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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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In polar coordinates, the equation for a circle is a function. When the circle is centered at the origin, the function is constant! – Blue Nov 27 '10 at 21:13
Yes, I know, but in polar coordinates even a simple sinusoidal or an exponential are not functions. I'm looking for a common way to represent all the 2D geometrical figures (or at least those that a non-mathematician mind imagines simple figures). – peoro Nov 27 '10 at 21:19
What about parametric equations? There is only one independent variable. – Raskolnikov Nov 27 '10 at 21:42
@peoro: Star-convex figures (with respect to the origin) are functions in polar coordinates. – Dario Nov 27 '10 at 21:48
"It sounds terribly weird that a given curve (say a sinusoidal) is a function only unless you rotate it by 45 or more degrees..." You are confounding the concept of function with the concept of a graph of a function. – yasmar Nov 27 '10 at 21:52
up vote 2 down vote accepted

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.

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

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