I wanted to post this note many months ago, but I guess I procastinated.

In this note, I would like to show how to do a similar graph like the above, to graph literally anything in a single function (As long as it is a function of x), and invite anybody to make a graph like this, depicting an object (be it a building, or an ant). Then, I will compile the graphs together such that graphing the single function depicts any object we have made.

In the image above, I have depicted the Empire State Building and the Singapore Parliament house in a single function.

\(\textbf{How to do it}\)

First, let me introduce this function: \[\frac{\left|\left(x-k\right)-\left|x-k\right|\right|}{2\left(k-x\right)}+\frac{\left|\left(x+a\right)-\left|x+a\right|\right|}{2\left(x+a\right)}\]

The above function, basically graphs \(y=0\) from \(x=-\infty \text{ to } x=-a\) and \(x=k \text{ to } x=\infty\), and \(y=1\) from \(x=-a \text{ to } x=k\), assuming \(a\) and \(k\) to be positive real numbers.

Now, if you can see where this is going, if I \(\times\) that above function by \(f(x)\), what would I get? I get \(y=0\) from \(x=-\infty \text{ to } x=-a\) and \(x=k \text{ to } x=\infty\), and \(y=f(x)\) from \(x=-a \text{ to } x=k\).

For the case of \(f(x) = x^{2}\), \(a=2\) and \(k=1\):

Graph of \(x^2\left(\frac{\left|\left(x-1\right)-\left|x-1\right|\right|}{2\left(1-x\right)}+\frac{\left|\left(x+2\right)-\left|x+2\right|\right|}{2\left(x+2\right)}\right)\)

You can try it yourself here

Now, what if we want to graph \(y=x^{2}\) from \(x=-2\) to \(x=1\) and then \(y=(x-2)^{3}\) from \(x=1\) to \(x=3\)? Easy, we just add \(2\) functions together:

Graph of \(x^2\left(\frac{\left|\left(x-1\right)-\left|x-1\right|\right|}{2\left(1-x\right)}+\frac{\left|\left(x+2\right)-\left|x+2\right|\right|}{2\left(x+2\right)}\right)+\left(x-2\right)^3\left(\frac{\left|\left(x-3\right)-\left|x-3\right|\right|}{2\left(3-x\right)}+\frac{\left|\left(x-1\right)-\left|x-1\right|\right|}{2\left(x-1\right)}\right)\)

And so we are done with the basis of how to graph any object you want, all you have to do is keep adding the functions together until you get the desired shape.

Now, what if your desired shape is referenced from a picture? No worries, Desmos enables you to add a picture for reference:

Sidenote: To achieve shading (For the picture above, the shading is red), just add an inequality. To know what I mean, try \(y>x\). The above picture took me about 45 minutes.

You might want to see this as an example

\(\textbf{Graph Submission details}\)

If you want to make one of these graphs, and want to submit yours too, just post the link of your graph into the comments section. However, for your graph to be accepted, it has to pass some requirements:

Requirements:

\(\bullet\) Your graph must be \(y=0\) from \(x=-\infty\) to \(x=k_{1}\) and from \(x=\infty\) to \(x=k_{2}\), where \(k_{1}\le x\le k_{2}\) is the section where your object exists. This is to allow me to add multiple objects to a single function's graph (Like how I added the Empire State Building and the Singapore Parliament House together into a single function.)

\(\bullet\) Your Graph should have a height of roughly between \(5\) to \(20\). This is to avoid graphs that are too small or too big to be added into the compilation.

Well that's about it, have fun making one of these graphs! If you have any questions, post them in the comments section.

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## Comments

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TopNewestIn response to Apple:\(\color{white}{\text{Excellent work!}}\)Log in to reply

In response to Lameness:\(\color{white}{\text{More Lameness}}\)Log in to reply

\(\color{white}{\text{I am not lame -_-}}\)

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On a slightly more practical level, NURBS, or "non-uniform rational B-spline", have been developed to provide functions of fairly arbitrary curves and surfaces for computer graphics and industrial uses. They are extremely powerful and amendable to mathematical methods, as compared to conventional "elementary functions" of mathematics. What you are proposing is something analogous, but for discretized shapes. It's a start, but needs more work to attain the same functionality as NURBS. See Bezier Curves for an easy-to-understand text on a subset of NURBS.

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Wow. I didn't know this kind of thing was practical

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