The heart has its reasons which reason knows not (Blaise Pascal)

You only need two functions to draw a heart mathematically. The upper part is generated by `(1-(|x|-1)`

and the lower one by ^{2})^{1/2}`acos(1-|x|)-PI`

. Here is how this heart is:

Whats the area of this heart? It’s easy: integrating `heart.up(x)-heart.dw(x)`

between -2 and 2 and you will obtain that heart measures 9.424778, but there is a simple and nice way to approximate to this value: shoot the heart.

The idea is very simple. Heart is delimited by a square with vertex in `(-2, heart.dw(0))`

, `(-2, 1)`

, `(2, heart.dw(0))`

and `(2, 1)`

. Generating a set of points uniformly distributed inside the square and counting how many of them fall into the heart in relation to the area of the square gives a very good approximation of the exact area of the heart. This is a plot representing a simulation of 2.000 shots (hits in red, fails in blue):

Given a simulation of n points, the estimated area of the heart is the *area of the square* by *percentage of points that falls inside the heart*. And of course, precision increases with the number of shots you make, as you can see in the following plot, where exact area is represented by the red horizontal line:

Here you have the code:

library("ggplot2") heart.up <- function(x) {sqrt(1-(abs(x)-1)^2)} #Upper part of the heart heart.dw <- function(x) {acos(1-abs(x))-pi} #Lower part of the heart #Plot of the heart ggplot(data.frame(x=c(-2,2)), aes(x)) + stat_function(fun=heart.up, geom="line", aes(colour="heart.up")) + stat_function(fun=heart.dw, geom="line", aes(colour="heart.dw")) + scale_colour_manual("Function", values=c("blue","red"), breaks=c("heart.up","heart.dw"))+ labs(x = "", y = "")+ theme(legend.position = c(.85, .15)) sims <- 2000 #Number of simulations rlts <- data.frame() for (i in 1:sims) { msm <- cbind(as.matrix(runif(i, min=-2, max=2)), as.matrix(runif(i, min=heart.dw(0), max=1))) nin <- 0 for (j in 1:nrow(msm)) {if (msm[j,2]<=heart.up(msm[j,1]) & msm[j,2]>=heart.dw(msm[j,1])) nin=nin+1} rlts <- rbind(c(i, 4*(1-heart.dw(0))*nin/i), rlts) } colnames(rlts) <- c("no.simulations","heart.area") exact.area <- integrate(function(x) {heart.up(x)-heart.dw(x)},-2,2)$value mean.area <- mean(rlts$heart.area) #Mean of All Estimated Areas ggplot(data = rlts, aes(x = no.simulations, y = heart.area))+ geom_point(size = 0.5, colour = "black", alpha=0.4)+ geom_abline(intercept = exact.area, slope = 0, size = 1, linetype=1, colour = "red", aes(color="My Line"), alpha=0.8, show_guide = TRUE)+ labs(list(x = "Number of Shots", y = "Estimated Area"))+ ggtitle("Shot The Heart With Monte Carlo") + theme(plot.title = element_text(size=20, face="bold"))+ scale_x_continuous(limits = c(0, sims), expand = c(0, 0))+ expand_limits(x = 0, y = 0)+ scale_y_continuous(limits = c(0, 2*exact.area), expand = c(0, 0), breaks=c(0, exact.area/4, exact.area/2, 3*exact.area/4, exact.area, 5*exact.area/4, 3*exact.area/2, 7*exact.area/4, 2*exact.area))+ geom_text(x = 1000, y = exact.area/2, label=paste("Exact Area =", sprintf("%7.6f", exact.area)), vjust=-1, colour="red", size=5)+ geom_text(x = 1000, y = exact.area/2, label=paste("Mean of All Estimated Areas=", sprintf("%7.6f", mean.area)), vjust=+1, colour="red", size=5)