495 lines
17 KiB
Markdown
495 lines
17 KiB
Markdown
---
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title: Diamond Square
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description: >-
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Tackling the problem of semi-realistic looking terrain generation in
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clojure.
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updated: 2018-09-06
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---
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![terrain][terrain]
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I recently started looking into the diamond-square algorithm (you can find a
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great article on it [here][diamondsquare]). The following is a short-ish
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walkthrough of how I tackled the problem in clojure and the results. You can
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find the [leiningen][lein] repo [here][repo] and follow along within that, or
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simply read the code below to get an idea.
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Also, Marco ported my code into clojurescript, so you can get random terrain
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in your browser. [Check it out!][marco]
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```clojure
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(ns diamond-square.core)
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; == The Goal ==
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; Create a fractal terrain generator using clojure
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; == The Algorithm ==
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; Diamond-Square. We start with a grid of points, each with a height of 0.
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;
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; 1. Take each corner point of the square, average the heights, and assign that
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; to be the height of the midpoint of the square. Apply some random error to
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; the midpoint.
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;
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; 2. Creating a line from the midpoint to each corner we get four half-diamonds.
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; Average the heights of the points (with some random error) and assign the
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; heights to the midpoints of the diamonds.
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;
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; 3. We now have four square sections, start at 1 for each of them (with
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; decreasing amount of error for each iteration).
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;
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; This picture explains it better than I can:
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; https://blog.mediocregopher.com/img/diamond-square/dsalg.png
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; (http://nbickford.wordpress.com/2012/12/21/creating-fake-landscapes/dsalg/)
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;
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; == The Strategy ==
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; We begin with a vector of vectors of numbers, and iterate over it, filling in
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; spots as they become available. Our grid will have the top-left being (0,0),
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; y being pointing down and x going to the right. The outermost vector
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; indicating row number (y) and the inner vectors indicate the column number (x)
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;
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; = Utility =
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; First we create some utility functions for dealing with vectors of vectors.
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(defn print-m
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"Prints a grid in a nice way"
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[m]
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(doseq [n m]
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(println n)))
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(defn get-m
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"Gets a value at the given x,y coordinate of the grid, with [0,0] being in the
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top left"
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[m x y]
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((m y) x))
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(defn set-m
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"Sets a value at the given x,y coordinat of the grid, with [0,0] being in the
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top left"
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[m x y v]
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(assoc m y
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(assoc (m y) x v)))
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(defn add-m
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"Like set-m, but adds the given value to the current on instead of overwriting
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it"
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[m x y v]
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(set-m m x y
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(+ (get-m m x y) v)))
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(defn avg
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"Returns the truncated average of all the given arguments"
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[& l]
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(int (/ (reduce + l) (count l))))
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; = Grid size =
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; Since we're starting with a blank grid we need to find out what sizes the
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; grids can be. For convenience the size (height and width) should be odd, so we
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; easily get a midpoint. And on each iteration we'll be halfing the grid, so
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; whenever we do that the two resultrant grids should be odd and halfable as
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; well, and so on.
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;
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; The algorithm that fits this is size = 2^n + 1, where 1 <= n. For the rest of
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; this guide I'll be referring to n as the "degree" of the grid.
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(def exp2-pre-compute
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(vec (map #(int (Math/pow 2 %)) (range 31))))
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(defn exp2
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"Returns 2^n as an integer. Uses pre-computed values since we end up doing
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this so much"
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[n]
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(exp2-pre-compute n))
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(def grid-sizes
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(vec (map #(inc (exp2 %)) (range 1 31))))
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(defn grid-size [degree]
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(inc (exp2 degree)))
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; Available grid heights/widths are as follows:
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;[3 5 9 17 33 65 129 257 513 1025 2049 4097 8193 16385 32769 65537 131073
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;262145 524289 1048577 2097153 4194305 8388609 16777217 33554433 67108865
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;134217729 268435457 536870913 1073741825])
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(defn blank-grid
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"Generates a grid of the given degree, filled in with zeros"
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[degree]
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(let [gsize (grid-size degree)]
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(vec (repeat gsize
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(vec (repeat gsize 0))))))
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(comment
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(print-m (blank-grid 3))
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)
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; = Coordinate Pattern (The Tricky Part) =
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; We now have to figure out which coordinates need to be filled in on each pass.
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; A pass is defined as a square step followed by a diamond step. The next pass
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; will be the square/dimaond steps on all the smaller squares generated in the
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; pass. It works out that the number of passes required to fill in the grid is
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; the same as the degree of the grid, where the first pass is 1.
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;
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; So we can easily find patterns in the coordinates for a given degree/pass,
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; I've laid out below all the coordinates for each pass for a 3rd degree grid
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; (which is 9x9).
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; Degree 3 Pass 1 Square
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . . . 1 . . . .] (4,4)
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; Degree 3 Pass 1 Diamond
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; [. . . . 2 . . . .] (4,0)
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [2 . . . . . . . 2] (0,4) (8,4)
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . . . 2 . . . .] (4,8)
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; Degree 3 Pass 2 Square
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . 3 . . . 3 . .] (2,2) (6,2)
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; [. . 3 . . . 3 . .] (2,6) (6,6)
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; [. . . . . . . . .]
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; [. . . . . . . . .]
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; Degree 3 Pass 2 Diamond
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; [. . 4 . . . 4 . .] (2,0) (6,0)
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; [. . . . . . . . .]
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; [4 . . . 4 . . . 4] (0,2) (4,2) (8,2)
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; [. . . . . . . . .]
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; [. . 4 . . . 4 . .] (2,4) (6,4)
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; [. . . . . . . . .]
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; [4 . . . 4 . . . 4] (0,6) (4,6) (8,6)
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; [. . . . . . . . .]
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; [. . 4 . . . 4 . .] (2,8) (6,8)
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; Degree 3 Pass 3 Square
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; [. . . . . . . . .]
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; [. 5 . 5 . 5 . 5 .] (1,1) (3,1) (5,1) (7,1)
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; [. . . . . . . . .]
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; [. 5 . 5 . 5 . 5 .] (1,3) (3,3) (5,3) (7,3)
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; [. . . . . . . . .]
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; [. 5 . 5 . 5 . 5 .] (1,5) (3,5) (5,5) (7,5)
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; [. . . . . . . . .]
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; [. 5 . 5 . 5 . 5 .] (1,7) (3,7) (5,7) (7,7)
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; [. . . . . . . . .]
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; Degree 3 Pass 3 Square
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; [. 6 . 6 . 6 . 6 .] (1,0) (3,0) (5,0) (7,0)
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; [6 . 6 . 6 . 6 . 6] (0,1) (2,1) (4,1) (6,1) (8,1)
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; [. 6 . 6 . 6 . 6 .] (1,2) (3,2) (5,2) (7,2)
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; [6 . 6 . 6 . 6 . 6] (0,3) (2,3) (4,3) (6,3) (8,3)
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; [. 6 . 6 . 6 . 6 .] (1,4) (3,4) (5,4) (7,4)
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; [6 . 6 . 6 . 6 . 6] (0,5) (2,5) (4,5) (6,5) (8,5)
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; [. 6 . 6 . 6 . 6 .] (1,6) (3,6) (5,6) (7,6)
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; [6 . 6 . 6 . 6 . 6] (0,7) (2,7) (4,7) (6,7) (8,7)
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; [. 6 . 6 . 6 . 6 .] (1,8) (3,8) (5,8) (7,8)
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;
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; I make two different functions, one to give the coordinates for the square
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; portion of each pass and one for the diamond portion of each pass. To find the
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; actual patterns it was useful to first look only at the pattern in the
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; y-coordinates, and figure out how that translated into the pattern for the
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; x-coordinates.
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(defn grid-square-coords
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"Given a grid degree and pass number, returns all the coordinates which need
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to be computed for the square step of that pass"
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[degree pass]
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(let [gsize (grid-size degree)
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start (exp2 (- degree pass))
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interval (* 2 start)
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coords (map #(+ start (* interval %))
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(range (exp2 (dec pass))))]
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(mapcat (fn [y]
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(map #(vector % y) coords))
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coords)))
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;
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; (grid-square-coords 3 2)
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; => ([2 2] [6 2] [2 6] [6 6])
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(defn grid-diamond-coords
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"Given a grid degree and a pass number, returns all the coordinates which need
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to be computed for the diamond step of that pass"
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[degree pass]
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(let [gsize (grid-size degree)
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interval (exp2 (- degree pass))
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num-coords (grid-size pass)
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coords (map #(* interval %) (range 0 num-coords))]
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(mapcat (fn [y]
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(if (even? (/ y interval))
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(map #(vector % y) (take-nth 2 (drop 1 coords)))
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(map #(vector % y) (take-nth 2 coords))))
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coords)))
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; (grid-diamond-coords 3 2)
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; => ([2 0] [6 0] [0 2] [4 2] [8 2] [2 4] [6 4] [0 6] [4 6] [8 6] [2 8] [6 8])
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; = Height Generation =
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; We now work on functions which, given a coordinate, will return what value
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; coordinate will have.
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(defn avg-points
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"Given a grid and an arbitrary number of points (of the form [x y]) returns
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the average of all the given points that are on the map. Any points which are
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off the map are ignored"
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[m & coords]
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(let [grid-size (count m)]
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(apply avg
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(map #(apply get-m m %)
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(filter
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(fn [[x y]]
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(and (< -1 x) (> grid-size x)
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(< -1 y) (> grid-size y)))
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coords)))))
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(defn error
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"Returns a number between -e and e, inclusive"
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[e]
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(- (rand-int (inc (* 2 e))) e))
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; The next function is a little weird. It primarily takes in a point, then
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; figures out the distance from that point to the points we'll take the average
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; of. The locf (locator function) is used to return back the actual points to
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; use. For the square portion it'll be the points diagonal from the given one,
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; for the diamond portion it'll be the points to the top/bottom/left/right from
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; the given one.
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;
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; Once it has those points, it finds the average and applies the error. The
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; error function is nothing more than a number between -interval and +interval,
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; where interval is the distance between the given point and one of the averaged
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; points. It is important that the error decreases the more passes you do, which
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; is why the interval is used.
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;
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; The error function is what should be messed with primarily if you want to
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; change what kind of terrain you generate (a giant mountain instead of
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; hills/valleys, for example). The one we use is uniform for all intervals, so
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; it generates a uniform terrain.
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(defn- grid-fill-point
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[locf m degree pass x y]
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(let [interval (exp2 (- degree pass))
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leftx (- x interval)
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rightx (+ x interval)
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upy (- y interval)
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downy (+ y interval)
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v (apply avg-points m
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(locf x y leftx rightx upy downy))]
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(add-m m x y (+ v (error interval)))))
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(def grid-fill-point-square
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"Given a grid, the grid's degree, the current pass number, and a point on the
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grid, fills in that point with the average (plus some error) of the
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appropriate corner points, and returns the resultant grid"
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(partial grid-fill-point
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(fn [_ _ leftx rightx upy downy]
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[[leftx upy]
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[rightx upy]
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[leftx downy]
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[rightx downy]])))
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(def grid-fill-point-diamond
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"Given a grid, the grid's degree, the current pass number, and a point on the
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grid, fills in that point with the average (plus some error) of the
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appropriate edge points, and returns the resultant grid"
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(partial grid-fill-point
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(fn [x y leftx rightx upy downy]
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[[leftx y]
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[rightx y]
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[x upy]
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[x downy]])))
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; = Filling in the Grid =
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; We finally compose the functions we've been creating to fill in the entire
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; grid
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(defn- grid-fill-point-passes
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"Given a grid, a function to fill in coordinates, and a function to generate
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those coordinates, fills in all coordinates for a given pass, returning the
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resultant grid"
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[m fill-f coord-f degree pass]
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(reduce
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(fn [macc [x y]] (fill-f macc degree pass x y))
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m
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(coord-f degree pass)))
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(defn grid-pass
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"Given a grid and a pass number, does the square then the diamond portion of
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the pass"
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[m degree pass]
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(-> m
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(grid-fill-point-passes
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grid-fill-point-square grid-square-coords degree pass)
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(grid-fill-point-passes
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grid-fill-point-diamond grid-diamond-coords degree pass)))
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; The most important function in this guide, does all the work
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(defn terrain
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"Given a grid degree, generates a uniformly random terrain on a grid of that
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degree"
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([degree]
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(terrain (blank-grid degree) degree))
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([m degree]
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(reduce
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#(grid-pass %1 degree %2)
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m
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(range 1 (inc degree)))))
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(comment
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(print-m
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(terrain 5))
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)
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; == The Results ==
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; We now have a generated terrain, probably. We should check it. First we'll
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; create an ASCII representation. But to do that we'll need some utility
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; functions.
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(defn max-terrain-height
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"Returns the maximum height found in the given terrain grid"
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[m]
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(reduce max
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(map #(reduce max %) m)))
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(defn min-terrain-height
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"Returns the minimum height found in the given terrain grid"
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[m]
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(reduce min
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(map #(reduce min %) m)))
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(defn norm
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"Given x in the range (A,B), normalizes it into the range (0,new-height)"
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[A B new-height x]
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(int (/ (* (- x A) new-height) (- B A))))
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(defn normalize-terrain
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"Given a terrain map and a number of \"steps\", normalizes the terrain so all
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heights in it are in the range (0,steps)"
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[m steps]
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(let [max-height (max-terrain-height m)
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min-height (min-terrain-height m)
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norm-f (partial norm min-height max-height steps)]
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(vec (map #(vec (map norm-f %)) m))))
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; We now define which ASCII characters we want to use for which heights. The
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; vector starts with the character for the lowest height and ends with the
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; character for the heighest height.
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(def tiles
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[\~ \~ \" \" \x \x \X \$ \% \# \@])
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(defn tile-terrain
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"Given a terrain map, converts it into an ASCII tile map"
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[m]
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(vec (map #(vec (map tiles %))
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(normalize-terrain m (dec (count tiles))))))
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(comment
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(print-m
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(tile-terrain
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(terrain 5)))
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; [~ ~ " " x x x X % $ $ $ X X X X X X $ x x x X X X x x x x " " " ~]
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; [" ~ " " x x X X $ $ $ X X X X X X X X X X X X X X x x x x " " " "]
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; [" " " x x x X X % $ % $ % $ $ X X X X $ $ $ X X X X x x x x " " "]
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; [" " " x x X $ % % % % % $ % $ $ X X $ $ $ $ X X x x x x x x " " x]
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; [" x x x x X $ $ # % % % % % % $ X $ X X % $ % X X x x x x x x x x]
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; [x x x X $ $ $ % % % % % $ % $ $ $ % % $ $ $ $ X X x x x x x x x x]
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; [X X X $ % $ % % # % % $ $ % % % % $ % $ $ X $ X $ X X x x x X x x]
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; [$ $ X $ $ % $ % % % % $ $ $ % # % % % X X X $ $ $ X X X x x x x x]
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; [% X X % % $ % % % $ % $ % % % # @ % $ $ X $ X X $ X x X X x x x x]
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; [$ $ % % $ $ % % $ $ X $ $ % % % % $ $ X $ $ X X X X X X x x x x x]
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; [% % % X $ $ % $ $ X X $ $ $ $ % % $ $ X X X $ X X X x x X x x X X]
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; [$ $ $ X $ $ X $ X X X $ $ $ $ % $ $ $ $ $ X $ X x X X X X X x X X]
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; [$ $ $ $ X X $ X X X X X $ % % % % % $ X $ $ $ X x X X X $ X X $ $]
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; [X $ $ $ $ $ X X X X X X X % $ % $ $ $ X X X X X x x X X x X X $ $]
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; [$ $ X X $ X X x X $ $ X X $ % X X X X X X X X X x X X x x X X X X]
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; [$ $ X X X X X X X $ $ $ $ $ X $ X X X X X X X x x x x x x x X X X]
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; [% % % $ $ X $ X % X X X % $ $ X X X X X X x x x x x x x x x X X $]
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; [$ % % $ $ $ X X $ $ $ $ $ $ X X X X x X x x x x " x x x " x x x x]
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; [$ X % $ $ $ $ $ X X X X X $ $ X X X X X X x x " " " " " " " " x x]
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; [$ X $ $ % % $ X X X $ X X X x x X X x x x x x " " " " " ~ " " " "]
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; [$ $ X X % $ % X X X X X X X X x x X X X x x x " " " " " " ~ " " "]
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; [$ $ X $ % $ $ X X X X X X x x x x x x x x x " " " " " " " " " ~ ~]
|
|
; [$ $ $ $ $ X X $ X X X X X x x x x x x x x " " " " " " " ~ " " " ~]
|
|
; [$ % X X $ $ $ $ X X X X x x x x x x x x x x " " " " ~ " " ~ " " ~]
|
|
; [% $ $ X $ X $ X $ X $ X x x x x x x x x x x " " " " ~ ~ ~ " ~ " ~]
|
|
; [$ X X X X $ $ $ $ $ X x x x x x x x x x x " " " " ~ ~ ~ ~ ~ ~ ~ ~]
|
|
; [X x X X x X X X X X X X X x x x x x x x x x " " " ~ ~ " " ~ ~ ~ ~]
|
|
; [x x x x x x X x X X x X X X x x x x x x x " x " " " " " ~ ~ ~ ~ ~]
|
|
; [x x x x x x x x X X X X $ X X x X x x x x x x x x " ~ ~ ~ ~ ~ ~ ~]
|
|
; [" x x x x x X x X X X X X X X X X x x x x x x " " " " ~ ~ ~ ~ ~ ~]
|
|
; [" " " x x x X X X X $ $ $ X X X X X X x x x x x x x x " " ~ ~ ~ ~]
|
|
; [" " " " x x x X X X X X $ $ X X x X X x x x x x x x " " " " " ~ ~]
|
|
; [~ " " x x x x X $ X $ X $ $ X x X x x x x x x x x x x x x " " " ~]
|
|
)
|
|
|
|
; = Pictures! =
|
|
; ASCII is cool, but pictures are better. First we import some java libraries
|
|
; that we'll need, then define the colors for each level just like we did tiles
|
|
; for the ascii representation.
|
|
|
|
(import
|
|
'java.awt.image.BufferedImage
|
|
'javax.imageio.ImageIO
|
|
'java.io.File)
|
|
|
|
(def colors
|
|
[0x1437AD 0x04859D 0x007D1C 0x007D1C 0x24913C
|
|
0x00C12B 0x38E05D 0xA3A3A4 0x757575 0xFFFFFF])
|
|
|
|
; Finally we reduce over a BufferedImage instance to output every tile as a
|
|
; single pixel on it.
|
|
|
|
(defn img-terrain
|
|
"Given a terrain map and a file name, outputs a png representation of the
|
|
terrain map to that file"
|
|
[m file]
|
|
(let [img (BufferedImage. (count m) (count m) BufferedImage/TYPE_INT_RGB)]
|
|
(reduce
|
|
(fn [rown row]
|
|
(reduce
|
|
(fn [coln tile]
|
|
(.setRGB img coln rown (colors tile))
|
|
(inc coln))
|
|
0 row)
|
|
(inc rown))
|
|
0 (normalize-terrain m (dec (count colors))))
|
|
(ImageIO/write img "png" (File. file))))
|
|
|
|
(comment
|
|
(img-terrain
|
|
(terrain 10)
|
|
"resources/terrain.png")
|
|
|
|
; https://blog.mediocregopher.com/img/diamond-square/terrain.png
|
|
)
|
|
|
|
; == Conclusion ==
|
|
; There's still a lot of work to be done. The algorithm starts taking a
|
|
; non-trivial amount of time around the 10th degree, which is only a 1025x1025px
|
|
; image. I need to profile the code and find out where the bottlenecks are. It's
|
|
; possible re-organizing the code to use pmaps instead of reduces in some places
|
|
; could help.
|
|
```
|
|
|
|
[marco]: http://marcopolo.io/diamond-square/
|
|
[terrain]: /img/diamond-square/terrain.png
|
|
[diamondsquare]: http://www.gameprogrammer.com/fractal.html
|
|
[lein]: https://github.com/technomancy/leiningen
|
|
[repo]: https://github.com/mediocregopher/diamond-square
|