the batman equation

August 13, 2011
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(This article was first published on YGC » R, and kindly contributed to R-bloggers)

HardOCP has an image with an equation which apparently draws the Batman logo.

((\frac{x}{7})^2 \cdot \sqrt{\frac{||x|-3|}{(|x|-3)}}+ (\frac{y}{3})^2 \cdot \sqrt{\frac{|y+3 \cdot \frac{\sqrt{33}}{7}|}{y+3 \cdot \frac{\sqrt{33}}{7}}}-1) \cdot (|\frac{x}{2}|-((3 \cdot \frac{\sqrt{33}-7)}{112}) \cdot x^2-3+\sqrt{1-(||x|-2|-1)^2}-y) \cdot (9 \cdot \sqrt{\frac{|(|x|-1) \cdot (|x|-0.75)|}{((1-|x|)*(|x|-0.75))}}-8 \cdot |x|-y) \cdot (3 \cdot |x|+0.75 \cdot \sqrt{\frac{|(|x|-0.75) \cdot (|x|-0.5)|}{((0.75-|x|) \cdot (|x|-0.5))}}-y) \cdot (2.25 \cdot \sqrt{\frac{|(x-0.5) \cdot (x+0.5)|}{((0.5-x) \cdot (0.5+x))}}-y) \cdot (\frac{6 \cdot \sqrt{10}}{7}+(1.5-0.5 \cdot |x|) \cdot \sqrt{\frac{||x|-1|}{|x|-1}}-(\frac{6 \cdot \sqrt{10}}{14}) \cdot \sqrt{4-(|x|-1)^2}-y) =0

This function is very delightful for drawing such a graph, but write it down in latex is very tedious.

As a product of factors is 0 if and only if any one of them is 0, multiplying these six factors puts the curves together. This graph is no more than the combination of six curves.

All these six curves are very simple, for instance, the first factor is the ellipse (\frac{x}{7})^2 + (\frac{y}{3})^2  , in the region where |x|>3 and y>-3\sqrt{33} /7; the region were restricted by \sqrt{\frac{||x|-3|}{(|x|-3)}} and \sqrt{\frac{|y+3 \cdot \frac{\sqrt{33}}{7}|}{y+3 \cdot \frac{\sqrt{33}}{7}}}-1

Here’s what I got from the equation using ggplot2…

?View Code RSPLUS
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require(ggplot2)
 
f1 <- function(x) {
    y1 <- 3*sqrt(1-(x/7)^2)
    y2 <- -3*sqrt(1-(x/7)^2)
    y <- c(y1,y2)
    d <- data.frame(x=x,y=y)
    d <- d[d$y > -3*sqrt(33)/7,]
    return(d)
}
 
x1 <- c(seq(3, 7, 0.001), seq(-7, -3, 0.001))
d1 <- f1(x1)
p1 <- ggplot(d1,aes(x,y)) + geom_point(color="red")
 
x2 <- seq(-4,4, 0.001)
y2 <- abs(x2/2)-(3*sqrt(33)-7)*x2^2/112-3 + sqrt(1-(abs(abs(x2)-2)-1)^2)
 
p2 <- p1 + geom_point(aes(x=x2,y=y2), color="yellow")
 
 
 
x3 <- c(seq(0.75,1,0.001), seq(-1,-0.75,0.001))
y3 <- 9-8*abs(x3)
p3 <- p2+geom_point(aes(x=x3,y=y3), color="green")
 
x4 <- c(seq(0.5,0.75,0.001), seq(-0.75,-0.5,0.001))
y4 <- 3*abs(x4)+0.75
p4 <- p3+geom_point(aes(x=x4,y=y4), color="steelblue")
 
x5 <- seq(-0.5,0.5,0.001)
y5 <- rep(2.25,length(x5))
p5 <- p4+geom_point(aes(x=x5,y=y5))
 
x6 <- c(seq(-3,-1,0.001), seq(1,3,0.001))
y6 <- 6 * sqrt(10)/7 +
    (1.5 - 0.5 * abs(x6)) * sqrt(abs(abs(x6)-1)/(abs(x6)-1)) -
    6 * sqrt(10) * sqrt(4-(abs(x6)-1)^2)/14
p6 <- p5+geom_point(aes(x=x6,y=y6), colour="blue")
 
p <- p6+theme_bw()
print(p)

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