# Ellipse Proof by Circle : Affine Transformation

By Affine Transformation the Ellipse to a Circle, the corresponding ratios preserved. Hence it is easier to prove in the circle below:

Similar triangles CQV ~ CTQ

https://m.toutiaoimg.cn/a6580683485986423309

Proof : Ellipse Area from Circle :

# Pi hiding in prime regularities

Three mysterious Math Objects:

• Pi,
• Complex Numbers,
• Prime Numbers

are hiding in circle.

# Circle in Different Representations

1-Dimensional Objects:
Affine Line: ${\mathbb {A}^1}$
Circle: ${\mathbb {S}^1}$

Six Representations of a Circle: ${\mathbb {S}^1}$
1) Euclidean Geometry (O-level Math)
Unit Circle : $x^2 + y^2 = 1$

2) Curve: (A-level Math)
Transcendental Parameterization :
$\boxed { e(\theta) = (\cos \theta, \sin \theta) \qquad 0 \leq \theta \leq 2\pi }$

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

3) Affine Plane (French Baccalaureate – equivalent A-level – Math) ${\mathbb {A}^2}$
1-Dim Sub-spaces = Projective Lines thru’ Origin

5) Identifying Intervals: (closed loop) (Undergraduate Math)

6) $\text {Translation } (\tau, {\tau}^{-1}) \text { on a Line }$ ${\mathbb {A}^1}$
$\boxed { {\mathbb {S}^1} \simeq {\mathbb {A}^1 } \Big/ { \langle \tau , {\tau}^{-1} \rangle} }$
${\mathbb {S}^1} = \text { Space of all orbits}$