Rational parametrization of circle in Wikipedia In http://en.wikipedia.org/wiki/Circle but also in the corresponding article in the German Wikipedia I find this formulation ( sorry, I exchange x and y as I am accustomed to it in this way ) :
"An alternative parametrisation of the circle is:
$$x=a+r\frac{1-t^2}{1+t^2}$$ and $$y=b+r\frac{2t}{1+t^2}$$ with real-valued parameter $t$."
But if I plot this curve I only get a half-circle with these formulas when $(a,b)\neq(0,0)$ which is obviously not what is searched for and expected.
I recently found without any elementary trigonometry e.g. for the unit circle with $(a,b)=(1,0)$ a closed rational parametrization which plots the "whole" circle perfectly if $\lim_{t\to\infty}$. 
Now my 2 questions:


*

*What is your formula for the case of the unit circle with $(a,b)=(1,0)$
and how is your way to derive it ?

*Is there a way to calculate a rational parametrization for the unit    circle with general $(a,b)$ which allows "whole"-circle plotting ? Maybe $a$ and $b$ must be rational numbers or even integers ? 

 A: Write
$$
f(t)=\frac{1-t^2}{1+t^2}
$$
and
$$
g(t)=\frac{2t}{1+t^2}
$$
The point should be that $(f(t))^2+(g(t))^2=1$ so that $(f(t),g(t))=(\cos \theta,\sin\theta)$ for some angle $\theta$. It is easy to see that 


*

*$f(t)=0\iff t=\pm 1$ and

*$f(t)>0\iff |t|<1$ and

*$f(t)=1\iff t=0$


It is also easy to see that


*$g(t)=0\iff t=0$ together with

*$g(t)>0\iff t>0$ and 

*$|g(t)|=1\iff |t|=1$


But $f(t)=-1$ is never reached. Analyzing $f$ and $g$, we see that
$$
t\to-\infty\implies f(t)\to -1, g(t)\to 0_-
$$
and
$$
t\to+\infty\implies f(t)\to -1, g(t)\to 0_+
$$
so to have a full circle (except for one point at $(-1,0)$) we must have $t\in\mathbb R$.



As you can see, it is like having one period of $\cos\theta$ and $\sin\theta$ reparametrized to $f(t)=\cos(\theta(t))$ and $g(t)=\sin(\theta(t))$ such that $\theta(t)\to-\pi$ for $t\to-\infty$ and $\theta(t)\to\pi$ for $t\to\infty$.
A: Wiki form is correct. However, along with the so-called "advantage" of rational parametric form comes the limitation of infinite argument requirement to complete a deep circle. 
For y to be zero or x to attain its extremum value you expect for  single parameter $t$ have a reasonable domain for full circular arc description. But it is not so.. as one root of $t$ is at $t=0$ and another one at $ t= \infty$. 
A trigonometric expression remedies the situation fully by expressing coordinates in terms of double angle $ 2 T$ ( double of half angle in Weierstrass form ) given in Wiki.( Actually I recommend its deletion or insertion of trig form as  Wiki is a central reference).
$$ x = a + r \cos (2 T) ;\;  y = b + r \sin (2 T) ;  $$

The Mathematica code is given in the image.I have slightly reduced rational circle radius to show full circle by trigonometrical parametrization by contrast.
As an aside the particular trig representation of Circle has significance in Mechanics of Structures. When y_offset = 0, it is the Mohr Circle, first used by famous German civil engineer Otto Mohr who defined static determinacy and constructed very useful diagrams of stress and strain. The angle $T$ represents angle between reference direction of stress and principal stress. 
Standard form with half angle in trigonometric $$x = h + r \sin \theta ; \; y = k + r \sin \theta $$ serves the same purpose, it is simpler.
