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I would like to find an elementary function of two variables such that the level set $f(x,y)=1$ is the ellipse with equation $2x^2+y^2=1$ and the level set $f(x,y)=2$ is the circle with equation $x^2+y^2=9.$

I have constructed non-elementary functions with those level sets in several ways, but I would like an elementary function to make it easier to give as an example.

Robert Z
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Johan
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    Do you mind to show the "non-elementary" functions that you have found? – Robert Z Nov 10 '23 at 11:54
  • How did you know meromorphic solutions existed ?? – mick Nov 10 '23 at 12:01
  • Robert Z: The easiest solution is probably to do interpolation on each half-line through the origin separately. Each half-line intersects both level curves, and between you interpolate in some nice way. (Say the value is zero at the origin.) Alternatively you can do $f(x,y) = g(2x^2+y^2)+h(x^2+y^2)$ where $g(1)=1$, g is increasing and $g(t)=1.5$ if $t\geq 2$, $h(9)=0.5$, $h$ is increasing and $h(t)=0$ if $t \leq 8$. – Johan Nov 12 '23 at 17:37

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I tried with a function $f$ of the form: $$f(x,y)=\frac{ax^2+by^2}{cx^2+dy^2+e}$$ with real coefficients $a,b,c,d,e$ to be found. Then the level set $f(x,y)=k$ is given by $$(a-kc)x^2+(b-kd)y^2=ke.$$ After a few calculations I got $$f(x,y)=\frac{34x^2+16y^2}{16x^2+7y^2+9}.$$ Verify the shape of the level sets of $f$ via this Desmos Demonstration.

Robert Z
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