Batman Equation
Published on 11.05.2025

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I came across the Batman Equation a few years ago, and but it's from the old times - Michael Keaton's Batman from 1989. About time it got an upgrade. 

I thought of doing the new Batman 2022, or the Afleck Batman logo, but they didn't look iconic in my opinion. They looked great on a suit, but maybe not on a poster. Or maybe it's just my brain. But I personally loved the Batman Trilogy logo. This logo as batarangs is just mad lit on god no cap.

Creating any art with just math is kinda fun. All you do is fireup Desmos, and start hacking!

The equation is not a continuous function. It's made up of multiple parts - piecewise-defined expressions.

Warning: You'll not find crazy big brain equations, they are all messy. Honestly they are just patiently tinkered equations. I'm not a math guy, so I prolly don't know what I'm doing apart from having some fun. And more importantly the logo isn't pixel perfect, it's does the job but not the best one could craft.

Here are the equations:

Head Curve:  
\[
y = \frac{1}{3} + \sqrt{1 - \frac{x^2}{4}} \quad \{-0.5 \leq x \leq 0.5\}
\]

Top Wings (Left):  
\[
y = 1.9 \quad \{-13.8 \leq x \leq -4.5\}
\]

Top Wings (Right):  
\[
y = 1.9 \quad \{4.5 \leq x \leq 13.8\}
\]

Left Ear Slope:  
\[
y = -1.55x + 0.525 \quad \{-0.8 \leq x \leq -0.5\}
\]

Right Ear Slope:  
\[
y = 1.55x + 0.525 \quad \{0.5 \leq x \leq 0.8\}
\]

Wing Dips:  
\[
y = 0.2 + x^{-2} \quad \{-1.117 < x < -0.8,\ 0.8 < x < 1.117\}
\]

Right Wing Connect:  
\[
y = 1 + \exp\left(0.69 \cdot (3x - 13.65)\right) \quad \{1.1167 < x < 4.5\}
\]

Left Wing Connect:  
\[
y = 1 + \exp\left(0.69 \cdot (-3x - 13.65)\right) \quad \{-4.5 < x < -1.1167\}
\]

Bottom Curve:  
\[
y = \left|\frac{x}{1.4}\right| - \left(\frac{\sqrt{0.85} - 0.85}{0.5}\right)x^{1.6} - 2.1 \quad \{-10 \leq x \leq 10\}
\]

Right Wing:  
\[
y = -0.69 + 2.6 \sqrt{1 - \left(\frac{x - 14.2}{4.2}\right)^2} \quad \{10 \leq x \leq 13.8\}
\]

Left Wing:  
\[
y = -0.69 + 2.6 \sqrt{1 - \left(\frac{x + 14.2}{4.2}\right)^2} \quad \{-13.8 \leq x \leq -10\}
\]

I can try to explain these equations individually, but I think it's better if you try it out yourself. Here's the Playground link.

Tinker and Learn ftw! See ya tomorrow, bat bat!



Got questions? Hit me up on twitter - @pwnfunction.