If R is the horizontal radius and S is the vertical radius, the Cartesian formula for an ellipse is:
f(x) = plusminus (S/R)(sqrt(R2-x2))
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If you start with the following equation of an ellipse: x2/4 + y2/9 = c2 and transform the equation to 9x2/36 + 4y2/36 = 36c2 the denominators are the same but the equation is unchanged and so the ellipse remains exactly as it was before.
It is the Cartesian equation of an ellipse.
x2/4^2+y2/12^2=1
There are different standard forms for different things. There is a standard form for scientific notation. There is a standard form for the equation of a line, circle, ellipse, hyperbola and so on.
It isn't possible to give a generalised formula for the circumference of an ellipse in terms of elementary functions.