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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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What do you mean by "most general"? –  Mariano Suárez-Alvarez Apr 1 '12 at 20:00
    
@MarianoSuárez-Alvarez: The most number of constants free for us to set, in other words, maximizing the number constants like $a,b,R, \omega, \theta$ to give an expression that still gives a circle. –  elementary Apr 1 '12 at 20:10
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You can always do silly things like $$t\mapsto\bigl(\cos(a t^5+bt^4+ct^3+dt^2+et+f),\sin(a t^5+bt^4+ct^3+dt^2+et+f)\bigr)$$ in order to involve more constants... –  Mariano Suárez-Alvarez Apr 1 '12 at 20:12
    
Does it matter whether the circles are different? The $\theta$ in your version is simply equivalent to a shift in the value of $t$. –  Mark Bennet Apr 1 '12 at 20:38

1 Answer 1

up vote 4 down vote accepted

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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The geometry of the manifold of circles in the plane and, more generally, of the manifolds of spheres in higher-dimensional euclidean spaces is a very beautiful and classical subject. "Lie sphere geometry" is a good keyword to search for information on this subject. –  Mariano Suárez-Alvarez Apr 1 '12 at 20:18
    
Thank you! By redundancies, I presume you mean factors that can be omitted and we still get the same circle, right? –  elementary Apr 1 '12 at 20:18
    
Indeed. The dimension of the manifold is, in essence, the number of parameters needed to perfectly describe a specific circle. –  Mariano Suárez-Alvarez Apr 1 '12 at 20:19
    
What if we are describing a circular path in the x,y-plane? Then I think there will be more non-redundant parameters? –  elementary Apr 1 '12 at 20:22
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If by «circular path» you mean an arbitrary way to trace the curve of a circle, then there are infinitely many paramateres needed: the set of all functions $\mathbb R\to\mathbb R^2$ whose image is a circle is an "infinite dimensional manifold", for an appropriate meaning of this term. –  Mariano Suárez-Alvarez Apr 1 '12 at 20:25

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