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There's one basic mathematical thing that keeps bugging me: the fact that a really simple 2D geometrical figure (like a circle) might not be a function.

I know what the definition of a function is. A circle is not a function (of one variable) because it would associate two values of the co-domain to a single value of the domain. But this doesn't help my intuition.

It sounds terribly weird that a given curve (say a sinusoidal) is a function only unless you rotate it through $45^o$ or more degrees...


Is there any simple way (a concept similar to that of how most people imagine a function: a curve on a graph) to represent 2D geometrical figures like a circle (or a rotated sinusoidal, or whatever)?

The only one I can think of is using a function of two or more variables, but this sounds pretty dirty to me: why should I use a function in three dimensions just to see its shadow on two dimensions?
Besides if I think of the function as a real object (in our real, 3D space), I cannot help thinking that it's not a 2D circle, it's a 3D weird object which can be seen as a circle when rotated in a particular way (just like the Penrose stairs look possible when rotated in a special way).

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    Raskolnikov has the right idea; in general lots of manipulations related to curves are eased when you possess parametric equations for them. Also, a curve that may not have a nice polar coordinate representation might still be amenable to conversion to a parametric form.2010-11-28

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Some possibilities:

First of all, you might also want to switch to polar coordinates, in which points are defined through angle $\varphi$ and radius $r$ rather than $x$/$y$ coordinates. For example, in a polar coordinate system, a circle (let's take the unit circle) $K$ actually is a function (now of form $r(\varphi)$ rather than $y(x)$).

As the radius is constant, we end up with

$K: r(\varphi) = 1$

But most figures still aren't functions in polar coordinates, so we might have to take a more general approach: A curve

A curve is a function that produces coordinates rather than a single value from some parameter, i.e. $\mathbb{R} \to \mathbb{R}^2$ in our case.

Let $K$ be our unit circle again - now parameterized by some angle $\varphi$

$K(\varphi) = (\cos \varphi, \sin \varphi)$

At least, this should work for most figures. The most general form though is simply an equation that the coordinates have to satisfy.

$K: x^2+y^2 = 1$