the batman equation

Posted by ygc on August 14, 2011 Leave a comment (7) Go to comments

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 $(\frac{x}{7})^2 + (\frac{y}{3})^2 = 1$ , in the region where |x|>3 and y>-3 $\sqrt{33}$ /7; the region were restricted by $\sqrt{\frac{||x|-3|}{(|x|-3)}}$ and $\sqrt{\frac{|y+3 \cdot \frac{\sqrt{33}}{7}|}{y+3 \cdot \frac{\sqrt{33}}{7}}}$

Here's what I got from the equation using ggplot2...

^?View Code RSPLUS

 
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">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)
 
#only work with ggplot2 <= 0.8.9
#p2 <- p1 + geom_point(aes(x=x2,y=y2), color="yellow") 
 
# in ggplot2 0.9.0, should be:
d2 <- data.frame">data.frame(x2=x2, y2=y2)
p2 <- p1 + geom_point(data=d2, 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")
d3 <- data.frame">data.frame(x3=x3, y3=y3)
p3 <- p2+geom_point(data=d3, 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")
d4 <- data.frame">data.frame(x4=x4,y4=y4)
p4 <- p3+geom_point(data=d4, 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))
d5 <- data.frame">data.frame(x5=x5,y5=y5)
p5 <- p4+geom_point(data=d5, 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")
d6 <- data.frame">data.frame(x6=x6,y6=y6)
p6 <- p5+geom_point(data=d6,aes(x=x6,y=y6), colour="blue")
 
p <- p6+theme_bw()
print(p)

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7 Comments.

Rudy Rooz August 14, 2011 at 11:22 pm
When I (mindlessly) run the above R script, I get the Console error output:
> print(p)
Error in data.frame(..., check.names = FALSE) :
arguments imply differing number of rows: 8001, 14002

and a blank plot canvass.

I was able to narrow this error down to p2:
> p2
Error in data.frame(..., check.names = FALSE) :
arguments imply differing number of rows: 8001, 14002

p1 plots the 2 base ellipse components (outer wings) correctly.

Reply
ygc Reply:
August 15th, 2011 at 12:47 am
you did not run the following command correctly.
```
p2 < - p1 + geom_point(aes(x=x2,y=y2), color="yellow")
```
Your error is due to the typo. You key in x or y instead of x2 or y2, and geom_point will parse it according to the dataset provided by p1.
```
> dim(d1)
[1] 14002     2
> length(x2)
[1] 8001
> length(y2)
[1] 8001
```
Reply
the batman equation | YGC | R | Scoop.it - pingback on August 15, 2011 at 7:08 am
happy valentine | YGC - pingback on February 15, 2012 at 3:13 pm
Berend Hasselman March 19, 2012 at 6:11 pm

I am trying to run the batman curve with R2.14.2 and ggplot2 0.9.0
I am now getting the same error message as Rudy Rooz reported.
I did not get that error message before.

What has changed with ggplot? Any idea how to repair this?

Reply
ygc Reply:
March 20th, 2012 at 11:33 am
corrected, now work with ggplot2 0.9.0.

Reply
Using Sweave | Sharp Statistics - pingback on October 29, 2012 at 12:28 am