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October 27th, 2013, 09:43 PM   #1
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centre circle in 3D

Who can calculate the coordinates of the centre of a circle given 3 points P1(x1,y1,z1),P2(x2,y2,z2),P3(x3,y3,z3)
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October 28th, 2013, 12:40 AM   #2
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Re: centre circle in 3D

Let the center of your circle be the point and its radius . Then you have the three simultaneous equations
for .
By expanding the squares in the three equations and making pairwise subtractions , your new set of equations will no longer have . Unfortunately, this new system of equations is not linearly independent so you need one more, which is the equation of the plane containing the three points.
Find the cross product and the plane's equation is where is uniquely determined by plugging any of into the plane's equation. Fourth equation is .
Using this last equation together with two of the pairwise subtractions will suffice in determining and then you may proceed to determine if you want.
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October 28th, 2013, 09:29 AM   #3
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Re: centre circle in 3D

Just for side info, three points are not sufficient to determine a 2-d sphere because 3 points in are always co-planar and so may determine a unique circle but infinitely many spheres. You will need a fourth point outside the plane of the other 3 to uniquely determine a sphere. Just fit the fourth point into the algorithm I first gave before the plane equation.
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