Least Squares Circle Fit

markraz

markraz

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Hi, I have a bunch of points in form of x,y (0, 1), (2, 5), (1, 6), and (7, 6)
and I'm trying to understand circle regression using least squares. I can't find any examples anywhere. But I did find this picture of a matrix that might do the trick.

1601163970209.png

So I think I first need to understand these summation terms within the matrix

row1 column1 it is Sum( xi^2) = does this mean I add all the x values up and square them? or do I square all the x values and then add them?
row1 column2 it is Sum( xi yi) = does this mean I multiply all the points first and then sum them? {(x1 * y1) + (x2 * y2) + (x3 * y3)+ (xn * yn).......} ??
so ..... sum{(0 * 1) + (2 * 5) + (1 * 6)+ (7 * 6)}

row 1 column3 it is Sum( xi) = does this mean I add all the x values? 0 + 2 + 1 +7
row 3 column 3 it is n = do I just put the number of points here? (4 in my case)

the rest are similar to above

thanks in advance
 
Last edited:
markraz said:
Hi, I have a bunch of points in form of x,y (0, 1), (2, 5), (1, 6), and (7, 6)
and I'm trying to understand circle regression using least squares. I can't find any examples anywhere. But I did find this picture of a matrix that might do the trick.

View attachment 21891

So I think I first need to understand these summation terms within the matrix

row1 column1 it is Sum( xi^2) = does this mean I add all the x values up and square them? or do I square all the x values and then add them?
row1 column2 it is Sum( xi yi) = does this mean I multiply all the points first and then sum them? {(x1 * y1) + (x2 * y2) + (x3 * y3)+ (xn * yn).......} ??
so ..... sum{(0 * 1) + (2 * 5) + (1 * 6)+ (7 * 6)}

row 1 column3 it is Sum( xi) = does this mean I add all the x values? 0 + 2 + 1 +7
row 3 column 3 it is n = do I just put the number of points here? (4 in my case)

the rest are similar to above

thanks in advance
Click to expand...
[MATH]\sum[/MATH]xi2=x12+x22+...+xn2. The other rows look right. If you want to understand what is going on you may look into this paper: https://www.emis.de/journals/BBMS/Bulletin/sup962/gander.pdf
or this one: http://www.dtcenter.org/sites/default/files/community-code/met/docs/write-ups/circle_fit.pdf
 
markraz said:
Hi, I have a bunch of points in form of x,y (0, 1), (2, 5), (1, 6), and (7, 6)
and I'm trying to understand circle regression using least squares. I can't find any examples anywhere. But I did find this picture of a matrix that might do the trick.

View attachment 21891

So I think I first need to understand these summation terms within the matrix

row1 column1 it is Sum( xi^2) = does this mean I add all the x values up and square them? or do I square all the x values and then add them?
row1 column2 it is Sum( xi yi) = does this mean I multiply all the points first and then sum them? {(x1 * y1) + (x2 * y2) + (x3 * y3)+ (xn * yn).......} ??
so ..... sum{(0 * 1) + (2 * 5) + (1 * 6)+ (7 * 6)}

row 1 column3 it is Sum( xi) = does this mean I add all the x values? 0 + 2 + 1 +7
row 3 column 3 it is n = do I just put the number of points here? (4 in my case)

the rest are similar to above

thanks in advance
Click to expand...
row1 column1 it is Sum( xi^2) = does this mean I add all the x values up and square them? or do I square all the x values and then add them?

row1 column2 it is Sum( xi yi) = does this mean I multiply all the points first and then sum them? {(x1 * y1) + (x2 * y2) + (x3 * y3)+ (xn * yn).......} ??
so ..... sum{(0 * 1) + (2 * 5) + (1 * 6)+ (7 * 6)} ............Correct

row 1 column3 it is Sum( xi) = does this mean I add all the x values? 0 + 2 + 1 +7............Correct

row 3 column 3 it is n = do I just put the number of points here? (4 in my case)............Correct
 
Subhotosh Khan said:
row1 column1 it is Sum( xi^2) = does this mean I add all the x values up and square them? or do I square all the x values and then add them?

row1 column2 it is Sum( xi yi) = does this mean I multiply all the points first and then sum them? {(x1 * y1) + (x2 * y2) + (x3 * y3)+ (xn * yn).......} ??
so ..... sum{(0 * 1) + (2 * 5) + (1 * 6)+ (7 * 6)} ............Correct

row 1 column3 it is Sum( xi) = does this mean I add all the x values? 0 + 2 + 1 +7............Correct

row 3 column 3 it is n = do I just put the number of points here? (4 in my case)............Correct
Click to expand...
appreciate it, thanks !!!
 
Ok thanks everyone. My next question is this top row of the 'b' matrix
1601228378436.png

again my set of points
(0, 1), (2, 5), (1, 6), and (7, 6)

first row in vector sum xi(xi^2 + yi^2)
0 * (0^2 + 1^2) +
2 * (2^2 + 5^2) +
1 * (1^2 + 6^2) +
7 * (7^2 + 6^2)

second row in vector sum yi(xi^2 + yi^2)
1 * (0^2 + 1^2) +
5 * (2^2 + 5^2) +
6 * (1^2 + 6^2) +
6 * (7^2 + 6^2)


thanks in advance
 
markraz said:
Ok thanks everyone. My next question is this top row of the 'b' matrix
View attachment 21894

again my set of points
(0, 1), (2, 5), (1, 6), and (7, 6)

first row in vector sum xi(xi^2 + yi^2)
0 * (0^2 + 1^2) +
2 * (2^2 + 5^2) +
1 * (1^2 + 6^2) +
7 * (7^2 + 6^2)

second row in vector sum yi(xi^2 + yi^2)
1 * (0^2 + 1^2) +
5 * (2^2 + 5^2) +
6 * (1^2 + 6^2) +
6 * (7^2 + 6^2)


thanks in advance
Click to expand...
Everything looks fine.
 
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