How to resolve the algorithm Hilbert curve step by step in the Haskell programming language

Published on 7 June 2024 03:52 AM
#Haskell

How to resolve the algorithm Hilbert curve step by step in the Haskell programming language

Table of Contents

Problem Statement

Produce a graphical or ASCII-art representation of a Hilbert curve of at least order 3.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Hilbert curve step by step in the Haskell programming language

The given Haskell code generates the Hilbert curve, a space-filling curve that visits every point in a given square grid. It also provides a function to convert the curve into an SVG image for visualization.

Here's a breakdown of the code:

Hilbert Curve Generation:

  • hilbertTree function:
    • Generates a Hilbert curve as a binary tree.
    • The input n specifies the order of the curve, and the output is a tree of characters representing the curve.

Hilbert Points Extraction:

  • hilbertPoints function:
    • Extracts the (x, y) coordinates of the points comprising the Hilbert curve.
    • The input is the side length w of the square grid, and the Hilbert tree generated by hilbertTree.
    • The output is a list of coordinates.

Production Rule and Vectors:

  • rule function:
    • Defines the production rule for the Hilbert curve as a string.
  • vectors function:
    • Maps characters in the production rule to unit vectors representing the direction of the curve.

SVG Generation:

  • svgFromPoints function:
    • Generates an SVG image from a list of points.
    • The input is the width of the square grid and the list of points.
    • The output is an SVG string representing the image.

Main Function:

  • The main function:
    • Generates a Hilbert curve with an order of 6 and a side length of 1024.
    • Calls the hilbertPoints function to extract coordinates.
    • Converts the points into an SVG image using the svgFromPoints function.

Example Usage:

  • The code demonstrates the generation of a Hilbert curve of order 6 with a grid size of 1024 x 1024.
  • The resulting SVG image is stored in a string and can be displayed using a web browser or other means.

Overall, this code provides a clear and concise implementation of the Hilbert curve generation algorithm in Haskell. It also includes a utility function for visualizing the curve as an SVG image.

Source code in the haskell programming language

import Data.Tree (Tree (..))

---------------------- HILBERT CURVE ---------------------

hilbertTree :: Int -> Tree Char
hilbertTree n
  | 0 < n = iterate go seed !! pred n
  | otherwise = seed
  where
    seed = Node 'a' []
    go tree
      | null xs = Node c (flip Node [] <$> rule c)
      | otherwise = Node c (go <$> xs)
      where
        c = rootLabel tree
        xs = subForest tree


hilbertPoints :: Int -> Tree Char -> [(Int, Int)]
hilbertPoints w = go r (r, r)
  where
    r = quot w 2
    go r xy tree
      | null xs = centres
      | otherwise = concat $ zipWith (go d) centres xs
      where
        d = quot r 2
        f g x = g xy + (d * g x)
        centres =
          ((,) . f fst)
            <*> f snd <$> vectors (rootLabel tree)
        xs = subForest tree


--------------------- PRODUCTION RULE --------------------

rule :: Char -> String
rule c =
  case c of
    'a' -> "daab"
    'b' -> "cbba"
    'c' -> "bccd"
    'd' -> "addc"
    _ -> []

vectors :: Char -> [(Int, Int)]
vectors c =
  case c of
    'a' -> [(-1, 1), (-1, -1), (1, -1), (1, 1)]
    'b' -> [(1, -1), (-1, -1), (-1, 1), (1, 1)]
    'c' -> [(1, -1), (1, 1), (-1, 1), (-1, -1)]
    'd' -> [(-1, 1), (1, 1), (1, -1), (-1, -1)]
    _ -> []


--------------------------- TEST -------------------------

main :: IO ()
main = do
  let w = 1024
  putStrLn $ svgFromPoints w $ hilbertPoints w (hilbertTree 6)

svgFromPoints :: Int -> [(Int, Int)] -> String
svgFromPoints w xys =
  let sw = show w
      points =
        (unwords . fmap (((<>) . show . fst) <*> ((' ' :) . show . snd))) xys
   in unlines
        [ "<svg xmlns=\"http://www.w3.org/2000/svg\"",
          unwords
            ["width=\"512\" height=\"512\" viewBox=\"5 5", sw, sw, "\"> "],
          "<path d=\"M" ++ points ++ "\" ",
          "stroke-width=\"2\" stroke=\"red\" fill=\"transparent\"/>",
          "</svg>"
        ]

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