Fastest way to fit a parabola to set of points?

Question

Given a set of points, what\'s the fastest way to fit a parabola to them? Is it doing the least squares calculation or is there an iterative way?

Thanks

Edit

Answer 1

I recently needed to find a parabola that passes through 3 points.

suppose you have (x1,y1), (x2,y2) and (x3,y3) and you want the parabola

y-y0 = a*(x-x0)^2

to pass through them: find y0, x0, and a.

You can do some algebra and get this solution (providing the points aren't all on a line) :

let c = (y1-y2) / (y2-y3)
x0    = ( -x1^2 + x2^2 + c*( x2^2 - x3^2 ) )  /  (2.0*( -x1+x2 + c*x2 - c*x3 ))
a     = (y1-y2)  /  ( (x1-x0)^2 - (x2-x0)^2 )
y0    = y1 - a*(x1-x0)^2

Note in the equation for c if y2==y3 then you've got a problem. So in my algorithm I check for this and swap say x1, y1 with x2, y2 and then proceed.

hope that helps!

Paul Probert