I'm trying to solve a problem and am currently stuck, so I came here in the hope that someone could help me.
I have a 10x3 matrix (call it "A") containing position data for points in space and I need to find out if these lie on a circle using QR-factorization (and, later on, the least square method). It's a matlab excercise, so the calculations themselves aren't a concern, but rather the theory behind them.
I've been staring at the problem for a long time now, but I can't quite figure it out.
I realize that the points must lie on a plane, i.e. a subspace R2 to the space R3 in which the data points are scattered.
The equation of a circle is (x - a)^2 + (y - b)^2 = r. Or, if the circle is centered around origo, x^2 + y^2 = r (but the given data points don't appear to be).
I know the principle of QR-factorization, and have managed to QR-factorize A. When QR-factoring A into QR, Q forms an orthonormal basis for col(A), which I think is the clue to determining if the points form a circle. But that's about as far as I get.
I can't quite see the full connection to QR-factorization and I'm not sure how to write the equation of the circle so that the least square approximation is applicable.
Can anyone please give me some help?