Euler's formula for off-center circle

Question

A circle with radius $R$ and center at $(a,b)$ is given by the formula $(x-a)^2 +(y-b)^2 = R^2$. A circle with radius $R$ whose center is at the origin is given by Euler's formula: $R e^{i \theta}$. What is the formula for a circle with radius $R$ and center $(a,b)$ in terms of Euler's formula?

Answer 1

$z \mapsto z+a+bi$ sends the origin to $(a,b)$. Hence the circle you want is given by $z=R e^{i \theta}+a+bi$.

Answer 2

There may be a bit of confusion here. First, you must distinguish between parametric and implicit formulae.

The first formula, $(x-a)^2+(y-b)^2=R^2$ is implicit. This means that the equation is the condition that points $(x,y)$ must hold in order to belong to the figure (a circle, in this case).

The second formula is parametric. The parameter is $\theta$, and this means that for each possible value of $\theta$ the formula gives you a point of the figure.

It is also important that the first formula is for $\Bbb R^2$ and the second one is for $\Bbb C$, that are similar in some senses, but not the same thing.

If what you want is a parametric formula for circles in $\Bbb C$ with arbitrary center $a+ib$, it is $$x+iy=a+ib+e^{i\theta},\quad\theta\in[0,2\pi)$$