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diff --git a/_posts/2014-01-11-diamond-square.md b/_posts/2014-01-11-diamond-square.md deleted file mode 100644 index 665e07c..0000000 --- a/_posts/2014-01-11-diamond-square.md +++ /dev/null @@ -1,494 +0,0 @@ ---- -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 |