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 ?
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
It is also easy to see that
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 $t\to\infty$.