From bcf9b230be6d74c71567fd0771b31d47d8dd39c7 Mon Sep 17 00:00:00 2001
From: Brian Picciano <mediocregopher@gmail.com>
Date: Thu, 21 Jan 2021 17:22:53 -0700
Subject: build the blog with nix

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