# DO you think this is real?

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

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## Answers and Replies

Mentallic
Homework Helper
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?

Mentallic
Homework Helper
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So it works?

statdad
Homework Helper
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
Homework Helper
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
Science Advisor
Gold Member
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.

uart
Science Advisor
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 explaination of the mysterious extra term. Thanks DrGreg. :)

Mentallic
Homework Helper
Yeah, nice explanation DrGreg! I also cannot comprehend why every term is being squared. What is the effect of this?