# How to extend rational parametrization of the circle to three dimensions?

I recently became aware of the rational parametrization of the circle in two dimensions:

$$\left(\frac{1-m^2}{1+m^2}, \frac{2m}{1+m^2}\right)$$

for a unit circle centered on the origin.

I'm interested in extending this to an arbitrary circle in three dimensions, in particular given the center, normal and radius.

I'd like the result to still be a rational parametrization.

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Note: still looking for a circle, $S^1$ not sphere $S^2$. – James Tauber Jan 30 '13 at 2:15

Pick two unit vectors $\mathbf u$ and $\mathbf v$ so that together with the specified normal they form an orthonormal frame. The parametrization you want is $$\mathbf c + \frac{1-m^2}{1+m^2}r\mathbf u+\frac{2m}{1+m^2}r\mathbf v,$$ where $\mathbf c$ and $r$ are the center and radius of the circle. What we're doing is simply taking the $xy$ plane that the original circle lies in, and mapping the origin to $\mathbf c$ and the $x$ and $y$ axes to $r\mathbf u$ and $r\mathbf v$.