Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Replies: 1   Last Post: Dec 8, 1996 4:26 PM

 Karine Lecarpentier Posts: 1 Registered: 12/15/04
Posted: Dec 8, 1996 4:26 PM

> In article <329CB599.62CD@worldnet.att.net>,
> Marcus <SCI.@worldnet.att.net> says:
>I have a number of x,y data points that form an arc of constant radius.
> Do you know a good technique for finding the center and radius of the arc?
>I've tried the following, but I'm not sure of it's reliability:
>take first data point, find line running through point but also tangent
>to the arc, take second data point and find line tangent again.
>Normalize both lines and find the intersection. This will give me the
>center, from here the radius is easy.
>
>Do you have any suggestions?
>Thanks, Marc

As some previous persons sugested, you shoud use an optimization method.

If C is your center, and M(i), the points of your trajectory,then

R = 1/N *sum_{i=1}^N {||M(i) - C ||},

You can find C by minimizing

m = sum_{k=1}^N {||M(k) - c|| - R}^2
= sum_{k=1}^N {||M(k) - c|| -
1/N *sum_{i=1}^N {||M(i) - C ||}^2

Or, if you are sure that your curve is a circle, you can use a linear
algorithm by setting :

R = sqrt (1/N * sum_{i=1}^N {||OM(i) - OC ||^2})
= sqrt (1/N * sum_{i=1}^N {|OM(i)|^2 - 2 t(OM(i)).OC + |OC|^2})

You want to minimize :

m = sum_{k=1}^N {||OM(k) - OC||^2 - R^2}^2
= sum_{k=1}^N { |OM(k)|^2 - 1/N*sum_{i=1}^N |OM(i)|^2 -
2*t(OM(k)-1/N*sum{i=1}^N OM(i)).OC}^2

You get then an easy to solve linear least squares paroblem where your
unknown is OC.

If you derive this expression with respect to OC, and set the derivation
to 0, you will get a simple equation A.OC = b, where A is a 2*2 matrix,
and b is a 2*1 vector.

I hope this helps. Feel free to contact me for more details.

Karine

----
Karine Lecarpentier, M.A.Sc student
UBC, Dept. of Electrical Engineering e-mail: karinel@ee.ubc.ca
2356 Main Mall phone: (604) 822-0532
Vancouver, B.C fax: (604) 822-5949