animation (version 2.0-5)

buffon.needle: Simulation of Buffon's Needle

Description

This function provides a simulation for the problem of Buffon's Needle, which is one of the oldest problems in the field of geometrical probability.

Usage

buffon.needle(l = 0.8, d = 1, redraw = TRUE, mat = matrix(c(1, 
    3, 2, 3), 2), heights = c(3, 2), col = c("lightgray", "red", 
    "gray", "red", "blue", "black", "red"), expand = 0.4, type = "l", 
    ...)

Arguments

l

numerical. length of the needle; shorter than d.

d

numerical. distances between lines; it should be longer than l.

redraw

logical. redraw former `needles' or not for each drop.

mat,heights

arguments passed to layout to set the layout of the three graphs.

col

a character vector of length 7 specifying the colors of: background of the area between parallel lines, the needles, the sin curve, points below / above the sin curve, estimated $\pi$ values, and the true $\pi$ value.

expand

a numerical value defining the expanding range of the y-axis when plotting the estimated $\pi$ values: the ylim will be (1 +/- expand) * pi.

type

an argument passed to plot when plotting the estimated $\pi$ values (default to be lines).

...

other arguments passed to plot when plotting the values of estimated $\pi$.

Value

• The values of estimated $\pi$ are returned as a numerical vector (of length nmax).

Details

This is quite an old problem in probability. For mathematical background, please refer to http://en.wikipedia.org/wiki/Buffon's_needle or http://www.mste.uiuc.edu/reese/buffon/buffon.html.

`Needles' are denoted by segments on the 2D plane, and dropped randomly to check whether they cross the parallel lines. Through many times of `dropping' needles, the approximate value of $\pi$ can be calculated out.

There are three graphs made in each step: the top-left one is a simulation of the scenario, the top-right one is to help us understand the connection between dropping needles and the mathematical method to estimate $\pi$, and the bottom one is the result for each drop.

References

Ramaley, J. F. (Oct 1969). Buffon's Noodle Problem. The American Mathematical Monthly 76 (8): 916-918.

http://animation.yihui.name/prob:buffon_s_needle

Examples

## it takes several seconds if 'redraw = TRUE'
oopt = ani.options(nmax = ifelse(interactive(), 500, 
    2), interval = 0.05)
par(mar = c(3, 2.5, 0.5, 0.2), pch = 20, mgp = c(1.5, 
    0.5, 0))
buffon.needle()

# this will be faster
buffon.needle(redraw = FALSE)

## create an HTML animation page
saveHTML({
    par(mar = c(3, 2.5, 1, 0.2), pch = 20, mgp = c(1.5, 0.5, 
        0))
    ani.options(nmax = ifelse(interactive(), 300, 10), interval = 0.1)
    buffon.needle(type = "S")
}, img.name = "buffon.needle", htmlfile = "buffon.needle.html", 
    ani.height = 500, ani.width = 600, title = "Simulation of Buffon's Needle", 
    description = c("There are three graphs made in each step: the", 
        "top-left, one is a simulation of the scenario, the top-right one", 
        "is to help us understand the connection between dropping needles", 
        "and the mathematical method to estimate pi, and the bottom one is", 
        "the result for each dropping."))

ani.options(oopt)