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What is the most general parametric representation of a circle?

The best I can come up with is $(x,y)=(a+R\cos(\omega t +\theta), b+R\sin(\omega t +\theta))$

I hope this question is not too elementary for this site!

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    Does it matter whether the circles are different? The $\theta$ in your version is simply equivalent to a shift in the value of $t$.2012-04-01

1 Answers 1

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The family of all circles in the plane can be viewed as a manifold of dimension $3$. Roughly, this is because we can specify each circle unequivocably using three real numbers (the two coordinates of its center and its radius)

This gives a specific sense in which "the most general equation of a circle" has exactly three non-redundant paramenters.

The parametrization you gave involves $\omega$ and $\theta$ which only introduce redundancies, and many more redundancies can be introduced as in my comment above: as a consequence, so it does not make much sense to consider that idea.

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    There is also stereographic projection. I'm just saying.2012-04-01