Why is a circle 1-dimensional?

In the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ position of a circle, but apparently a circle can be described with only the $x$-coordinate? How is this possible without the $y$-coordinate also?

Answer

Suppose we’re talking about a unit circle. We could specify any point on it as:
$$(\sin(\theta),\cos(\theta))$$
which uses only one parameter. We could also notice that there are only $2$ points with a given $x$ coordinate:
$$(x,\pm\sqrt{1-x^2})$$
and we would generally not consider having to specify a sign as being an additional parameter, since it is discrete, whereas we consider only continuous parameters for dimension.

That said, a Hilbert curve or Z-order curve parameterizes a square in just one parameter, but we would certainly not say a square is one dimensional. The definition of dimension that you were given is kind of sloppy – really, the fact that the circle is of dimension one can be taken more to mean “If you zoom in really close to a circle, it looks basically like a line” and this happens, more or less, to mean that it can be paramaterized in one variable.

Attribution
Source : Link , Question Author : mr eyeglasses , Answer Author : Milo Brandt

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