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