Is the Inverse Graphing Calculator Accurate and Effective?

[LucasGB]

Do you guys think this really works?
http://www.xamuel.com/inverse-graphing-calculator.php
I wasn't able to verify it with my graphing software.

 

[Mentallic]

It looks incredible! Really mind-boggling!
I always wondered if words and pictures could be made out of equations in cartesian form.

I tried the letter O which should have just been a circle radius 1 unit centre (3,3) by the looks of it, and this is what it gives:

$$\left((x-3)^2+(y-3)^2-1\right)^2+\left(y^2-6y+8+\sqrt{y^4-12y^3+52y^2-96y+64}\right)^2=0$$

Now, the first part is the correct equation for the circle, but what's this extra nonsense that's tacked on?

 

[statdad]

Notice this has been addressed here

https://www.physicsforums.com/showthread.php?t=384729

 

[Mentallic]

While that portion of the graph is zero for $-2\leq y\leq 2$ (with respect to redbelly's post https://www.physicsforums.com/showpost.php?p=2614013&postcount=2"), it doesn't explain why it is required for to show the equation of this circle. The same thing is added to all letters.

 

[LucasGB]

So it works?

 

[statdad]

That expression is added to the ``formula'' for words and sentences as well. Perhaps it's important to the work; perhaps it's a private joke by the programmer.

 

[Mentallic]

Yes I believe it does work. And once you start to realize the pattern (go for 1 simple letter at first, such as O and X and work your way from there), it's not as unbelievably impossible as you first might have thought - including myself.

 

[DrGreg]

If you look at the structure of the equation for several letters, you will see a pattern emerging, e.g. for 3 letters

$$z_1^2 \, z_2^2 \, z_3^2 \, + \, w^2 \, = \, 0$$​

where z₁ is a formula for the first letter, z₂ is a formula for the second letter, z₃ is a formula for the third letter, and w is always the same each time, and all of them functions of x and y.

Note that the equation is true if and only if w is zero and at least one of the other expressions is zero.

w is zero if $2 \leq y \leq 4$ and otherwise non-zero, so the effect of adding w² is to prevent any curves being drawn above y=4 or below y=2. In the case of a circle ("O") it doesn't matter, but for some of the other letters, the formula given creates the correct symbol between heights 2 and 4 but also creates extra lines outside that range, so adding w² suppresses the unwanted lines.

 

[Tac-Tics]

http://mathworld.wolfram.com/TuppersSelf-ReferentialFormula.html

 

[uart]

If you look at the structure of the equation for several letters, you will see a pattern emerging, e.g. for 3 letters

$$z_1^2 \, z_2^2 \, z_3^2 \, + \, w^2 \, = \, 0$$​

where z₁ is a formula for the first letter, z₂ is a formula for the second letter, z₃ is a formula for the third letter, and w is always the same each time, and all of them functions of x and y.

Note that the equation is true if and only if w is zero and at least one of the other expressions is zero.

w is zero if $2 \leq y \leq 4$ and otherwise non-zero, so the effect of adding w² is to prevent any curves being drawn above y=4 or below y=2. In the case of a circle ("O") it doesn't matter, but for some of the other letters, the formula given creates the correct symbol between heights 2 and 4 but also creates extra lines outside that range, so adding w² suppresses the unwanted lines.

Nice explanation of the mysterious extra term. Thanks DrGreg. :)

 

[Mentallic]

Yeah, nice explanation DrGreg!

I also cannot comprehend why every term is being squared. What is the effect of this?