Rose in Mathematics

Source: vignettes/mathematical_rose.Rmd

In this article, we will make a mathematical rose like this:

1. Implement the rose function

The mathematical function that draws the rose is: \[(x,y) = \left(a \cos(kt) \cos(t), a \cos(kt) \sin(t)\right)\] where \(a\) and \(k\) are the parameters controlling the size and the shape of the rose respectively.

rose <- function(theta, a = 1, k = 3) {
    # `a` controls the size, and `k` controls the shape of the rose
    list(x = a * cos(k * theta) * cos(theta), y = a * cos(k * theta) * sin(theta))
}

The parameter \(k\) is specified with \(k = p/q\), where \(p\) and \(q\) should be coprime, e.g. 1/3, 2/7, not 2/6, 3/6. If \(p\) and \(q\) are not coprime, then the rose will be drawn more than once.

p <- 2
q <- 7
k <- p / q

Some calculation is needed to work out the number of time steps required to get to the end of the drawing of the rose. Assuming \(p\) and \(q\) are coprime, then the period is \(\pi q\) if both \(p\) and \(q\) are odd, and \(2 \pi q\) otherwise.

is_odd <- function(x) x %% 2 == 1
period <- ifelse(is_odd(p) && is_odd(q), pi * q, 2 * pi * q)

Now we compute the points on the rose with the parameters a and k for time=0 to time=period.

time <- seq(from = 0, to = period, length.out = 400)  # 400 time steps
pts <- rose(time, a = 1, k = k)

2. Plot the Rose

Set up the device 

library(animate)
border_with_round_corners <- "border:1px solid lightgray; border-radius:5px;"
device <- animate$new(600, 600, virtual = TRUE, attr = list(style = border_with_round_corners))
attach(device)

Notes

  • line 1: library(animate) loads the animate library.

  • line 2: We define a style to add a border around the plot so that we know it is there even when nothing has been drawn. The style is specified using CSS; if you haven’t used it before, don’t worry too much about that, the package website maintains a list of the most used ones for you to copy and paste to your code.

  • line 3: animate$new(600, 600, ...) sets the canvas size to be 600 by 600 (width by height).

  • line 3: An additional flag virtual = TRUE is needed only for in-line RMD usage, i.e. you are animating the plot directly in the code chunk rather than importing it from a file.

  • line 4: attach(device) lets us use the functions in device directly rather than typing device$some_function every time. To detach from the device, use detach(device) afterwards.

Making the plot 

set_max_stacksize(0)
par(xlim = extendrange(pts$x), ylim = extendrange(pts$y))
for (step in seq_along(time)) {
    points(pts$x[step], pts$y[step], id = "point-1")
    lines(pts$x[1:step], pts$y[1:step], id = "line-1")
}
rmd_animate(device, click_to_loop(start = 3, wait = 10))

Notes

  • line 1: set_max_stacksize(0) sets the internal memory of the device to 0, so that it does not use any memory.
  • line 2: par(xlim = ..., ylim = ...) sets the ranges of the x-axis and the y-axis to be fixed at the specified values.
  • line 3-6: points(...) and lines(...) draw points and lines on the screen. The argument id needs to be provided so that the graphics elements are updated, rather than redrawn, on the screen.
  • line 7: this line is needed only for in-line RMD usage. It executes the animation using the virtual device and embeds the output in the RMD file.

Clean up

If you are making the plot in the R console (not in RMD), you may want to clear the plot, switch off and detach the device when you are done.

clear()
off()
detach(device)

3. [Advanced] Coordinating multiple plots

Continuing from above, we will draw a 4 x 4 grid of roses. First, we generate the roses using the previous code but looping through a range of \(p\) and \(q\) values.

ps <- 1:4
qs <- c(5, 7, 11, 13)
roses <- list()
for (p in ps) {
  for (q in qs) {
    period <- ifelse(is_odd(p) && is_odd(q), pi * q, 2 * pi * q)
    time <- seq(from = 0, to = period, length.out = 400)
    roses[[length(roses) + 1]] <- rose(time, a = 1, k = p / q)
  }
}

Next, we create 15 additional plot areas using the svg function. Then when we want to make a plot at a specific area, we simply call set(PLOT_AREA_ID) just before the drawing commands. As the generated plot is quite large in size, we will do this directly in the R console.

library(animate)
device <- animate$new(150, 150)
attach(device)
set_max_stacksize(0)


# Create 15 additional plot areas. The id of the first plot area is by default "SVG_1";
# we follow the convention and name the rest as "SVG_i", where i = 2, ..., 16.
for (i in 2:16) {
  svg(150, 150, id = paste0("SVG_", i))
}


# Loop over the plot area and draw the corresponding rose.
for (step in 1:400) {
  for (i in 1:16) {
    set(paste0("SVG_", i))  # Make plot area named "SVG_i" active
    pts <- roses[[i]]
    par(xlim = extendrange(pts$x), ylim = extendrange(pts$y))
    points(pts$x[step], pts$y[step], id = "point-1")
    lines(pts$x[1:step], pts$y[1:step], id = "line-1")
  }
}