Affine Line: $latex {mathbb {A}^1}&s=3$

** Six Representations of a Circle**: $latex {mathbb {S}^1}&s=3$

1) Euclidean Geometry

Unit Circle : $latex x^2 + y^2 = 1$

2) Curve:**Transcendental Parameterization :**

$latex boxed { e(theta) = (cos theta, sin theta) qquad

0 leq theta leq 2pi }&fg=aa0000&s=3

$

**Rational Parameterisation :**

$latex boxed {

e(h) = left(frac {1-h^2} {1+h^2} : , : frac {2h} {1+h^2}right) quad text { h any number or } infty

} &fg=aa0000&s=2

$

3)** Affine Plane** $latex {mathbb {A}^2}&s=3$

1-dim sub-Space = Lines thru Origin

5) Identifying Intervals: (closed loop)

6) $latex text {Translation } (tau, {tau}^{-1}) text { on a Line } $ $latex {mathbb {A}^1}&s=3$

$latex boxed {

{mathbb {S}^1} = {mathbb {A}^1 } Big/ { langle tau , {tau}^{-1} rangle}

}&fg=aa0000&s=3

$

$latex {mathbb {S}^1} = text { Space of all orbits} $