From geometric numbers to function

There is one standard mathematical point that maintains badgering me : the reality that an actually straightforward 2D geometric number (like a circle) could not be a function.

I recognize what the definition of a function is. A circle is not a function (of one variable) due to the fact that it would certainly associate 2 values of the carbon monoxide - domain name to a solitary value of the domain name. Yet this does not aid my instinct.

It appears horribly unusual that an offered contour (claim a sinusoidal) is a function just unless you revolve it via $45^o$ or even more levels ...


Is there any kind of straightforward means (a principle comparable to that of just how most individuals visualize a function : a contour on a chart) to stand for 2D geometric numbers like a circle (or a revolved sinusoidal, or whatever)?

The just one I can consider is making use of a function of 2 or even more variables, yet this appears rather unclean to me : why should I make use of a function in 3 measurements simply to see its darkness on 2 measurements?
Besides if I consider the function as an actual object (in our actual, 3D room), I can not aid assuming that it is not a 2D circle, it is a 3D unusual object which can be viewed as a circle when revolved in a certain means (similar to the Penrose stairs appearance feasible when revolved in an unique means).

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2019-12-02 02:49:54
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Answers: 1

Some opportunities :

First of all, you could additionally intend to switch over to polar coordinates, in which factors are specified via angle $\varphi$ and also distance $r$ as opposed to $x$/ $y$ works with. As an example, in a polar coordinate system, a circle (allow is take the device circle) $K$ in fact is a function (currently of kind $r(\varphi)$ as opposed to $y(x)$).

As the distance is constant, we wind up with

$$K: r(\varphi) = 1$$

But the majority of numbers still aren't functions in polar works with, so we could need to take an extra basic strategy : A contour

A contour is a function that generates works with as opposed to a solitary value from some parameter, i.e. $\mathbb{R} \to \mathbb{R}^2$ in our instance.

Allow $K$ be our device circle once more - currently parameterized by some angle $\varphi$

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

At the very least, this need to benefit the majority of numbers. One of the most basic kind though is merely a formula that the works with need to please.

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

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2019-12-03 04:24:57
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