~~~
Jonas
I seem to think 5 variables are needed to place an ellipse in the
2 dimensional plane. Say for example: the coordinates of 2 foci,
and the sum of the distances form the foci to the ellipse for a total of
5 variables, say [fu1,fv1,fu2,fv2,D]. Another possibility as you describe
is: the coordinates of the center, the semi major axis ratio,
the ellipse orientation angle, and the constant D, also for 5 variables
[uc,vc,a,k,D]. I seem to think that with given 3 points on the ellipse
6 variables are determined and so in principle one should be able to solve
for the 5 variables that define the solution ellipse.
What I do not understand how only 3 proposed solution variables [uc,vc,D]
can define a generalized ellipse on the plane. I guess I do not understand D.
Interesting problem.
Ed
~~~
At 10:18 AM 1/19/2008, Jonas Forssell wrote:
>I am using Maxima to find an algebraic solution to an ellipse passing through three points.
>
>The implicit form of an ellipse with center (uc,vc), angle a and ratio k between major and minor axis lengths is:
>
>G := k^2*((u-uc)^2*sin(a)^2-(u-uc)*(v-vc)*sin(2*a)+(v-vc)^2*cos(a)^2)+(u-uc)^2*cos(a)^2+(v-vc)^2*sin(a)^2+(u-uc)*(v-vc)*sin(2*a)-D=0
>
>Assume three points with coordinates (u1,v1) (u2,v2) (u3,v3) which the ellipse passes through with k and a given. I then need to determine uc,vc and D
>
>So, I generate three equations substituting u for u1 and v for v1 and so on.
>After that I use linsolve to get the solution for uc,vc and D.
>The equations get very long.
>
>My problem is that these equations does not seem to give the right answer numerically. I try three points and using the derived values for uc,vc and D does not solve G to be equal to 0.
>
>Could this be a bug in Maxima or am I missing something fundamental?
>
>Thanks
>/Jonas Forssell, Gothenburg, Sweden
>
>
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