# Center of a circle in polar coordinates

So I was attempting to solve a topology exercise when the following question came to me.

The objective was trying to fine a bijection $$f$$ between the following disks in $$\Bbb R^2$$:

$$D_1 :=\{(x,y)\in \Bbb R^2: x^2 + y^2 \leq1\}$$ $$D_2 :=\{(x,y)\in \Bbb R^2: x^2 + y^2 \leq 4\}$$

So $$D_1$$ is a disk of radius $$1$$ centered at the origin, and $$D_2$$ is a disk of radius $$2$$ centered at the origin.

The first thing that came to my mind was to use polar coordinates, so let's redefine both disks as:

$$D_1 :=\{(r,\varphi): r\in [0,1] \wedge \varphi \in [0,2\pi]\}$$ $$D_2 :=\{(r,\varphi): r\in [0,2] \wedge \varphi \in [0,2\pi]\}$$

Now we can just simply scale disk 1 into disk 2:

$$f:D_1\to D_2$$ $$f(r,\varphi)=(2r,\varphi)$$

My question about the injectivity of this function, more concretely in the center of the disks.

Let $$\varphi_1,\varphi_2 \in [0,2\pi]$$, with $$\varphi_1 \neq \varphi_2$$. Then how do we treat points like $$(0,\varphi_1)$$ and $$(0,\varphi_2)$$. In the disk they represent the same point: the center of the disk. But, when learning about double integrals with polar coordinates, my teacher taught us that when we use polar coordinates to describe a disk we are just defining an rectangle in the $$rO\varphi$$ plane, instead of in the $$xOy$$ plane:

And all the points of the form $$(0,\varphi)$$ are in that line in $$\varphi-$$axis and are indeed different points.

So how do we treat the points with this form? Are they considered all the same point and thus $$(0,\varphi_1) = (0,\varphi_2)$$? Or are they considered different points as seen in the $$rO\varphi$$ plane?

• They are different names for the same point in $\Bbb R^2$, just as $\langle 1,0\rangle$ and $\langle 1,2\pi\rangle$ denote the same point. (You might want to take $\varphi\in[0,2\pi)$ rather than in $[0,2\pi]$.) Aug 22, 2020 at 17:06

Your function is simply $$(x,y)\mapsto2(x,y)$$, and therefore it is injective (indeed, bijective).
Concerning your final questions, all the pairs $$(0,\varphi_1)$$, with $$\varphi\in[0,2\pi]$$, describe the same point (the origin), and therefore there is no problem there.
The correct way to define polar co-ordidinates is using the domain $$r > 0$$ , $$\varphi$$ in $$(0, 2\pi)$$ so that the transformation between cartesian and polar co-ordinates is a diffeomorphism (has a differentiable inverse).
• This way there is no representation for $(0,0)$ or any point on the positive $x$ axis, so we cannot actually transform one disk to the other. Aug 22, 2020 at 17:34