There's one basilar math 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 codomain to a single value of the domain. But this doesn't help my intuitivity.
It sounds terribly weird that a given curve (say a sinusoidal) is a function only unless you rotate it by 45 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).