# Is there a way to represent the interior of a circle with a curve?

As you already know, the interior of a circle is represented by an inequality. For example,

$$x^2+y^2\leq1$$

for the unit circle. Today I was thinking by myself and I wondered if there is a curve that could represent every point inside of a circle. Maybe with a spiral like this,

If you can’t represent it perfectly with a curve, what would be the closest way to represent it?

This question is asked merely out of curiosity, it may be completely irrelevant or meaningless 🙂

Since topologically a disc and a square are the same, most of what you might want to know about this falls under the heading of Space-filling curves. To summarize, the answer to your main question is that the disc $D^2$ is the image of the interval $[0,1]$ under a continuous map, but not a one-to-one (non-intersecting) continuous map. So it depends on exactly what you mean by curve.