Parameter Form

x:=a⋅\cos t
y:=b⋅\sin t

where:

\sin x:=÷{exp(i⋅x)-\exp(-i⋅x)}{2⋅i}
\cos x:=÷{exp(i⋅x)+\exp(-i⋅x)}{2}

where: i²=-1

\tan x:=÷{\sin x}{\cos x}

Properties

Circumference

The Circumference s is:

s:=∫\limits_{∂} ds

where ds :: short_arc.

Since:

(÷{ds}{dt})²=(÷{dx}{dt})²+(÷{dy}{dt})²

... it follows that:

s:=∫\limits_{0}^{2⋅π} √{ẋ²+ẏ²}⋅dt
s=a⋅∫ \limits_{0}^{2⋅π} √{1-k²⋅(\sin s)²}⋅ds

Numeric Eccentricity

ε:=√{÷{a²-b²}{a²}} Excentricity.

And thus:

b=a⋅√{1-ε²}

Parameter Form in Polar Coordinates, center of the ellipse as center of coordinate system

r(φ)=a⋅√{1-ε²⋅(\sin φ)²}

Because:

x=a⋅\cos φ
y=a⋅√{1-ε²}⋅\sin φ
r²=x²+y²
r²=a²⋅((\cos φ)²+(1-ε²)⋅(\sin φ)²)
r²=a²⋅((\cos φ)²+(\sin φ)²-ε²⋅(\sin φ)²)
r²=a²⋅(1-ε²⋅(\sin φ)²)

... after choosing the positive root:

r=a⋅√{1-ε²⋅(\sin φ)²}