Understanding the grid structure • hexmap

library(hexmap)
library(sf)
library(ggplot2)

Consider a map with boundaries \((x_1, y_1, x_2, y_2)\) such that

\(x_1\) and \(x_2\) denote the minimum and maximum longitude of the map, respectively,
\(y_1\) and \(y_2\) denote the minimum and maximum latitude of the map, respectively,
\(A = (y_2 - y_1)(x_2 - x_1)\) is the total area of the map,
\(L\) is the total land area of the map,
\(n_{xe}\) and \(n_{xo}\) is the total number of hexagons on even and odd rows respectively,
\(n_{ye}\) and \(n_{yo}\) is the total number of hexagons on even and odd columns respectively,
\(n_r\) and \(n_c\) are the maximum number of rows and columns in the grid structure,
\(n\) is the desired total number of hexagons,
\(c\) is the cell size, which is the same as the width of the non-flat topped hexagon.

We assume that the hexagon is not flat topped. Note that \(A\), \(L\), \(n\), \(x_1\), \(x_2\), \(y_1\) and \(y_2\) are known.

Below is an example where the “map” is a unit square. In this case \(A = L\).

square <- matrix(c(0, 0, 
                   0, 1, 
                   1, 1, 
                   1, 0, 
                   0, 0), 
                 ncol = 2, 
                 byrow = TRUE) %>% 
  list() %>% 
  st_polygon()

plot_hex_tile <- function(cellsize = NULL, n = NULL) {
  title <- ifelse(is.null(cellsize),
                  paste0("n = ", n),
                  paste0("Cell size = ", cellsize))
  if(is.null(cellsize)) {
    title <- paste0("n = ", n)
    grid <- st_make_grid(square, n = n, square = FALSE)
  } else {
    title <- paste0("Cell size = ", cellsize)
    grid <- st_make_grid(square, 
                         cellsize = cellsize,
                         square = FALSE)
  }
  
  ggplot(grid) +
    geom_sf(fill = "transparent") + 
    geom_sf(data = square, fill = "transparent", color = "red") +
    ggtitle(title)
}

plot_hex_tile(0.25)

plot_hex_tile(0.3)

plot_hex_tile(0.5)

plot_hex_tile(1)

plot_hex_tile(n = 3)

plot_hex_tile(n = 5)

Using basic trigonometry for a regular hexagon of height \(h\), width \(w\) and length \(b\) as shown in the diagram below, you can find that \(h = 2 w / \sqrt{3}\) and \(b = w / \sqrt{3}\) .

Putting it all together, you can find that the maximum and minimum number of hexagons in a row is \(\lfloor x_2 - x_1 \rfloor / c + 2\) (even rows) and \(\lfloor x_2 - x_1 \rfloor / c + 1\) (odd rows), respectively. The maximum and minimum number of hexagons in a column (which we define as the number of hexagon where its center passes through a vertical line) is \(\lfloor (y_2 - y_1 + h / 2 - b) / (h + b) \rfloor + 1\) and \(\lfloor (y_2 - y_1 + h / 2 - b) / (h + b) \rfloor\) (need to double check this).