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diff --git a/static/src/_posts/2014-01-11-diamond-square.md b/static/src/_posts/2014-01-11-diamond-square.md new file mode 100644 index 0000000..528c953 --- /dev/null +++ b/static/src/_posts/2014-01-11-diamond-square.md @@ -0,0 +1,495 @@ +--- +title: Diamond Square +description: >- + Tackling the problem of semi-realistic looking terrain generation in + clojure. +updated: 2018-09-06 +tags: tech art +--- + +![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 |