below.
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