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For example, if someone draws a circle onto a graph, this will fit an equation of $(x-a)^2+(y-b)^2=r^2$.

However, if someone were to draw another shape, made up of seemingly random points (such as a portrait), would there be an equation that would form the same image if drawn onto a graph?

Chris Godsil
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PhysicsGuy123
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    Depends on what you consider an equation. And of course this question needs a reference to https://math.stackexchange.com/questions/54506/is-this-batman-equation-for-real – Hagen von Eitzen Apr 14 '18 at 13:01
  • The quick answer is yes, but not for any very interesting reason: if $X$ is any subset whatsoever of the plane, you can define a function $\chi_X$ (called the characteristic function of $X$) such that $\chi_X(x) = 1$ if $x \in X$ and $\chi_X(x) = 0$ otherwise. So $\chi_X(x) = 1$ is an equation whose solution set is $X$. If you place some more requirements on the equation (e.g., if you require it to involve only continuous functions or polynomials), then the question becomes more interesting. – Rob Arthan Apr 14 '18 at 13:03
  • Look a Tuppers formula for some interesting results. – Karl Apr 14 '18 at 18:04

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There are many, many more graphs in the plane than there are formulas we can write down. In fact, since equations must be written in finitely many characters from a finite alphabet, the number of possible equations - or definite descriptions of any kind - we can write is a countable infinity. However, there are uncountably many functions with different graphs on the plane.

Even the most jagged of curves can be given a corresponding Piecewise function,for certain lengths of well known functions . One can also transform known functions to fit the given curve. You could also take some points and try to brute force a function which is true for all the points sampled

The Integrator
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