inscribed circles and regular polygons

circles-and-polygons.R

# objective - plot circles and polygons such that
# 0. A circle
# 1. An equilateral triangle such that the edges are tangent to the circle
# 2. A circle such that the vertices of the triangle are on the circumference
# 3. A square with edges tangent to the circle drown in step 2.
# 4. A circle with the vertices of the square on the circumference
# 5. A regular pentagon....
#
# A vertex of the regular polygons will all be co-linear.  That is, with the
# initial circle centered at the Cartesian point (0, 0), then on vertex for each
# of the regular polygons will have an x-coordinate of 0.

library(ggplot2)
library(ggforce)  # needed for geom_circle

# Radius given length of side:
# r = s / (2 * sin(pi / n))
# where s is the length of the side of regular polygon
# n is the number of sides

# Apothem (inradius)
# r = a / cos(pi / n)
# a is the apothem
# n is the number of sides

# Get the vertices of a regular polygon centered at the origin given the apothem
# and at least one vertex with x = 0 (one vertex if n is odd, two if n is even)

vertices <- function(apothem = NULL, radius = NULL, n = 3, rotate = 0) {
  if (!xor(is.null(apothem), is.null(radius)) ) {
    stop("apothem xor radius needs to be defined")
  } 

  if (!is.null(apothem)) {
    radius <- apothem / cos(pi / n)
  } else {
    apothem <- radius * cos(pi / n)
  }

  pts <- matrix(c(0, radius), ncol = 1)
  theta <- seq(0, 2 * pi, length = n + 1)[-(n + 1)] + rotate
  rtn <- do.call(rbind, lapply(theta, function(th) {data.frame(x = - radius * sin (th), y = radius * cos(th))}))
  rtn$n <- n
  rtn$apothem <- apothem
  rtn$radius  <- radius
  rtn
}


vertices(radius = 2)

circles <- data.frame(x0 = 0, y0 = 0, r = 1)
polygons <- data.frame <- vertices(apothem = 1)

for(i in 2:30) {
  circles <- rbind(circles, data.frame(x0 = 0, y0 = 0, r = tail(polygons$radius, 1)))
  polygons <- rbind(polygons,
                    vertices(apothem = circles[i, "r"], n = tail(polygons$n, 1) + 1)
  )
}

# build up layers

p1 <-
lapply(
       1:nrow(circles),
       function(i) {
         eval(
         substitute(
                    list(
                         geom_polygon(data = subset(polygons, n == nn), mapping = aes(x = x, y = y, group = n), fill = "black"),
                         geom_circle(data = subset(circles, r == rr), mapping = aes(x0 = x0, y0 = y0, r = r), fill = "white", color = NA)
                    )
                    ,
                    list(nn = rev(unique(polygons$n))[i],
                         rr = rev(unique(circles$r))[i]))
         )
       })

g1 <-
  ggplot() +
  p1 +
  coord_fixed() +
  theme_no_axes() +
  theme(legend.position = "none",
        panel.background = element_rect(fill = "black"))

# Another way
circles <- data.frame(x0 = 0, y0 = 0, r = 1)
polygons <- vertices(radius = 1, n = 3, rotate = pi)

for(i in 2:10) {
  circles  <- rbind(circles,  data.frame(x0 = 0, y0 = 0, r = tail(polygons$apothem, 1)))
  polygons <- rbind(polygons, vertices(radius = circles[i, "r"], n = tail(polygons$n, 1) + 1, rotate = pi)
  )
}

p2 <-
lapply(
       1:nrow(circles),
       function(i) {
         eval(
              substitute(
                         list(
                              geom_circle(data = subset(circles, r == rr), mapping = aes(x0 = x0, y0 = y0, r = r), fill = "white", color = NA),
                              geom_polygon(data = subset(polygons, n == nn), mapping = aes(x = x, y = y, group = n), fill = "black")
                         )
                         ,
                         list(nn = unique(polygons$n)[i],
                              rr = unique(circles$r)[i]))
         )
       })

g2 <-
  ggplot() +
  p2 +
  coord_fixed() +
  theme_no_axes() +
  theme(legend.position = "none",
        panel.background = element_rect(fill = "black"))


g3 <-
  ggplot() +
    p1 + p2 +
  coord_fixed() +
  theme_no_axes() +
  theme(legend.position = "none",
        panel.background = element_rect(fill = "black"))


gridExtra::grid.arrange(g1, g2, g3, nrow  = 1)

ggsave(filename = "g1.pdf", g1, width = 8, height = 8)

# end of file

g1.pdf