1639 lines
50 KiB
Markdown
1639 lines
50 KiB
Markdown
-----
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isHidden: false
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menupriority: 1
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kind: article
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created_at: 2012-04-30T19:17:53+02:00
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title: Un programme Haskell fonctionnel
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subtitle: Une extension de l'ensemble de Mandelbrot en 3D et en OpenGL
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author_name: Yann Esposito
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author_uri: yannesposito.com
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tags:
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- Haskell
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- programming
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- functional
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- tutorial
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-----
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blogimage("BenoitBMandelbrot.jpg","The B in Benoît B. Mandelbrot stand for Benoît B. Mandelbrot")
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begindiv(intro)
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%tlal Un exemple progressif d'utilisation d'Haskell.
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> <center><hr style="width:30%;float:left;border-color:#CCCCD0;margin-top:1em"/><span class="sc"><b>Table of Content</b></span><hr style="width:30%;float:right;border-color:#CCCCD0;margin-top:1em"/></center>
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>
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> * This will be replaced by the ToC
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> {:toc}
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>
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enddiv
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## Introduction
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I wanted to go further than my
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[preceding article](/Scratch/en/blog/Haskell-the-Hard-Way/) in which I introduced Haskell.
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Instead of arguing that Haskell is better, because it is functional and "Functional Programming! Yeah!", I'll give an example of what benefit
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functional programming can provide.
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This article is more about functional paradigm than functional language.
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The code organization can be used in most imperative language.
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As Haskell is designed for functional paradigm, it is easier to talk about functional paradigm using it.
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In reality, in the firsts sections I use an imperative paradigm.
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As you can use functional paradigm in imperative language,
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you can also use imperative paradigm in functional languages.
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This article is about creating a useful program.
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It can interact with the user in real time.
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It uses OpenGL, a library with imperative programming foundations.
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But the final code will be quite clean.
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Most of the code will remain in the pure part (no `IO`).
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I believe the main audience for this article are:
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- Haskell programmer looking for an OpengGL tutorial.
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- People interested in program organization (programming language agnostic).
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- Fractal lovers and in particular 3D fractal.
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- Game programmers (any language)
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I wanted to talk about something cool.
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For example I always wanted to make a Mandelbrot set explorer.
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I had written a [command line Mandelbrot set generator in Haskell](http://github.com/yogsototh/mandelbrot.git).
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The cool part of this utility is that it use all the cores to make the computation (it uses the `repa` package)[^001].
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[^001]: Unfortunately, I couldn't make this program to work on my Mac. More precisely, I couldn't make the [DevIL](http://openil.sourceforge.net/) library work on Mac to output the image. Yes I have done a `brew install libdevil`. But even a minimal program who simply write some `jpg` didn't worked.
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This time, we will display the Mandelbrot set extended in 3D using OpenGL and Haskell.
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You will be able to move it using your keyboard.
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This object is a Mandelbrot set in the plan (z=0),
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and something nice to see in 3D.
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Here is what you'll end with:
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blogfigure("GoldenMandelbulb.png","The entire Mandelbulb")
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blogfigure("3DMandelbulbDetail.png","A Mandelbulb detail")
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blogfigure("3DMandelbulbDetail2.png","Another detail of the Mandelbulb")
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And here are the intermediate steps:
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blogimage("HGL_Plan.png","The parts of the article")
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From the 2nd section to the 4th it will be _dirtier_ and _dirtier_.
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We start cleaning everything at the 5th section.
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<hr/><a href="code/01_Introduction/hglmandel.lhs" class="cut">Download the source code of this section → 01_Introduction/<strong>hglmandel.lhs</strong></a>
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## First version
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We can consider two parts.
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The first being mostly some boilerplate[^011].
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And the second part more focused on OpenGL and content.
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[^011]: Generally in Haskell you need to declare a lot of import lines.
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This is something I find annoying.
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In particular, it should be possible to create a special file, Import.hs
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which make all the necessary import for you, as you generally need them all.
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I understand why this is cleaner to force the programmer not to do so,
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but, each time I do a copy/paste, I feel something is wrong.
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I believe this concern can be generalized to the lack of namespace in Haskell.
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### Let's play the song of our people
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<div class="codehighlight">
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<code class="haskell">
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import Graphics.Rendering.OpenGL
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import Graphics.UI.GLUT
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import Data.IORef
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</code>
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</div>
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For efficiency reason, I won't use the default Haskell `Complex` data type.
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<div class="codehighlight">
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<code class="haskell">
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newtype Complex = C (Float,Float) deriving (Show,Eq)
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</code>
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</div>
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<div class="codehighlight">
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<code class="haskell">
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instance Num Complex where
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fromInteger n = C (fromIntegral n,0.0)
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C (x,y) * C (z,t) = C (z*x - y*t, y*z + x*t)
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C (x,y) + C (z,t) = C (x+z, y+t)
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abs (C (x,y)) = C (sqrt (x*x + y*y),0.0)
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signum (C (x,y)) = C (signum x , 0.0)
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</code>
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</div>
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We declare some useful functions for manipulating complex numbers:
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<div class="codehighlight">
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<code class="haskell">
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complex :: Float -> Float -> Complex
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complex x y = C (x,y)
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real :: Complex -> Float
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real (C (x,y)) = x
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im :: Complex -> Float
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im (C (x,y)) = y
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magnitude :: Complex -> Float
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magnitude = real.abs
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</code>
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</div>
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### Let us start
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We start by giving the main architecture of our program:
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<div class="codehighlight">
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<code class="haskell">
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main :: IO ()
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main = do
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-- GLUT need to be initialized
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(progname,_) <- getArgsAndInitialize
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-- We will use the double buffered mode (GL constraint)
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initialDisplayMode $= [DoubleBuffered]
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-- We create a window with some title
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createWindow "Mandelbrot Set with Haskell and OpenGL"
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-- Each time we will need to update the display
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-- we will call the function 'display'
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displayCallback $= display
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-- We enter the main loop
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mainLoop
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</code>
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</div>
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Mainly, we initialize our OpenGL application.
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We declared that the function `display` will be used to render the graphics:
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<div class="codehighlight">
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<code class="haskell">
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display = do
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clear [ColorBuffer] -- make the window black
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loadIdentity -- reset any transformation
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preservingMatrix drawMandelbrot
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swapBuffers -- refresh screen
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</code>
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</div>
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Also here, there is only one interesting line;
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the draw will occur in the function `drawMandelbrot`.
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This function will provide a list of draw actions.
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Remember that OpenGL is imperative by design.
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Then, one of the consequence is you must write the actions in the right order.
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No easy parallel drawing here.
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Here is the function which will render something on the screen:
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<div class="codehighlight">
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<code class="haskell">
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drawMandelbrot =
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-- We will print Points (not triangles for example)
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renderPrimitive Points $ do
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mapM_ drawColoredPoint allPoints
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where
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drawColoredPoint (x,y,c) = do
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color c -- set the current color to c
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-- then draw the point at position (x,y,0)
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-- remember we're in 3D
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vertex $ Vertex3 x y 0
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</code>
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</div>
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The `mapM_` function is mainly the same as map but inside a monadic context.
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More precisely, this can be transformed as a list of actions where the order is important:
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~~~
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drawMandelbrot =
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renderPrimitive Points $ do
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color color1
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vertex $ Vertex3 x1 y1 0
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...
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color colorN
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vertex $ Vertex3 xN yN 0
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~~~
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We also need some kind of global variables.
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In fact, global variable are a proof of a design problem.
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We will get rid of them later.
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<div class="codehighlight">
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<code class="haskell">
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width = 320 :: GLfloat
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height = 320 :: GLfloat
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</code>
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</div>
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And of course our list of colored points.
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In OpenGL the default coordinate are from -1 to 1.
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<div class="codehighlight">
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<code class="haskell">
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allPoints :: [(GLfloat,GLfloat,Color3 GLfloat)]
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allPoints = [ (x/width,y/height,colorFromValue $ mandel x y) |
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x <- [-width..width],
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y <- [-height..height]]
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</code>
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</div>
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We need a function which transform an integer value to some color:
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<div class="codehighlight">
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<code class="haskell">
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colorFromValue n =
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let
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t :: Int -> GLfloat
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t i = 0.5 + 0.5*cos( fromIntegral i / 10 )
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in
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Color3 (t n) (t (n+5)) (t (n+10))
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</code>
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</div>
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And now the `mandel` function.
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Given two coordinates in pixels, it returns some integer value:
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<div class="codehighlight">
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<code class="haskell">
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mandel x y =
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let r = 2.0 * x / width
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i = 2.0 * y / height
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in
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f (complex r i) 0 64
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</code>
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</div>
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It uses the main Mandelbrot function for each complex \\(c\\).
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The Mandelbrot set is the set of complex number \\(c\\) such that the following sequence does not escape to infinity.
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Let us define \\(f_c: \mathbb{C} \to \mathbb{C}\\)
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$$ f_c(z) = z^2 + c $$
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The sequence is:
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$$ 0 \rightarrow f_c(0) \rightarrow f_c(f_c(0)) \rightarrow \cdots \rightarrow f^n_c(0) \rightarrow \cdots $$
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Of course, instead of trying to test the real limit, we just make a test after a finite number of occurrences.
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<div class="codehighlight">
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<code class="haskell">
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f :: Complex -> Complex -> Int -> Int
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f c z 0 = 0
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f c z n = if (magnitude z > 2 )
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then n
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else f c ((z*z)+c) (n-1)
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</code>
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</div>
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Well, if you download this file (look at the bottom of this section), compile it and run it this is the result:
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blogimage("hglmandel_v01.png","The mandelbrot set version 1")
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A first very interesting property of this program is that the computation for all the points is done only once.
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It is a bit long before the first image appears, but if you resize the window, it updates instantaneously.
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This property is a direct consequence of purity.
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If you look closely, you see that `allPoints` is a pure list.
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Therefore, calling `allPoints` will always render the same result and Haskell is clever enough to use this property.
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While Haskell doesn't garbage collect `allPoints` the result is reused for free.
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We didn't specified this value should be saved for later use.
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It is saved for us.
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See what occurs if we make the window bigger:
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blogimage("hglmandel_v01_too_wide.png","The mandelbrot too wide, black lines and columns")
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We see some black lines because we drawn less point than there is on the surface.
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We can repair this by drawing little squares instead of just points.
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But, instead we will do something a bit different and unusual.
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<a href="code/01_Introduction/hglmandel.lhs" class="cut">Download the source code of this section → 01_Introduction/<strong>hglmandel.lhs</strong> </a>
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<hr/><a href="code/02_Edges/HGLMandelEdge.lhs" class="cut">Download the source code of this section → 02_Edges/<strong>HGLMandelEdge.lhs</strong></a>
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## Only the edges
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<div style="display:none">
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<div class="codehighlight">
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<code class="haskell">
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import Graphics.Rendering.OpenGL
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import Graphics.UI.GLUT
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import Data.IORef
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newtype Complex = C (Float,Float) deriving (Show,Eq)
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instance Num Complex where
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fromInteger n = C (fromIntegral n,0.0)
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C (x,y) * C (z,t) = C (z*x - y*t, y*z + x*t)
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C (x,y) + C (z,t) = C (x+z, y+t)
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abs (C (x,y)) = C (sqrt (x*x + y*y),0.0)
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signum (C (x,y)) = C (signum x , 0.0)
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complex :: Float -> Float -> Complex
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complex x y = C (x,y)
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real :: Complex -> Float
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real (C (x,y)) = x
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im :: Complex -> Float
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im (C (x,y)) = y
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magnitude :: Complex -> Float
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magnitude = real.abs
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main :: IO ()
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main = do
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||
-- GLUT need to be initialized
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(progname,_) <- getArgsAndInitialize
|
||
-- We will use the double buffered mode (GL constraint)
|
||
initialDisplayMode $= [DoubleBuffered]
|
||
-- We create a window with some title
|
||
createWindow "Mandelbrot Set with Haskell and OpenGL"
|
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-- Each time we will need to update the display
|
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-- we will call the function 'display'
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displayCallback $= display
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-- We enter the main loop
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mainLoop
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display = do
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-- set the background color (dark solarized theme)
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clearColor $= Color4 0 0.1686 0.2117 1
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clear [ColorBuffer] -- make the window black
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loadIdentity -- reset any transformation
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preservingMatrix drawMandelbrot
|
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swapBuffers -- refresh screen
|
||
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width = 320 :: GLfloat
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height = 320 :: GLfloat
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</code>
|
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</div>
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|
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</div>
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This time, instead of drawing all points, I'll simply want to draw the edges of the Mandelbrot set.
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The method I use is a rough approximation.
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I consider the Mandelbrot set to be almost convex.
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The result will be good enough.
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We change slightly the `drawMandelbrot` function.
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We replace the `Points` by `LineLoop`
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<div class="codehighlight">
|
||
<code class="haskell">
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drawMandelbrot =
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-- We will print Points (not triangles for example)
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renderPrimitive LineLoop $ do
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mapM_ drawColoredPoint allPoints
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where
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drawColoredPoint (x,y,c) = do
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||
color c -- set the current color to c
|
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-- then draw the point at position (x,y,0)
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||
-- remember we're in 3D
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vertex $ Vertex3 x y 0
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</code>
|
||
</div>
|
||
|
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And now, we should change our list of points.
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Instead of drawing every point of the visible surface,
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we will choose only point on the surface.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
allPoints = positivePoints ++
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map (\(x,y,c) -> (x,-y,c)) (reverse positivePoints)
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</code>
|
||
</div>
|
||
|
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We only need to compute the positive point.
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The Mandelbrot set is symmetric relatively to the abscisse axis.
|
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|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
positivePoints :: [(GLfloat,GLfloat,Color3 GLfloat)]
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positivePoints = do
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x <- [-width..width]
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let y = maxZeroIndex (mandel x) 0 height (log2 height)
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if y < 1 -- We don't draw point in the absciss
|
||
then []
|
||
else return (x/width,y/height,colorFromValue $ mandel x y)
|
||
where
|
||
log2 n = floor ((log n) / log 2)
|
||
</code>
|
||
</div>
|
||
|
||
This function is interesting.
|
||
For those not used to the list monad here is a natural language version of this function:
|
||
|
||
<code class="no-highlight">
|
||
positivePoints =
|
||
for all x in the range [-width..width]
|
||
let y be smallest number s.t. mandel x y > 0
|
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if y is on 0 then don't return a point
|
||
else return the value corresonding to (x,y,color for (x+iy))
|
||
</code>
|
||
|
||
In fact using the list monad you write like if you consider only one element at a time and the computation is done non deterministically.
|
||
To find the smallest number such that `mandel x y > 0` we use a simple dichotomy:
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
-- given f min max nbtest,
|
||
-- considering
|
||
-- - f is an increasing function
|
||
-- - f(min)=0
|
||
-- - f(max)≠0
|
||
-- then maxZeroIndex f min max nbtest returns x such that
|
||
-- f(x - ε)=0 and f(x + ε)≠0
|
||
-- where ε=(max-min)/2^(nbtest+1)
|
||
maxZeroIndex func minval maxval 0 = (minval+maxval)/2
|
||
maxZeroIndex func minval maxval n =
|
||
if (func medpoint) /= 0
|
||
then maxZeroIndex func minval medpoint (n-1)
|
||
else maxZeroIndex func medpoint maxval (n-1)
|
||
where medpoint = (minval+maxval)/2
|
||
</code>
|
||
</div>
|
||
|
||
No rocket science here. See the result now:
|
||
|
||
blogimage("HGLMandelEdges.png","The edges of the mandelbrot set")
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
colorFromValue n =
|
||
let
|
||
t :: Int -> GLfloat
|
||
t i = 0.5 + 0.5*cos( fromIntegral i / 10 )
|
||
in
|
||
Color3 (t n) (t (n+5)) (t (n+10))
|
||
</code>
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
mandel x y =
|
||
let r = 2.0 * x / width
|
||
i = 2.0 * y / height
|
||
in
|
||
f (complex r i) 0 64
|
||
</code>
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
f :: Complex -> Complex -> Int -> Int
|
||
f c z 0 = 0
|
||
f c z n = if (magnitude z > 2 )
|
||
then n
|
||
else f c ((z*z)+c) (n-1)
|
||
</code>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<a href="code/02_Edges/HGLMandelEdge.lhs" class="cut">Download the source code of this section → 02_Edges/<strong>HGLMandelEdge.lhs</strong> </a>
|
||
|
||
<hr/><a href="code/03_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 03_Mandelbulb/<strong>Mandelbulb.lhs</strong></a>
|
||
|
||
## 3D Mandelbrot?
|
||
|
||
Now we will we extend to a third dimension.
|
||
But, there is no 3D equivalent to complex.
|
||
In fact, the only extension known are quaternions (in 4D).
|
||
As I know almost nothing about quaternions, I will use some extended complex,
|
||
instead of using a 3D projection of quaternions.
|
||
I am pretty sure this construction is not useful for numbers.
|
||
But it will be enough for us to create something that look nice.
|
||
|
||
This section is quite long, but don't be afraid,
|
||
most of the code is some OpenGL boilerplate.
|
||
If you just want to skim this section,
|
||
here is a high level representation:
|
||
|
||
> - OpenGL Boilerplate
|
||
>
|
||
> - set some IORef (understand variables) for states
|
||
> - Drawing:
|
||
>
|
||
> - set doubleBuffer, handle depth, window size...
|
||
> - Use state to apply some transformations
|
||
>
|
||
> - Keyboard: hitting some key change the state of IORef
|
||
>
|
||
> - Generate 3D Object
|
||
>
|
||
> ~~~
|
||
> allPoints :: [ColoredPoint]
|
||
> allPoints =
|
||
> for all (x,y), -width<x<width, 0<y<height
|
||
> Let z be the minimal depth such that
|
||
> mandel x y z > 0
|
||
> add the points
|
||
> (x, y, z,color)
|
||
> (x,-y, z,color)
|
||
> (x, y,-z,color)
|
||
> (x,-y,-z,color)
|
||
> + neighbors to make triangles
|
||
> ~~~
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
import Graphics.Rendering.OpenGL
|
||
import Graphics.UI.GLUT
|
||
import Data.IORef
|
||
type ColoredPoint = (GLfloat,GLfloat,GLfloat,Color3 GLfloat)
|
||
</code>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
We declare a new type `ExtComplex` (for extended complex).
|
||
An extension of complex numbers with a third component:
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
data ExtComplex = C (GLfloat,GLfloat,GLfloat)
|
||
deriving (Show,Eq)
|
||
instance Num ExtComplex where
|
||
-- The shape of the 3D mandelbrot
|
||
-- will depend on this formula
|
||
C (x,y,z) * C (x',y',z') = C (x*x' - y*y' - z*z',
|
||
x*y' + y*x' + z*z',
|
||
x*z' + z*x' )
|
||
-- The rest is straightforward
|
||
fromInteger n = C (fromIntegral n, 0, 0)
|
||
C (x,y,z) + C (x',y',z') = C (x+x', y+y', z+z')
|
||
abs (C (x,y,z)) = C (sqrt (x*x + y*y + z*z), 0, 0)
|
||
signum (C (x,y,z)) = C (signum x, 0, 0)
|
||
</code>
|
||
</div>
|
||
|
||
The most important part is the new multiplication instance.
|
||
Modifying this formula will change radically the shape of the result.
|
||
Here is the formula written in a more mathematical notation.
|
||
I called the third component of these extended complex _strange_.
|
||
|
||
$$ \mathrm{real} ((x,y,z) * (x',y',z')) = xx' - yy' - zz' $$
|
||
|
||
$$ \mathrm{im} ((x,y,z) * (x',y',z')) = xy' - yx' + zz' $$
|
||
|
||
$$ \mathrm{strange} ((x,y,z) * (x',y',z')) = xz' + zx' $$
|
||
|
||
Note how if `z=z'=0` then the multiplication is the same to the complex one.
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
extcomplex :: GLfloat -> GLfloat -> GLfloat -> ExtComplex
|
||
extcomplex x y z = C (x,y,z)
|
||
|
||
real :: ExtComplex -> GLfloat
|
||
real (C (x,y,z)) = x
|
||
|
||
im :: ExtComplex -> GLfloat
|
||
im (C (x,y,z)) = y
|
||
|
||
strange :: ExtComplex -> GLfloat
|
||
strange (C (x,y,z)) = z
|
||
|
||
magnitude :: ExtComplex -> GLfloat
|
||
magnitude = real.abs
|
||
</code>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
### From 2D to 3D
|
||
|
||
As we will use some 3D, we add some new directive in the boilerplate.
|
||
But mainly, we simply state that will use some depth buffer.
|
||
And also we will listen the keyboard.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
main :: IO ()
|
||
main = do
|
||
-- GLUT need to be initialized
|
||
(progname,_) <- getArgsAndInitialize
|
||
-- We will use the double buffered mode (GL constraint)
|
||
-- We also Add the DepthBuffer (for 3D)
|
||
initialDisplayMode $=
|
||
[WithDepthBuffer,DoubleBuffered,RGBMode]
|
||
-- We create a window with some title
|
||
createWindow "3D HOpengGL Mandelbrot"
|
||
-- We add some directives
|
||
depthFunc $= Just Less
|
||
windowSize $= Size 500 500
|
||
-- Some state variables (I know it feels BAD)
|
||
angle <- newIORef ((35,0)::(GLfloat,GLfloat))
|
||
zoom <- newIORef (2::GLfloat)
|
||
campos <- newIORef ((0.7,0)::(GLfloat,GLfloat))
|
||
-- Function to call each frame
|
||
idleCallback $= Just idle
|
||
-- Function to call when keyboard or mouse is used
|
||
keyboardMouseCallback $=
|
||
Just (keyboardMouse angle zoom campos)
|
||
-- Each time we will need to update the display
|
||
-- we will call the function 'display'
|
||
-- But this time, we add some parameters
|
||
displayCallback $= display angle zoom campos
|
||
-- We enter the main loop
|
||
mainLoop
|
||
</code>
|
||
</div>
|
||
|
||
The `idle` is here to change the states.
|
||
There should never be any modification done in the `display` function.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
idle = postRedisplay Nothing
|
||
</code>
|
||
</div>
|
||
|
||
We introduce some helper function to manipulate
|
||
standard `IORef`.
|
||
Mainly `modVar x f` is equivalent to the imperative `x:=f(x)`,
|
||
`modFst (x,y) (+1)` is equivalent to `(x,y) := (x+1,y)`
|
||
and `modSnd (x,y) (+1)` is equivalent to `(x,y) := (x,y+1)`
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
modVar v f = do
|
||
v' <- get v
|
||
v $= (f v')
|
||
mapFst f (x,y) = (f x, y)
|
||
mapSnd f (x,y) = ( x,f y)
|
||
</code>
|
||
</div>
|
||
|
||
And we use them to code the function handling keyboard.
|
||
We will use the keys `hjkl` to rotate,
|
||
`oi` to zoom and `sedf` to move.
|
||
Also, hitting space will reset the view.
|
||
Remember that `angle` and `campos` are pairs and `zoom` is a scalar.
|
||
Also note `(+0.5)` is the function `\x->x+0.5`
|
||
and `(-0.5)` is the number `-0.5` (yes I share your pain).
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
keyboardMouse angle zoom campos key state modifiers position =
|
||
-- We won't use modifiers nor position
|
||
kact angle zoom campos key state
|
||
where
|
||
-- reset view when hitting space
|
||
kact a z p (Char ' ') Down = do
|
||
a $= (0,0) -- angle
|
||
z $= 1 -- zoom
|
||
p $= (0,0) -- camera position
|
||
-- use of hjkl to rotate
|
||
kact a _ _ (Char 'h') Down = modVar a (mapFst (+0.5))
|
||
kact a _ _ (Char 'l') Down = modVar a (mapFst (+(-0.5)))
|
||
kact a _ _ (Char 'j') Down = modVar a (mapSnd (+0.5))
|
||
kact a _ _ (Char 'k') Down = modVar a (mapSnd (+(-0.5)))
|
||
-- use o and i to zoom
|
||
kact _ z _ (Char 'o') Down = modVar z (*1.1)
|
||
kact _ z _ (Char 'i') Down = modVar z (*0.9)
|
||
-- use sdfe to move the camera
|
||
kact _ _ p (Char 's') Down = modVar p (mapFst (+0.1))
|
||
kact _ _ p (Char 'f') Down = modVar p (mapFst (+(-0.1)))
|
||
kact _ _ p (Char 'd') Down = modVar p (mapSnd (+0.1))
|
||
kact _ _ p (Char 'e') Down = modVar p (mapSnd (+(-0.1)))
|
||
-- any other keys does nothing
|
||
kact _ _ _ _ _ = return ()
|
||
</code>
|
||
</div>
|
||
|
||
Note `display` take some parameters this time.
|
||
This function if full of boilerplate:
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
display angle zoom position = do
|
||
-- set the background color (dark solarized theme)
|
||
clearColor $= Color4 0 0.1686 0.2117 1
|
||
clear [ColorBuffer,DepthBuffer]
|
||
-- Transformation to change the view
|
||
loadIdentity -- reset any transformation
|
||
-- tranlate
|
||
(x,y) <- get position
|
||
translate $ Vector3 x y 0
|
||
-- zoom
|
||
z <- get zoom
|
||
scale z z z
|
||
-- rotate
|
||
(xangle,yangle) <- get angle
|
||
rotate xangle $ Vector3 1.0 0.0 (0.0::GLfloat)
|
||
rotate yangle $ Vector3 0.0 1.0 (0.0::GLfloat)
|
||
|
||
-- Now that all transformation were made
|
||
-- We create the object(s)
|
||
preservingMatrix drawMandelbrot
|
||
|
||
swapBuffers -- refresh screen
|
||
</code>
|
||
</div>
|
||
|
||
Not much to say about this function.
|
||
Mainly there are two parts: apply some transformations, draw the object.
|
||
|
||
### The 3D Mandelbrot
|
||
|
||
We have finished with the OpenGL section, let's talk about how we
|
||
generate the 3D points and colors.
|
||
First, we will set the number of details to 200 pixels in the three dimensions.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
nbDetails = 200 :: GLfloat
|
||
width = nbDetails
|
||
height = nbDetails
|
||
deep = nbDetails
|
||
</code>
|
||
</div>
|
||
|
||
This time, instead of just drawing some line or some group of points,
|
||
we will show triangles.
|
||
The function `allPoints` will provide a multiple of three points.
|
||
Each three successive point representing the coordinate of each vertex of a triangle.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
drawMandelbrot = do
|
||
-- We will print Points (not triangles for example)
|
||
renderPrimitive Triangles $ do
|
||
mapM_ drawColoredPoint allPoints
|
||
where
|
||
drawColoredPoint (x,y,z,c) = do
|
||
color c
|
||
vertex $ Vertex3 x y z
|
||
</code>
|
||
</div>
|
||
|
||
In fact, we will provide six ordered points.
|
||
These points will be used to draw two triangles.
|
||
|
||
blogimage("triangles.png","Explain triangles")
|
||
|
||
The next function is a bit long.
|
||
Here is an approximative English version:
|
||
|
||
~~~
|
||
forall x from -width to width
|
||
forall y from -height to height
|
||
forall the neighbors of (x,y)
|
||
let z be the smalled depth such that (mandel x y z)>0
|
||
let c be the color given by mandel x y z
|
||
add the point corresponding to (x,y,z,c)
|
||
~~~
|
||
|
||
Also, I added a test to hide points too far from the border.
|
||
In fact, this function show points close to the surface of the modified mandelbrot set. But not the mandelbrot set itself.
|
||
|
||
<code class="haskell">
|
||
depthPoints :: [ColoredPoint]
|
||
depthPoints = do
|
||
x <- [-width..width]
|
||
y <- [-height..height]
|
||
let
|
||
depthOf x' y' = maxZeroIndex (mandel x' y') 0 deep logdeep
|
||
logdeep = floor ((log deep) / log 2)
|
||
z1 = depthOf x y
|
||
z2 = depthOf (x+1) y
|
||
z3 = depthOf (x+1) (y+1)
|
||
z4 = depthOf x (y+1)
|
||
c1 = mandel x y (z1+1)
|
||
c2 = mandel (x+1) y (z2+1)
|
||
c3 = mandel (x+1) (y+1) (z3+1)
|
||
c4 = mandel x (y+1) (z4+1)
|
||
p1 = ( x /width, y /height, z1/deep, colorFromValue c1)
|
||
p2 = ((x+1)/width, y /height, z2/deep, colorFromValue c2)
|
||
p3 = ((x+1)/width,(y+1)/height, z3/deep, colorFromValue c3)
|
||
p4 = ( x /width,(y+1)/height, z4/deep, colorFromValue c4)
|
||
if (and $ map (>=57) [c1,c2,c3,c4])
|
||
then []
|
||
else [p1,p2,p3,p1,p3,p4]
|
||
</code>
|
||
|
||
If you look at the function above, you see a lot of common patterns.
|
||
Haskell is very efficient to make this better.
|
||
Here is a harder to read but shorter and more generic rewritten function:
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
depthPoints :: [ColoredPoint]
|
||
depthPoints = do
|
||
x <- [-width..width]
|
||
y <- [-height..height]
|
||
let
|
||
neighbors = [(x,y),(x+1,y),(x+1,y+1),(x,y+1)]
|
||
depthOf (u,v) = maxZeroIndex (mandel u v) 0 deep logdeep
|
||
logdeep = floor ((log deep) / log 2)
|
||
-- zs are 3D points with found depth
|
||
zs = map (\(u,v) -> (u,v,depthOf (u,v))) neighbors
|
||
-- ts are 3D pixels + mandel value
|
||
ts = map (\(u,v,w) -> (u,v,w,mandel u v (w+1))) zs
|
||
-- ps are 3D opengl points + color value
|
||
ps = map (\(u,v,w,c') ->
|
||
(u/width,v/height,w/deep,colorFromValue c')) ts
|
||
-- If the point diverged too fast, don't display it
|
||
if (and $ map (\(_,_,_,c) -> c>=57) ts)
|
||
then []
|
||
-- Draw two triangles
|
||
else [ps!!0,ps!!1,ps!!2,ps!!0,ps!!2,ps!!3]
|
||
</code>
|
||
</div>
|
||
|
||
If you prefer the first version, then just imagine how hard it will be to change the enumeration of the point from (x,y) to (x,z) for example.
|
||
|
||
Also, we didn't searched for negative values.
|
||
This modified Mandelbrot is no more symmetric relatively to the plan `y=0`.
|
||
But it is symmetric relatively to the plan `z=0`.
|
||
Then I mirror these values.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
allPoints :: [ColoredPoint]
|
||
allPoints = planPoints ++ map inverseDepth planPoints
|
||
where
|
||
planPoints = depthPoints
|
||
inverseDepth (x,y,z,c) = (x,y,-z+1/deep,c)
|
||
</code>
|
||
</div>
|
||
|
||
The rest of the program is very close to the preceding one.
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
-- given f min max nbtest,
|
||
-- considering
|
||
-- - f is an increasing function
|
||
-- - f(min)=0
|
||
-- - f(max)≠0
|
||
-- then maxZeroIndex f min max nbtest returns x such that
|
||
-- f(x - ε)=0 and f(x + ε)≠0
|
||
-- where ε=(max-min)/2^(nbtest+1)
|
||
maxZeroIndex :: (Fractional a,Num a,Num b,Eq b) =>
|
||
(a -> b) -> a -> a -> Int -> a
|
||
maxZeroIndex func minval maxval 0 = (minval+maxval)/2
|
||
maxZeroIndex func minval maxval n =
|
||
if (func medpoint) /= 0
|
||
then maxZeroIndex func minval medpoint (n-1)
|
||
else maxZeroIndex func medpoint maxval (n-1)
|
||
where medpoint = (minval+maxval)/2
|
||
</code>
|
||
</div>
|
||
|
||
I made the color slightly brighter
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
colorFromValue n =
|
||
let
|
||
t :: Int -> GLfloat
|
||
t i = 0.7 + 0.3*cos( fromIntegral i / 10 )
|
||
in
|
||
Color3 (t n) (t (n+5)) (t (n+10))
|
||
</code>
|
||
</div>
|
||
|
||
We only changed from `Complex` to `ExtComplex` of the main `f` function.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
f :: ExtComplex -> ExtComplex -> Int -> Int
|
||
f c z 0 = 0
|
||
f c z n = if (magnitude z > 2 )
|
||
then n
|
||
else f c ((z*z)+c) (n-1)
|
||
</code>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
We simply add a new dimension to the `mandel` function
|
||
and change the type signature of `f` from `Complex` to `ExtComplex`.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
mandel x y z =
|
||
let r = 2.0 * x / width
|
||
i = 2.0 * y / height
|
||
s = 2.0 * z / deep
|
||
in
|
||
f (extcomplex r i s) 0 64
|
||
</code>
|
||
</div>
|
||
|
||
Here is the result:
|
||
|
||
blogimage("mandelbrot_3D.png","A 3D mandelbrot like")
|
||
|
||
<a href="code/03_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 03_Mandelbulb/<strong>Mandelbulb.lhs</strong> </a>
|
||
|
||
<hr/><a href="code/04_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 04_Mandelbulb/<strong>Mandelbulb.lhs</strong></a>
|
||
|
||
## Naïve code cleaning
|
||
|
||
The first thing to do is to separate the GLUT/OpenGL
|
||
part from the computation of the shape.
|
||
Here is the cleaned version of the preceding section.
|
||
Most boilerplate was put in external files.
|
||
|
||
- [`YBoiler.hs`](code/04_Mandelbulb/YBoiler.hs), the 3D rendering
|
||
- [`Mandel`](code/04_Mandelbulb/Mandel.hs), the mandel function
|
||
- [`ExtComplex`](code/04_Mandelbulb/ExtComplex.hs), the extended complexes
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
import YBoiler -- Most the OpenGL Boilerplate
|
||
import Mandel -- The 3D Mandelbrot maths
|
||
</code>
|
||
</div>
|
||
|
||
The `yMainLoop` takes two arguments:
|
||
the title of the window
|
||
and a function from time to triangles
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
main :: IO ()
|
||
main = yMainLoop "3D Mandelbrot" (\_ -> allPoints)
|
||
</code>
|
||
</div>
|
||
|
||
We set some global constant (this is generally bad).
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
nbDetails = 200 :: GLfloat
|
||
width = nbDetails
|
||
height = nbDetails
|
||
deep = nbDetails
|
||
</code>
|
||
</div>
|
||
|
||
We then generate colored points from our function.
|
||
This is similar to the preceding section.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
allPoints :: [ColoredPoint]
|
||
allPoints = planPoints ++ map inverseDepth planPoints
|
||
where
|
||
planPoints = depthPoints ++ map inverseHeight depthPoints
|
||
inverseHeight (x,y,z,c) = (x,-y,z,c)
|
||
inverseDepth (x,y,z,c) = (x,y,-z+1/deep,c)
|
||
</code>
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
depthPoints :: [ColoredPoint]
|
||
depthPoints = do
|
||
x <- [-width..width]
|
||
y <- [0..height]
|
||
let
|
||
neighbors = [(x,y),(x+1,y),(x+1,y+1),(x,y+1)]
|
||
depthOf (u,v) = maxZeroIndex (ymandel u v) 0 deep 7
|
||
-- zs are 3D points with found depth
|
||
zs = map (\(u,v) -> (u,v,depthOf (u,v))) neighbors
|
||
-- ts are 3D pixels + mandel value
|
||
ts = map (\(u,v,w) -> (u,v,w,ymandel u v (w+1))) zs
|
||
-- ps are 3D opengl points + color value
|
||
ps = map (\(u,v,w,c') ->
|
||
(u/width,v/height,w/deep,colorFromValue c')) ts
|
||
-- If the point diverged too fast, don't display it
|
||
if (and $ map (\(_,_,_,c) -> c>=57) ts)
|
||
then []
|
||
-- Draw two triangles
|
||
else [ps!!0,ps!!1,ps!!2,ps!!0,ps!!2,ps!!3]
|
||
|
||
|
||
-- given f min max nbtest,
|
||
-- considering
|
||
-- - f is an increasing function
|
||
-- - f(min)=0
|
||
-- - f(max)≠0
|
||
-- then maxZeroIndex f min max nbtest returns x such that
|
||
-- f(x - ε)=0 and f(x + ε)≠0
|
||
-- where ε=(max-min)/2^(nbtest+1)
|
||
maxZeroIndex func minval maxval 0 = (minval+maxval)/2
|
||
maxZeroIndex func minval maxval n =
|
||
if (func medpoint) /= 0
|
||
then maxZeroIndex func minval medpoint (n-1)
|
||
else maxZeroIndex func medpoint maxval (n-1)
|
||
where medpoint = (minval+maxval)/2
|
||
|
||
colorFromValue n =
|
||
let
|
||
t :: Int -> GLfloat
|
||
t i = 0.7 + 0.3*cos( fromIntegral i / 10 )
|
||
in
|
||
((t n),(t (n+5)),(t (n+10)))
|
||
|
||
ymandel x y z = mandel (2*x/width) (2*y/height) (2*z/deep) 64
|
||
</code>
|
||
</div>
|
||
|
||
This code is cleaner but many things doesn't feel right.
|
||
First, all the user interaction code is outside our main file.
|
||
I feel it is okay to hide the detail for the rendering.
|
||
But I would have preferred to control the user actions.
|
||
|
||
On the other hand, we continue to handle a lot rendering details.
|
||
For example, we provide ordered vertices.
|
||
|
||
<a href="code/04_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 04_Mandelbulb/<strong>Mandelbulb.lhs</strong> </a>
|
||
|
||
<hr/><a href="code/05_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 05_Mandelbulb/<strong>Mandelbulb.lhs</strong></a>
|
||
|
||
## Functional organization?
|
||
|
||
Some points:
|
||
|
||
1. OpenGL and GLUT is done in C.
|
||
In particular the `mainLoop` function is a direct link to the C library (FFI).
|
||
This function is clearly far from the functional paradigm.
|
||
Could we make this better?
|
||
We will have two choices:
|
||
|
||
- create our own `mainLoop` function to make it more functional.
|
||
- deal with the imperative nature of the GLUT `mainLoop` function.
|
||
|
||
As one of the goal of this article is to understand how to deal with existing libraries and particularly the one coming from imperative languages, we will continue to use the `mainLoop` function.
|
||
2. Our main problem come from user interaction.
|
||
If you ask "the Internet",
|
||
about how to deal with user interaction with a functional paradigm,
|
||
the main answer is to use _functional reactive programming_ (FRP).
|
||
I won't use FRP in this article.
|
||
Instead, I'll use a simpler while less effective way to deal with user interaction.
|
||
But The method I'll use will be as pure and functional as possible.
|
||
|
||
Here is how I imagine things should go.
|
||
First, what the main loop should look like if we could make our own:
|
||
|
||
<code class="no-highlight">
|
||
functionalMainLoop =
|
||
Read user inputs and provide a list of actions
|
||
Apply all actions to the World
|
||
Display one frame
|
||
repetere aeternum
|
||
</code>
|
||
|
||
Clearly, ideally we should provide only three parameters to this main loop function:
|
||
|
||
- an initial World state
|
||
- a mapping between the user interaction and function which modify the world
|
||
- a function taking two parameters: time and world state and render a new world without user interaction.
|
||
|
||
Here is a real working code, I've hidden most display functions.
|
||
The YGL, is a kind of framework to display 3D functions.
|
||
But it can easily be extended to many kind of representation.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
import YGL -- Most the OpenGL Boilerplate
|
||
import Mandel -- The 3D Mandelbrot maths
|
||
</code>
|
||
</div>
|
||
|
||
We first set the mapping between user input and actions.
|
||
The type of each couple should be of the form
|
||
`(user input, f)` where (in a first time) `f:World -> World`.
|
||
It means, the user input will transform the world state.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
-- Centralize all user input interaction
|
||
inputActionMap :: InputMap World
|
||
inputActionMap = inputMapFromList [
|
||
(Press 'k' , rotate xdir 5)
|
||
,(Press 'i' , rotate xdir (-5))
|
||
,(Press 'j' , rotate ydir 5)
|
||
,(Press 'l' , rotate ydir (-5))
|
||
,(Press 'o' , rotate zdir 5)
|
||
,(Press 'u' , rotate zdir (-5))
|
||
,(Press 'f' , translate xdir 0.1)
|
||
,(Press 's' , translate xdir (-0.1))
|
||
,(Press 'e' , translate ydir 0.1)
|
||
,(Press 'd' , translate ydir (-0.1))
|
||
,(Press 'z' , translate zdir 0.1)
|
||
,(Press 'r' , translate zdir (-0.1))
|
||
,(Press '+' , zoom 1.1)
|
||
,(Press '-' , zoom (1/1.1))
|
||
,(Press 'h' , resize 1.2)
|
||
,(Press 'g' , resize (1/1.2))
|
||
]
|
||
</code>
|
||
</div>
|
||
|
||
And of course a type design the World State.
|
||
The important part is that it is our World State type.
|
||
We could have used any kind of data type.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
-- I prefer to set my own name for these types
|
||
data World = World {
|
||
angle :: Point3D
|
||
, scale :: Scalar
|
||
, position :: Point3D
|
||
, shape :: Scalar -> Function3D
|
||
, box :: Box3D
|
||
, told :: Time -- last frame time
|
||
}
|
||
</code>
|
||
</div>
|
||
|
||
The important part to glue our own type to the framework
|
||
is to make our type an instance of the type class `DisplayableWorld`.
|
||
We simply have to provide the definition of some functions.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
instance DisplayableWorld World where
|
||
winTitle _ = "The YGL Mandelbulb"
|
||
camera w = Camera {
|
||
camPos = position w,
|
||
camDir = angle w,
|
||
camZoom = scale w }
|
||
-- objects for world w
|
||
-- is the list of one unique element
|
||
-- The element is an YObject
|
||
-- more precisely the XYFunc Function3D Box3D
|
||
-- where the Function3D is the type
|
||
-- Point -> Point -> Maybe (Point,Color)
|
||
-- and its value here is ((shape w) res)
|
||
-- and the Box3D value is defbox
|
||
objects w = [XYFunc ((shape w) res) defbox]
|
||
where
|
||
res = resolution $ box w
|
||
defbox = box w
|
||
</code>
|
||
</div>
|
||
|
||
The `camera` function will retrieve an object of type `Camera` which contains
|
||
most necessary information to set our camera.
|
||
The `objects` function will returns a list of objects.
|
||
Their type is `YObject`. Note the generation of triangles is no more in this file.
|
||
Until here we only used declarative pattern.
|
||
|
||
We also need to set all our transformation functions.
|
||
These function are used to update the world state.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
xdir :: Point3D
|
||
xdir = makePoint3D (1,0,0)
|
||
ydir :: Point3D
|
||
ydir = makePoint3D (0,1,0)
|
||
zdir :: Point3D
|
||
zdir = makePoint3D (0,0,1)
|
||
</code>
|
||
</div>
|
||
|
||
Note `(-*<)` is the scalar product (`α -*< (x,y,z) = (αx,αy,αz)`).
|
||
Also note we could add two Point3D.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
rotate :: Point3D -> Scalar -> World -> World
|
||
rotate dir angleValue world =
|
||
world {
|
||
angle = (angle world) + (angleValue -*< dir) }
|
||
|
||
translate :: Point3D -> Scalar -> World -> World
|
||
translate dir len world =
|
||
world {
|
||
position = (position world) + (len -*< dir) }
|
||
|
||
zoom :: Scalar -> World -> World
|
||
zoom z world = world {
|
||
scale = z * scale world }
|
||
|
||
resize :: Scalar -> World -> World
|
||
resize r world = world {
|
||
box = (box world) {
|
||
resolution = sqrt ((resolution (box world))**2 * r) }}
|
||
</code>
|
||
</div>
|
||
|
||
The resize is used to generate the 3D function.
|
||
As I wanted the time spent to generate a more detailed view
|
||
to grow linearly I use this not so straightforward formula.
|
||
|
||
The `yMainLoop` takes three arguments.
|
||
|
||
- A map between user Input and world transformation
|
||
- A timed world transformation
|
||
- An initial world state
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
main :: IO ()
|
||
main = yMainLoop inputActionMap idleAction initialWorld
|
||
</code>
|
||
</div>
|
||
|
||
Here is our initial world state.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
-- We initialize the world state
|
||
-- then angle, position and zoom of the camera
|
||
-- And the shape function
|
||
initialWorld :: World
|
||
initialWorld = World {
|
||
angle = makePoint3D (-30,-30,0)
|
||
, position = makePoint3D (0,0,0)
|
||
, scale = 0.8
|
||
, shape = shapeFunc
|
||
, box = Box3D { minPoint = makePoint3D (-2,-2,-2)
|
||
, maxPoint = makePoint3D (2,2,2)
|
||
, resolution = 0.16 }
|
||
, told = 0
|
||
}
|
||
</code>
|
||
</div>
|
||
|
||
We will define `shapeFunc` later.
|
||
Here is the function which transform the world even without user action.
|
||
Mainly it makes some rotation.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
idleAction :: Time -> World -> World
|
||
idleAction tnew world = world {
|
||
angle = (angle world) + (delta -*< zdir)
|
||
, told = tnew
|
||
}
|
||
where
|
||
anglePerSec = 5.0
|
||
delta = anglePerSec * elapsed / 1000.0
|
||
elapsed = fromIntegral (tnew - (told world))
|
||
</code>
|
||
</div>
|
||
|
||
Now the function which will generate points in 3D.
|
||
The first parameter (`res`) is the resolution of the vertex generation.
|
||
More precisely, `res` is distance between two points on one direction.
|
||
We need it to "close" our shape.
|
||
|
||
The type `Function3D` is `Point -> Point -> Maybe Point`.
|
||
Because we consider partial functions
|
||
(for some `(x,y)` our function can be undefined).
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
shapeFunc :: Scalar -> Function3D
|
||
shapeFunc res x y =
|
||
let
|
||
z = maxZeroIndex (ymandel x y) 0 1 20
|
||
in
|
||
if and [ maxZeroIndex (ymandel (x+xeps) (y+yeps)) 0 1 20 < 0.000001 |
|
||
val <- [res], xeps <- [-val,val], yeps<-[-val,val]]
|
||
then Nothing
|
||
else Just (z,colorFromValue ((ymandel x y z) * 64))
|
||
</code>
|
||
</div>
|
||
|
||
With the color function.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
colorFromValue :: Point -> Color
|
||
colorFromValue n =
|
||
let
|
||
t :: Point -> Scalar
|
||
t i = 0.7 + 0.3*cos( i / 10 )
|
||
in
|
||
makeColor (t n) (t (n+5)) (t (n+10))
|
||
</code>
|
||
</div>
|
||
|
||
The rest is similar to the preceding sections.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
-- given f min max nbtest,
|
||
-- considering
|
||
-- - f is an increasing function
|
||
-- - f(min)=0
|
||
-- - f(max)≠0
|
||
-- then maxZeroIndex f min max nbtest returns x such that
|
||
-- f(x - ε)=0 and f(x + ε)≠0
|
||
-- where ε=(max-min)/2^(nbtest+1)
|
||
maxZeroIndex :: (Fractional a,Num a,Num b,Eq b) =>
|
||
(a -> b) -> a -> a -> Int -> a
|
||
maxZeroIndex _ minval maxval 0 = (minval+maxval)/2
|
||
maxZeroIndex func minval maxval n =
|
||
if (func medpoint) /= 0
|
||
then maxZeroIndex func minval medpoint (n-1)
|
||
else maxZeroIndex func medpoint maxval (n-1)
|
||
where medpoint = (minval+maxval)/2
|
||
|
||
ymandel :: Point -> Point -> Point -> Point
|
||
ymandel x y z = fromIntegral (mandel x y z 64) / 64
|
||
</code>
|
||
</div>
|
||
|
||
I won't explain how the magic occurs here.
|
||
If you are interested, just read the file [`YGL.hs`](code/05_Mandelbulb/YGL.hs).
|
||
It is commented a lot.
|
||
|
||
- [`YGL.hs`](code/05_Mandelbulb/YGL.hs), the 3D rendering framework
|
||
- [`Mandel`](code/05_Mandelbulb/Mandel.hs), the mandel function
|
||
- [`ExtComplex`](code/05_Mandelbulb/ExtComplex.hs), the extended complexes
|
||
|
||
<a href="code/05_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 05_Mandelbulb/<strong>Mandelbulb.lhs</strong> </a>
|
||
|
||
<hr/><a href="code/06_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 06_Mandelbulb/<strong>Mandelbulb.lhs</strong></a>
|
||
|
||
## Optimization
|
||
|
||
From the architecture stand point all is clear.
|
||
If you read the code of `YGL.hs`, you'll see I didn't made everything perfect.
|
||
For example, I didn't finished the code of the lights.
|
||
But I believe it is a good first step and it will be easy to go further.
|
||
The separation between rendering and world behavior is clear.
|
||
Unfortunately the program of the preceding session is extremely slow.
|
||
We compute the Mandelbulb for each frame now.
|
||
|
||
Before we had
|
||
|
||
<code class="no-highlight">
|
||
Constant Function -> Constant List of Triangles -> Display
|
||
</code>
|
||
|
||
Now we have
|
||
|
||
<code class="no-highlight">
|
||
World -> Function -> List of Objects -> Atoms -> Display
|
||
</code>
|
||
|
||
And the World state could change.
|
||
Then it is no more straightforward for the compiler to understand
|
||
when not to recompute the entire list of atoms.
|
||
|
||
Then to optimize we will have to make things a little less separate.
|
||
We must control the flow of atom generation.
|
||
|
||
Mostly the program is the same as before, but instead of providing a
|
||
function, we will provide the list of atoms directly.
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
import YGL -- Most the OpenGL Boilerplate
|
||
import Mandel -- The 3D Mandelbrot maths
|
||
|
||
-- Centralize all user input interaction
|
||
inputActionMap :: InputMap World
|
||
inputActionMap = inputMapFromList [
|
||
(Press ' ' , switchRotation)
|
||
,(Press 'k' , rotate xdir 5)
|
||
,(Press 'i' , rotate xdir (-5))
|
||
,(Press 'j' , rotate ydir 5)
|
||
,(Press 'l' , rotate ydir (-5))
|
||
,(Press 'o' , rotate zdir 5)
|
||
,(Press 'u' , rotate zdir (-5))
|
||
,(Press 'f' , translate xdir 0.1)
|
||
,(Press 's' , translate xdir (-0.1))
|
||
,(Press 'e' , translate ydir 0.1)
|
||
,(Press 'd' , translate ydir (-0.1))
|
||
,(Press 'z' , translate zdir 0.1)
|
||
,(Press 'r' , translate zdir (-0.1))
|
||
,(Press '+' , zoom 1.1)
|
||
,(Press '-' , zoom (1/1.1))
|
||
,(Press 'h' , resize 2.0)
|
||
,(Press 'g' , resize (1/2.0))
|
||
]
|
||
</code>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
data World = World {
|
||
angle :: Point3D
|
||
, anglePerSec :: Scalar
|
||
, scale :: Scalar
|
||
, position :: Point3D
|
||
, box :: Box3D
|
||
, told :: Time
|
||
-- We replace shape by cache
|
||
, cache :: [YObject]
|
||
}
|
||
</code>
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
instance DisplayableWorld World where
|
||
winTitle _ = "The YGL Mandelbulb"
|
||
camera w = Camera {
|
||
camPos = position w,
|
||
camDir = angle w,
|
||
camZoom = scale w }
|
||
-- We update our objects instanciation
|
||
objects = cache
|
||
</code>
|
||
</div>
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
xdir :: Point3D
|
||
xdir = makePoint3D (1,0,0)
|
||
ydir :: Point3D
|
||
ydir = makePoint3D (0,1,0)
|
||
zdir :: Point3D
|
||
zdir = makePoint3D (0,0,1)
|
||
|
||
rotate :: Point3D -> Scalar -> World -> World
|
||
rotate dir angleValue world =
|
||
world {
|
||
angle = angle world + (angleValue -*< dir) }
|
||
|
||
switchRotation :: World -> World
|
||
switchRotation world =
|
||
world {
|
||
anglePerSec = if anglePerSec world > 0 then 0 else 5.0 }
|
||
|
||
translate :: Point3D -> Scalar -> World -> World
|
||
translate dir len world =
|
||
world {
|
||
position = position world + (len -*< dir) }
|
||
|
||
zoom :: Scalar -> World -> World
|
||
zoom z world = world {
|
||
scale = z * scale world }
|
||
</code>
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
main :: IO ()
|
||
main = yMainLoop inputActionMap idleAction initialWorld
|
||
</code>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
Our initial world state is slightly changed:
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
-- We initialize the world state
|
||
-- then angle, position and zoom of the camera
|
||
-- And the shape function
|
||
initialWorld :: World
|
||
initialWorld = World {
|
||
angle = makePoint3D (30,30,0)
|
||
, anglePerSec = 5.0
|
||
, position = makePoint3D (0,0,0)
|
||
, scale = 1.0
|
||
, box = Box3D { minPoint = makePoint3D (0-eps, 0-eps, 0-eps)
|
||
, maxPoint = makePoint3D (0+eps, 0+eps, 0+eps)
|
||
, resolution = 0.02 }
|
||
, told = 0
|
||
-- We declare cache directly this time
|
||
, cache = objectFunctionFromWorld initialWorld
|
||
}
|
||
where eps=2
|
||
</code>
|
||
</div>
|
||
|
||
We use the `YGL.getObject3DFromShapeFunction` function directly.
|
||
This way instead of providing `XYFunc`, we provide directly a list of Atoms.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
objectFunctionFromWorld :: World -> [YObject]
|
||
objectFunctionFromWorld w = [Atoms atomList]
|
||
where atomListPositive =
|
||
getObject3DFromShapeFunction
|
||
(shapeFunc (resolution (box w))) (box w)
|
||
atomList = atomListPositive ++
|
||
map negativeTriangle atomListPositive
|
||
negativeTriangle (ColoredTriangle (p1,p2,p3,c)) =
|
||
ColoredTriangle (negz p1,negz p3,negz p2,c)
|
||
where negz (P (x,y,z)) = P (x,y,-z)
|
||
</code>
|
||
</div>
|
||
|
||
We know that resize is the only world change that necessitate to
|
||
recompute the list of atoms (triangles).
|
||
Then we update our world state accordingly.
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
resize :: Scalar -> World -> World
|
||
resize r world =
|
||
tmpWorld { cache = objectFunctionFromWorld tmpWorld }
|
||
where
|
||
tmpWorld = world { box = (box world) {
|
||
resolution = sqrt ((resolution (box world))**2 * r) }}
|
||
</code>
|
||
</div>
|
||
|
||
All the rest is exactly the same.
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
<code class="haskell">
|
||
idleAction :: Time -> World -> World
|
||
idleAction tnew world =
|
||
world {
|
||
angle = angle world + (delta -*< zdir)
|
||
, told = tnew
|
||
}
|
||
where
|
||
delta = anglePerSec world * elapsed / 1000.0
|
||
elapsed = fromIntegral (tnew - (told world))
|
||
|
||
shapeFunc :: Scalar -> Function3D
|
||
shapeFunc res x y =
|
||
let
|
||
z = maxZeroIndex (ymandel x y) 0 1 20
|
||
in
|
||
if and [ maxZeroIndex (ymandel (x+xeps) (y+yeps)) 0 1 20 < 0.000001 |
|
||
val <- [res], xeps <- [-val,val], yeps<-[-val,val]]
|
||
then Nothing
|
||
else Just (z,colorFromValue 0)
|
||
|
||
colorFromValue :: Point -> Color
|
||
colorFromValue n =
|
||
let
|
||
t :: Point -> Scalar
|
||
t i = 0.0 + 0.5*cos( i /10 )
|
||
in
|
||
makeColor (t n) (t (n+5)) (t (n+10))
|
||
|
||
-- given f min max nbtest,
|
||
-- considering
|
||
-- - f is an increasing function
|
||
-- - f(min)=0
|
||
-- - f(max)≠0
|
||
-- then maxZeroIndex f min max nbtest returns x such that
|
||
-- f(x - ε)=0 and f(x + ε)≠0
|
||
-- where ε=(max-min)/2^(nbtest+1)
|
||
maxZeroIndex :: (Fractional a,Num a,Num b,Eq b) =>
|
||
(a -> b) -> a -> a -> Int -> a
|
||
maxZeroIndex _ minval maxval 0 = (minval+maxval)/2
|
||
maxZeroIndex func minval maxval n =
|
||
if func medpoint /= 0
|
||
then maxZeroIndex func minval medpoint (n-1)
|
||
else maxZeroIndex func medpoint maxval (n-1)
|
||
where medpoint = (minval+maxval)/2
|
||
|
||
ymandel :: Point -> Point -> Point -> Point
|
||
ymandel x y z = fromIntegral (mandel x y z 64) / 64
|
||
</code>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
And you can also consider small changes in other source files.
|
||
|
||
- [`YGL.hs`](code/06_Mandelbulb/YGL.hs), the 3D rendering framework
|
||
- [`Mandel`](code/06_Mandelbulb/Mandel.hs), the mandel function
|
||
- [`ExtComplex`](code/06_Mandelbulb/ExtComplex.hs), the extended complexes
|
||
|
||
<a href="code/06_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 06_Mandelbulb/<strong>Mandelbulb.lhs</strong> </a>
|
||
|
||
## Conclusion
|
||
|
||
As we can use imperative style in a functional language,
|
||
know you can use functional style in imperative languages.
|
||
This article exposed a way to organize some code in a functional way.
|
||
I'd like to stress the usage of Haskell made it very simple to achieve this.
|
||
|
||
Once you are used to pure functional style,
|
||
it is hard not to see all advantages it offers.
|
||
|
||
The code in the two last sections is completely pure and functional.
|
||
Furthermore I don't use `GLfloat`, `Color3` or any other OpenGL type.
|
||
If I want to use another library in the future,
|
||
I would be able to keep all the pure code and simply update the YGL module.
|
||
|
||
The `YGL` module can be seen as a "wrapper" around 3D display and user interaction.
|
||
It is a clean separator between the imperative paradigm and functional paradigm.
|
||
|
||
If you want to go further, it shouldn't be hard to add parallelism.
|
||
This should be easy mainly because most of the visible code is pure.
|
||
Such an optimization would have been harder by using directly the OpenGL library.
|
||
|
||
You should also want to make a more precise object. Because, the Mandelbulb is
|
||
clearly not convex. But a precise rendering might be very long from
|
||
O(n².log(n)) to O(n³).
|