---
title: "On the computation of pi"
author: "Wenzhuo LIU"
date: "2021/03/10"
output: html_document
---


```{r setup, include=FALSE}
pi
```

## Some explanations

```{r}
pi
```

```{r}
set.seed(42)
N = 100000
x = runif(N)
theta = pi/2*runif(N)
2/(mean(x+sin(theta)>1))
```

```{r}
set.seed(42)
N = 1000
df = data.frame(X = runif(N), Y = runif(N))
df$Accept = (df$X**2 + df$Y**2 <=1)
library(ggplot2)
ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw()
```

```{r}
4*mean(df$Accept)
```