🚀 go-pugleaf

In article <3523A24A.F9C10F3B@conversionsinc.com>, Kurt Feldbush <kurt@conversionsinc.com> writes:
|>     I am looking for an equation or algorithm to find the best fit
|> circle or ellipse to a set of data points.  I have searched everywhere I
|> could think of with no luck.  Can anyone help?
|> 
take odrpack from netlib. there is an ellipse-fit in the user-guide,
and, of course, the circle is the simpler case. you could do the
circle fit in a special manner (not using general orthogonal regression)
simply by minimizing 

   sum_i { ( (x_i-xm)**2+(y_i-ym)**2-r**2 )**2 }

with respect to xm,ym and r (midpoint and radius), with x_i,y_i the
given points
hope this helps
peter

In article <3523A24A.F9C10F3B@conversionsinc.com>, Kurt Feldbush
<kurt@conversionsinc.com> wrote:

>    I am looking for an equation or algorithm to find the best fit
>circle or ellipse to a set of data points.  I have searched everywhere I
>could think of with no luck.  Can anyone help?

Have a look at

  "Least-squares fitting of circles and ellipses", W.Gander, G.H. Golub,
  and R. Strebel, BIT, vol 34 (1994), pp. 448-578

Gander et al compare different methods.  MATLAB source codes are
available at ftp.inf.etch.ch.

Gerry

-- 
+-+-+-+-+-+-+-+-...
Gerald Recktenwald    (503) 725-4296
gerry@me.pdx.edu      http://www.me.pdx.edu/~gerry/
PSU Mechanical Engineering, P.O. Box 751, Portland, OR 97207

Kurt Feldbush wrote:

>    I am looking for an equation or algorithm to find the best fit
>circle or ellipse to a set of data points.  I have searched everywhere I
>could think of with no luck.  Can anyone help?

Kurt,


I used the Mathematica function onlinearRegress

	<<Statistics`NonlinearFit

I made 50 perturbed points, a list points,  for an ellipse center {.2, .4}
with semi-axes 3 in direction -Pi/3 and 5 at right angles to this.

First I need a model (for a circle this can be much simpler)

u ={Cos[t],Sin[t]}; (*t = angle of one axis*)
p= {x,y};          (*p is a generic data point*)
c = {cx,cy};        (*c is the center of the ellipse*)
pc = p-c;                         
model = ((pc).u)^2 /a^2 + (pc.pc - (pc.u)^2)/b^2 - 1 ;
(* a is semi-axis in direction of u , a the other one*)
(*equation of ellipse is model == 0*)

vars = {x,y};
params = {a,b,t,cx,cy};
data = Append[#,0.0]&/@points;

We calculate the optimal model

bestparams=
BestFitParameters/.NonlinearRegress[data, model, vars, params,
RegressionReport ->BestFitParameters]

{a -> 3.33709, b -> 4.87806, t -> 1.03938, cx -> -0.24978, 
 
  cy -> 0.486313}

Put thes values into the model

fmla = Expand[model /. bestparams]

                                      2
0.0166528 + 0.00682348 x + 0.0542927 x  - 0.0649814 y + 
 
                             2
  0.0417406 x y + 0.0775296 y

Examine the fit graphically

<<Graphics`ImplicitPlot`

ft=ImplicitPlot[fmla==0,{x,-5,4.5},{y,-4.5,4.5}, DisplayFunction -> Identity]

(*DisplayFunction -> Identity turns display off*)

pts =Show[Graphics[
		{PointSize[.02],Hue[0],Point/@points}
	
], DisplayFunction -> Identity]


Show[ft, pts, DisplayFunction -> $DisplayFunction]

(*DisplayFunction -> $DisplayFunction turns display on*)
-- 
Allan Hayes
Mathematica Training and Consulting
Leicester, UK
hay@haystack.demon.co.uk
http://www.haystack.demon.co.uk
voice: +44 (0)116 271 4198
fax: +44 (0)116 271 8642

Kurt Feldbush wrote:

>    I am looking for an equation or algorithm to find the best fit
>circle or ellipse to a set of data points.  I have searched everywhere I
>could think of with no luck.  Can anyone help?

Kurt,


I used the Mathematica function NonlinearRegress

	<<Statistics`NonlinearFit

I made 50 perturbed points, a list points,  for an ellipse center {.2, .4}
with semi-axes 3 in direction -Pi/3 and 5 at right angles to this.

First I need a model (for a circle this can be much simpler)

u ={Cos[t],Sin[t]}; (*t = angle of one axis*)
p= {x,y};          (*p is a generic data point*)
c = {cx,cy};        (*c is the center of the ellipse*)
pc = p-c;                         
model = ((pc).u)^2 /a^2 + (pc.pc - (pc.u)^2)/b^2 - 1 ;
(* a is semi-axis in direction of u , a the other one*)
(*equation of ellipse is model == 0*)

vars = {x,y};
params = {a,b,t,cx,cy};
data = Append[#,0.0]&/@points;

We calculate the optimal model

bestparams=
BestFitParameters/.NonlinearRegress[data, model, vars, params,
RegressionReport ->BestFitParameters]

{a -> 3.33709, b -> 4.87806, t -> 1.03938, cx -> -0.24978, 
 
  cy -> 0.486313}

Put thes values into the model

fmla = Expand[model /. bestparams]

                                      2
0.0166528 + 0.00682348 x + 0.0542927 x  - 0.0649814 y + 
 
                             2
  0.0417406 x y + 0.0775296 y

Examine the fit graphically

<<Graphics`ImplicitPlot`

ft=ImplicitPlot[fmla==0,{x,-5,4.5},{y,-4.5,4.5}, DisplayFunction -> Identity]

(*DisplayFunction -> Identity turns display off*)

pts =Show[Graphics[
		{PointSize[.02],Hue[0],Point/@points}
	
], DisplayFunction -> Identity]


Show[ft, pts, DisplayFunction -> $DisplayFunction]

(*DisplayFunction -> $DisplayFunction turns display on*)
-- 
Allan Hayes
Mathematica Training and Consulting
Leicester, UK
hay@haystack.demon.co.uk
http://www.haystack.demon.co.uk
voice: +44 (0)116 271 4198
fax: +44 (0)116 271 8642