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<h1>
|
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Un example progressif avec Haskell
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</h1>
|
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|
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<h2>
|
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Une extension de l'ensemble de Mandelbrot en 3D et en OpenGL
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</h2>
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<div class="corps">
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<p><img alt="The B in Benoît B. Mandelbrot stand for Benoît B. Mandelbrot" src="/Scratch/img/blog/Haskell-OpenGL-Mandelbrot/BenoitBMandelbrot.jpg" /></p>
|
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|
||
<div class="intro">
|
||
|
||
|
||
<p><span class="sc"><abbr title="Trop long à lire">tlàl</abbr> : </span> Un exemple progressif d’utilisation d’Haskell.
|
||
Vous pourrez voir un ensemble de Mandelbrot étendu à la troisième dimension.
|
||
De plus le code sera très propre.
|
||
Les détails de rendu sont séparés dans un module externe.
|
||
Le code descriptif intéressant est concentré dans un environnement pur et fonctionnel.
|
||
Vous pouvez vous inspirer de ce code utilisant le paradigme fonctional dans tous les languages.</p>
|
||
|
||
<blockquote>
|
||
<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>
|
||
|
||
<ul id="markdown-toc">
|
||
<li><a href="#introduction">Introduction</a></li>
|
||
<li><a href="#first-version">First version</a> <ul>
|
||
<li><a href="#lets-play-the-song-of-our-people">Let’s play the song of our people</a></li>
|
||
<li><a href="#let-us-start">Let us start</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#only-the-edges">Only the edges</a></li>
|
||
<li><a href="#d-mandelbrot">3D Mandelbrot?</a> <ul>
|
||
<li><a href="#from-2d-to-3d">From 2D to 3D</a></li>
|
||
<li><a href="#the-3d-mandelbrot">The 3D Mandelbrot</a></li>
|
||
</ul>
|
||
</li>
|
||
<li><a href="#nave-code-cleaning">Naïve code cleaning</a></li>
|
||
<li><a href="#functional-organization">Functional organization?</a></li>
|
||
<li><a href="#optimization">Optimization</a></li>
|
||
<li><a href="#conclusion">Conclusion</a></li>
|
||
</ul>
|
||
|
||
</blockquote>
|
||
|
||
|
||
</div>
|
||
|
||
|
||
<h2 id="introduction">Introduction</h2>
|
||
|
||
<p>In my
|
||
<a href="/Scratch/en/blog/Haskell-the-Hard-Way/">preceding article</a> I introduced Haskell. </p>
|
||
|
||
<p>This article goes further.
|
||
It will show how to use functional programming with interactive programs.
|
||
But more than that, it will show how to organize your code in a functional way.
|
||
This article is more about functional paradigm than functional language.
|
||
The code organization can be used in most imperative language.</p>
|
||
|
||
<p>As Haskell is designed for functional paradigm, it is easier to use in this context.
|
||
In reality, the firsts sections will use an imperative paradigm.
|
||
As you can use functional paradigm in imperative language,
|
||
you can also use imperative paradigm in functional languages.</p>
|
||
|
||
<p>This article is about creating an useful and clean program.
|
||
It can interact with the user in real time.
|
||
It uses OpenGL, a library with imperative programming foundations.
|
||
Despite this fact,
|
||
most of the final code will remain in the pure part (no <code>IO</code>).</p>
|
||
|
||
<p>I believe the main audience for this article are:</p>
|
||
|
||
<ul>
|
||
<li>Haskell programmer looking for an OpengGL tutorial.</li>
|
||
<li>People interested in program organization (programming language agnostic).</li>
|
||
<li>Fractal lovers and in particular 3D fractal.</li>
|
||
<li>People interested in user interaction in a functional paradigm.</li>
|
||
</ul>
|
||
|
||
<p>I had in mind for some time now to make a Mandelbrot set explorer.
|
||
I had already written a <a href="http://github.com/yogsototh/mandelbrot.git">command line Mandelbrot set generator in Haskell</a>.
|
||
This utility is highly parallel; it uses the <code>repa</code> package<sup id="fnref:001"><a href="#fn:001" rel="footnote">1</a></sup>.</p>
|
||
|
||
<p>This time, we will not parallelize the computation.
|
||
Instead, we will display the Mandelbrot set extended in 3D using OpenGL and Haskell.
|
||
You will be able to move it using your keyboard.
|
||
This object is a Mandelbrot set in the plan (z=0),
|
||
and something nice to see in 3D.</p>
|
||
|
||
<p>Here are some screenshots of the result:</p>
|
||
|
||
<figure><img alt="The entire Mandelbulb" src="/Scratch/img/blog/Haskell-OpenGL-Mandelbrot/GoldenMandelbulb.png" /><figcaption>The entire Mandelbulb</figcaption></figure>
|
||
<figure><img alt="A Mandelbulb detail" src="/Scratch/img/blog/Haskell-OpenGL-Mandelbrot/3DMandelbulbDetail.png" /><figcaption>A Mandelbulb detail</figcaption></figure>
|
||
<figure><img alt="Another detail of the Mandelbulb" src="/Scratch/img/blog/Haskell-OpenGL-Mandelbrot/3DMandelbulbDetail2.png" /><figcaption>Another detail of the Mandelbulb</figcaption></figure>
|
||
|
||
<p>And you can see the intermediate steps to reach this goal:</p>
|
||
|
||
<p><img alt="The parts of the article" src="/Scratch/img/blog/Haskell-OpenGL-Mandelbrot/HGL_Plan.png" /></p>
|
||
|
||
<p>From the 2<sup>nd</sup> section to the 4<sup>th</sup> it will be <em>dirtier</em> and <em>dirtier</em>.
|
||
We start cleaning the code at the 5<sup>th</sup> section.</p>
|
||
|
||
<hr />
|
||
<p><a href="code/01_Introduction/hglmandel.lhs" class="cut">Download the source code of this section → 01_Introduction/<strong>hglmandel.lhs</strong></a></p>
|
||
|
||
<h2 id="first-version">First version</h2>
|
||
|
||
<p>We can consider two parts.
|
||
The first being mostly some boilerplate<sup id="fnref:011"><a href="#fn:011" rel="footnote">2</a></sup>.
|
||
And the second part more focused on OpenGL and content.</p>
|
||
|
||
<h3 id="lets-play-the-song-of-our-people">Let’s play the song of our people</h3>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">import Graphics.Rendering.OpenGL
|
||
import Graphics.UI.GLUT
|
||
import Data.IORef
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>For efficiency reason, I will not use the default Haskell <code>Complex</code> data type.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">data Complex = C (Float,Float) deriving (Show,Eq)
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">instance Num Complex where
|
||
fromInteger n = C (fromIntegral n,0.0)
|
||
C (x,y) * C (z,t) = C (z*x - y*t, y*z + x*t)
|
||
C (x,y) + C (z,t) = C (x+z, y+t)
|
||
abs (C (x,y)) = C (sqrt (x*x + y*y),0.0)
|
||
signum (C (x,y)) = C (signum x , 0.0)
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>We declare some useful functions for manipulating complex numbers:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">complex :: Float -> Float -> Complex
|
||
complex x y = C (x,y)
|
||
|
||
real :: Complex -> Float
|
||
real (C (x,y)) = x
|
||
|
||
im :: Complex -> Float
|
||
im (C (x,y)) = y
|
||
|
||
magnitude :: Complex -> Float
|
||
magnitude = real.abs
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<h3 id="let-us-start">Let us start</h3>
|
||
|
||
<p>We start by giving the main architecture of our program:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">main :: IO ()
|
||
main = do
|
||
-- GLUT need to be initialized
|
||
(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"
|
||
-- Each time we will need to update the display
|
||
-- we will call the function 'display'
|
||
displayCallback $= display
|
||
-- We enter the main loop
|
||
mainLoop
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>Mainly, we initialize our OpenGL application.
|
||
We declared that the function <code>display</code> will be used to render the graphics:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">display = do
|
||
clear [ColorBuffer] -- make the window black
|
||
loadIdentity -- reset any transformation
|
||
preservingMatrix drawMandelbrot
|
||
swapBuffers -- refresh screen
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>Also here, there is only one interesting line;
|
||
the draw will occur in the function <code>drawMandelbrot</code>.</p>
|
||
|
||
<p>This function will provide a list of draw actions.
|
||
Remember that OpenGL is imperative by design.
|
||
Then, one of the consequence is you must write the actions in the right order.
|
||
No easy parallel drawing here.
|
||
Here is the function which will render something on the screen:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">drawMandelbrot =
|
||
-- We will print Points (not triangles for example)
|
||
renderPrimitive Points $ do
|
||
mapM_ drawColoredPoint allPoints
|
||
where
|
||
drawColoredPoint (x,y,c) = do
|
||
color c -- set the current color to c
|
||
-- then draw the point at position (x,y,0)
|
||
-- remember we're in 3D
|
||
vertex $ Vertex3 x y 0
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>The <code>mapM_</code> function is mainly the same as map but inside a monadic context.
|
||
More precisely, this can be transformed as a list of actions where the order is important:</p>
|
||
|
||
<pre><code>drawMandelbrot =
|
||
renderPrimitive Points $ do
|
||
color color1
|
||
vertex $ Vertex3 x1 y1 0
|
||
...
|
||
color colorN
|
||
vertex $ Vertex3 xN yN 0
|
||
</code></pre>
|
||
|
||
<p>We also need some kind of global variables.
|
||
In fact, global variable are a proof of a design problem.
|
||
We will get rid of them later.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">width = 320 :: GLfloat
|
||
height = 320 :: GLfloat
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>And of course our list of colored points.
|
||
In OpenGL the default coordinate are from -1 to 1.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">allPoints :: [(GLfloat,GLfloat,Color3 GLfloat)]
|
||
allPoints = [ (x/width,y/height,colorFromValue $ mandel x y) |
|
||
x <- [-width..width],
|
||
y <- [-height..height]]
|
||
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>We need a function which transform an integer value to some color:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>And now the <code>mandel</code> function.
|
||
Given two coordinates in pixels, it returns some integer value:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>It uses the main Mandelbrot function for each complex \(c\).
|
||
The Mandelbrot set is the set of complex number \(c\) such that the following sequence does not escape to infinity.</p>
|
||
|
||
<p>Let us define \(f_c: \mathbb{C} \to \mathbb{C}\)</p>
|
||
|
||
<script type="math/tex; mode=display"> f_c(z) = z^2 + c </script>
|
||
|
||
<p>The sequence is: </p>
|
||
|
||
<script type="math/tex; mode=display"> 0 \rightarrow f_c(0) \rightarrow f_c(f_c(0)) \rightarrow \cdots \rightarrow f^n_c(0) \rightarrow \cdots </script>
|
||
|
||
<p>Of course, instead of trying to test the real limit, we just make a test after a finite number of occurrences.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>Well, if you download this file (look at the bottom of this section), compile it and run it this is the result:</p>
|
||
|
||
<p><img alt="The mandelbrot set version 1" src="/Scratch/img/blog/Haskell-OpenGL-Mandelbrot/hglmandel_v01.png" /></p>
|
||
|
||
<p>A first very interesting property of this program is that the computation for all the points is done only once.
|
||
It is a bit long before the first image appears, but if you resize the window, it updates instantaneously.
|
||
This property is a direct consequence of purity.
|
||
If you look closely, you see that <code>allPoints</code> is a pure list.
|
||
Therefore, calling <code>allPoints</code> will always render the same result and Haskell is clever enough to use this property.
|
||
While Haskell doesn’t garbage collect <code>allPoints</code> the result is reused for free.
|
||
We did not specified this value should be saved for later use.
|
||
It is saved for us.</p>
|
||
|
||
<p>See what occurs if we make the window bigger:</p>
|
||
|
||
<p><img alt="The mandelbrot too wide, black lines and columns" src="/Scratch/img/blog/Haskell-OpenGL-Mandelbrot/hglmandel_v01_too_wide.png" /></p>
|
||
|
||
<p>We see some black lines because we have drawn less point than there is on the surface.
|
||
We can repair this by drawing little squares instead of just points.
|
||
But, instead we will do something a bit different and unusual.</p>
|
||
|
||
<p><a href="code/01_Introduction/hglmandel.lhs" class="cut">Download the source code of this section → 01_Introduction/<strong>hglmandel.lhs</strong> </a></p>
|
||
|
||
<hr />
|
||
<p><a href="code/02_Edges/HGLMandelEdge.lhs" class="cut">Download the source code of this section → 02_Edges/<strong>HGLMandelEdge.lhs</strong></a></p>
|
||
|
||
<h2 id="only-the-edges">Only the edges</h2>
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">import Graphics.Rendering.OpenGL
|
||
import Graphics.UI.GLUT
|
||
import Data.IORef
|
||
data Complex = C (Float,Float) deriving (Show,Eq)
|
||
instance Num Complex where
|
||
fromInteger n = C (fromIntegral n,0.0)
|
||
C (x,y) * C (z,t) = C (z*x - y*t, y*z + x*t)
|
||
C (x,y) + C (z,t) = C (x+z, y+t)
|
||
abs (C (x,y)) = C (sqrt (x*x + y*y),0.0)
|
||
signum (C (x,y)) = C (signum x , 0.0)
|
||
complex :: Float -> Float -> Complex
|
||
complex x y = C (x,y)
|
||
|
||
real :: Complex -> Float
|
||
real (C (x,y)) = x
|
||
|
||
im :: Complex -> Float
|
||
im (C (x,y)) = y
|
||
|
||
magnitude :: Complex -> Float
|
||
magnitude = real.abs
|
||
main :: IO ()
|
||
main = do
|
||
-- GLUT need to be initialized
|
||
(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"
|
||
-- Each time we will need to update the display
|
||
-- we will call the function 'display'
|
||
displayCallback $= display
|
||
-- We enter the main loop
|
||
mainLoop
|
||
display = do
|
||
-- set the background color (dark solarized theme)
|
||
clearColor $= Color4 0 0.1686 0.2117 1
|
||
clear [ColorBuffer] -- make the window black
|
||
loadIdentity -- reset any transformation
|
||
preservingMatrix drawMandelbrot
|
||
swapBuffers -- refresh screen
|
||
|
||
width = 320 :: GLfloat
|
||
height = 320 :: GLfloat
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<p>This time, instead of drawing all points,
|
||
we will simply draw the edges of the Mandelbrot set.
|
||
The method I use is a rough approximation.
|
||
I consider the Mandelbrot set to be almost convex.
|
||
The result will be good enough for the purpose of this tutorial.</p>
|
||
|
||
<p>We change slightly the <code>drawMandelbrot</code> function.
|
||
We replace the <code>Points</code> by <code>LineLoop</code></p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">drawMandelbrot =
|
||
-- We will print Points (not triangles for example)
|
||
renderPrimitive LineLoop $ do
|
||
mapM_ drawColoredPoint allPoints
|
||
where
|
||
drawColoredPoint (x,y,c) = do
|
||
color c -- set the current color to c
|
||
-- then draw the point at position (x,y,0)
|
||
-- remember we're in 3D
|
||
vertex $ Vertex3 x y 0
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>And now, we should change our list of points.
|
||
Instead of drawing every point of the visible surface,
|
||
we will choose only point on the surface.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">allPoints = positivePoints ++
|
||
map (\(x,y,c) -> (x,-y,c)) (reverse positivePoints)
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>We only need to compute the positive point.
|
||
The Mandelbrot set is symmetric relatively to the abscisse axis.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">positivePoints :: [(GLfloat,GLfloat,Color3 GLfloat)]
|
||
positivePoints = do
|
||
x <- [-width..width]
|
||
let y = maxZeroIndex (mandel x) 0 height (log2 height)
|
||
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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>This function is interesting.
|
||
For those not used to the list monad here is a natural language version of this function:</p>
|
||
|
||
<pre><code class="no-highlight">positivePoints =
|
||
for all x in the range [-width..width]
|
||
let y be smallest number s.t. mandel x y > 0
|
||
if y is on 0 then don't return a point
|
||
else return the value corresonding to (x,y,color for (x+iy))
|
||
</code></pre>
|
||
|
||
<p>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 <code>mandel x y > 0</code> we use a simple dichotomy:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>No rocket science here. See the result now:</p>
|
||
|
||
<p><img alt="The edges of the mandelbrot set" src="/Scratch/img/blog/Haskell-OpenGL-Mandelbrot/HGLMandelEdges.png" /></p>
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<p><a href="code/02_Edges/HGLMandelEdge.lhs" class="cut">Download the source code of this section → 02_Edges/<strong>HGLMandelEdge.lhs</strong> </a></p>
|
||
|
||
<hr />
|
||
<p><a href="code/03_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 03_Mandelbulb/<strong>Mandelbulb.lhs</strong></a></p>
|
||
|
||
<h2 id="d-mandelbrot">3D Mandelbrot?</h2>
|
||
|
||
<p>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.</p>
|
||
|
||
<p>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:</p>
|
||
|
||
<blockquote>
|
||
<ul>
|
||
<li>
|
||
<p>OpenGL Boilerplate</p>
|
||
|
||
<ul>
|
||
<li>set some IORef (understand variables) for states </li>
|
||
<li>
|
||
<p>Drawing: </p>
|
||
|
||
<ul>
|
||
<li>set doubleBuffer, handle depth, window size…</li>
|
||
<li>Use state to apply some transformations</li>
|
||
</ul>
|
||
</li>
|
||
<li>Keyboard: hitting some key change the state of IORef</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>Generate 3D Object</p>
|
||
|
||
<pre><code>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
|
||
</code></pre>
|
||
</li>
|
||
</ul>
|
||
</blockquote>
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">import Graphics.Rendering.OpenGL
|
||
import Graphics.UI.GLUT
|
||
import Data.IORef
|
||
type ColoredPoint = (GLfloat,GLfloat,GLfloat,Color3 GLfloat)
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<p>We declare a new type <code>ExtComplex</code> (for extended complex).
|
||
An extension of complex numbers with a third component:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>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 <em>strange</em>.</p>
|
||
|
||
<script type="math/tex; mode=display"> \mathrm{real} ((x,y,z) * (x',y',z')) = xx' - yy' - zz' </script>
|
||
|
||
<script type="math/tex; mode=display"> \mathrm{im} ((x,y,z) * (x',y',z')) = xy' - yx' + zz' </script>
|
||
|
||
<script type="math/tex; mode=display"> \mathrm{strange} ((x,y,z) * (x',y',z')) = xz' + zx' </script>
|
||
|
||
<p>Note how if <code>z=z'=0</code> then the multiplication is the same to the complex one.</p>
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<h3 id="from-2d-to-3d">From 2D to 3D</h3>
|
||
|
||
<p>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.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>The <code>idle</code> is here to change the states.
|
||
There should never be any modification done in the <code>display</code> function.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">idle = postRedisplay Nothing
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>We introduce some helper function to manipulate
|
||
standard <code>IORef</code>.
|
||
Mainly <code>modVar x f</code> is equivalent to the imperative <code>x:=f(x)</code>,
|
||
<code>modFst (x,y) (+1)</code> is equivalent to <code>(x,y) := (x+1,y)</code>
|
||
and <code>modSnd (x,y) (+1)</code> is equivalent to <code>(x,y) := (x,y+1)</code></p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>And we use them to code the function handling keyboard.
|
||
We will use the keys <code>hjkl</code> to rotate,
|
||
<code>oi</code> to zoom and <code>sedf</code> to move.
|
||
Also, hitting space will reset the view.
|
||
Remember that <code>angle</code> and <code>campos</code> are pairs and <code>zoom</code> is a scalar.
|
||
Also note <code>(+0.5)</code> is the function <code>\x->x+0.5</code>
|
||
and <code>(-0.5)</code> is the number <code>-0.5</code> (yes I share your pain).</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>Note <code>display</code> takes some parameters this time.
|
||
This function if full of boilerplate:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>Not much to say about this function.
|
||
Mainly there are two parts: apply some transformations, draw the object.</p>
|
||
|
||
<h3 id="the-3d-mandelbrot">The 3D Mandelbrot</h3>
|
||
|
||
<p>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.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">nbDetails = 200 :: GLfloat
|
||
width = nbDetails
|
||
height = nbDetails
|
||
deep = nbDetails
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>This time, instead of just drawing some line or some group of points,
|
||
we will show triangles.
|
||
The function <code>allPoints</code> will provide a multiple of three points.
|
||
Each three successive point representing the coordinate of each vertex of a triangle.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>In fact, we will provide six ordered points.
|
||
These points will be used to draw two triangles.</p>
|
||
|
||
<p><img alt="Explain triangles" src="/Scratch/img/blog/Haskell-OpenGL-Mandelbrot/triangles.png" /></p>
|
||
|
||
<p>The next function is a bit long.
|
||
Here is an approximative English version:</p>
|
||
|
||
<pre><code>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)
|
||
</code></pre>
|
||
|
||
<p>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.</p>
|
||
|
||
<pre><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></pre>
|
||
|
||
<p>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:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>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.</p>
|
||
|
||
<p>Also, we didn’t searched for negative values.
|
||
This modified Mandelbrot is no more symmetric relatively to the plan <code>y=0</code>.
|
||
But it is symmetric relatively to the plan <code>z=0</code>.
|
||
Then I mirror these values. </p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>The rest of the program is very close to the preceding one.</p>
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
I made the color slightly brighter
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
We only changed from `Complex` to `ExtComplex` of the main `f` function.
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<p>We simply add a new dimension to the <code>mandel</code> function
|
||
and change the type signature of <code>f</code> from <code>Complex</code> to <code>ExtComplex</code>.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>Here is the result:</p>
|
||
|
||
<p><img alt="A 3D mandelbrot like" src="/Scratch/img/blog/Haskell-OpenGL-Mandelbrot/mandelbrot_3D.png" /></p>
|
||
|
||
<p><a href="code/03_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 03_Mandelbulb/<strong>Mandelbulb.lhs</strong> </a></p>
|
||
|
||
<hr />
|
||
<p><a href="code/04_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 04_Mandelbulb/<strong>Mandelbulb.lhs</strong></a></p>
|
||
|
||
<h2 id="nave-code-cleaning">Naïve code cleaning</h2>
|
||
|
||
<p>The first approach to clean the code 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.</p>
|
||
|
||
<ul>
|
||
<li><a href="code/04_Mandelbulb/YBoiler.hs"><code>YBoiler.hs</code></a>, the 3D rendering</li>
|
||
<li><a href="code/04_Mandelbulb/Mandel.hs"><code>Mandel</code></a>, the mandel function</li>
|
||
<li><a href="code/04_Mandelbulb/ExtComplex.hs"><code>ExtComplex</code></a>, the extended complexes</li>
|
||
</ul>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">import YBoiler -- Most the OpenGL Boilerplate
|
||
import Mandel -- The 3D Mandelbrot maths
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>The <code>yMainLoop</code> takes two arguments:
|
||
the title of the window
|
||
and a function from time to triangles</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">main :: IO ()
|
||
main = yMainLoop "3D Mandelbrot" (\_ -> allPoints)
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>We set some global constant (this is generally bad).</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">nbDetails = 200 :: GLfloat
|
||
width = nbDetails
|
||
height = nbDetails
|
||
deep = nbDetails
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>We then generate colored points from our function.
|
||
This is similar to the preceding section.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>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.</p>
|
||
|
||
<p>On the other hand, we continue to handle a lot rendering details.
|
||
For example, we provide ordered vertices.</p>
|
||
|
||
<p><a href="code/04_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 04_Mandelbulb/<strong>Mandelbulb.lhs</strong> </a></p>
|
||
|
||
<hr />
|
||
<p><a href="code/05_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 05_Mandelbulb/<strong>Mandelbulb.lhs</strong></a></p>
|
||
|
||
<h2 id="functional-organization">Functional organization?</h2>
|
||
|
||
<p>Some points:</p>
|
||
|
||
<ol>
|
||
<li>
|
||
<p>OpenGL and GLUT is done in C.
|
||
In particular the <code>mainLoop</code> 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: </p>
|
||
|
||
<ul>
|
||
<li>create our own <code>mainLoop</code> function to make it more functional.</li>
|
||
<li>deal with the imperative nature of the GLUT <code>mainLoop</code> function.</li>
|
||
</ul>
|
||
|
||
<p>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 <code>mainLoop</code> function.</p>
|
||
</li>
|
||
<li>
|
||
<p>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 <em>functional reactive programming</em> (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.</p>
|
||
</li>
|
||
</ol>
|
||
|
||
<p>Here is how I imagine things should go.
|
||
First, what the main loop should look like if we could make our own:</p>
|
||
|
||
<pre><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></pre>
|
||
|
||
<p>Clearly, ideally we should provide only three parameters to this main loop function:</p>
|
||
|
||
<ul>
|
||
<li>an initial World state</li>
|
||
<li>a mapping between the user interactions and functions which modify the world</li>
|
||
<li>a function taking two parameters: time and world state and render a new world without user interaction.</li>
|
||
</ul>
|
||
|
||
<p>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.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">import YGL -- Most the OpenGL Boilerplate
|
||
import Mandel -- The 3D Mandelbrot maths
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>We first set the mapping between user input and actions.
|
||
The type of each couple should be of the form
|
||
<code>(user input, f)</code> where (in a first time) <code>f:World -> World</code>.
|
||
It means, the user input will transform the world state.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>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.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>The important part to glue our own type to the framework
|
||
is to make our type an instance of the type class <code>DisplayableWorld</code>.
|
||
We simply have to provide the definition of some functions.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>The <code>camera</code> function will retrieve an object of type <code>Camera</code> which contains
|
||
most necessary information to set our camera.
|
||
The <code>objects</code> function will returns a list of objects.
|
||
Their type is <code>YObject</code>. Note the generation of triangles is no more in this file.
|
||
Until here we only used declarative pattern.</p>
|
||
|
||
<p>We also need to set all our transformation functions.
|
||
These function are used to update the world state.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>Note <code>(-*<)</code> is the scalar product (<code>α -*< (x,y,z) = (αx,αy,αz)</code>).
|
||
Also note we could add two Point3D. </p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>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.</p>
|
||
|
||
<p>The <code>yMainLoop</code> takes three arguments.</p>
|
||
|
||
<ul>
|
||
<li>A map between user Input and world transformation</li>
|
||
<li>A timed world transformation</li>
|
||
<li>An initial world state</li>
|
||
</ul>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">main :: IO ()
|
||
main = yMainLoop inputActionMap idleAction initialWorld
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>Here is our initial world state.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>We will define <code>shapeFunc</code> later.
|
||
Here is the function which transform the world even without user action.
|
||
Mainly it makes some rotation.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>Now the function which will generate points in 3D.
|
||
The first parameter (<code>res</code>) is the resolution of the vertex generation.
|
||
More precisely, <code>res</code> is distance between two points on one direction.
|
||
We need it to “close” our shape.</p>
|
||
|
||
<p>The type <code>Function3D</code> is <code>Point -> Point -> Maybe Point</code>.
|
||
Because we consider partial functions
|
||
(for some <code>(x,y)</code> our function can be undefined).</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>With the color function.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>The rest is similar to the preceding sections.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>I won’t explain how the magic occurs here.
|
||
If you are interested, just read the file <a href="code/05_Mandelbulb/YGL.hs"><code>YGL.hs</code></a>.
|
||
It is commented a lot.</p>
|
||
|
||
<ul>
|
||
<li><a href="code/05_Mandelbulb/YGL.hs"><code>YGL.hs</code></a>, the 3D rendering framework</li>
|
||
<li><a href="code/05_Mandelbulb/Mandel.hs"><code>Mandel</code></a>, the mandel function</li>
|
||
<li><a href="code/05_Mandelbulb/ExtComplex.hs"><code>ExtComplex</code></a>, the extended complexes</li>
|
||
</ul>
|
||
|
||
<p><a href="code/05_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 05_Mandelbulb/<strong>Mandelbulb.lhs</strong> </a></p>
|
||
|
||
<hr />
|
||
<p><a href="code/06_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 06_Mandelbulb/<strong>Mandelbulb.lhs</strong></a></p>
|
||
|
||
<h2 id="optimization">Optimization</h2>
|
||
|
||
<p>Our code architecture feel very clean.
|
||
All the meaningful code is in our main file and all display details are
|
||
externalized.
|
||
If you read the code of <code>YGL.hs</code>, 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.
|
||
Unfortunately the program of the preceding session is extremely slow.
|
||
We compute the Mandelbulb for each frame now.</p>
|
||
|
||
<p>Before our program structure was:</p>
|
||
|
||
<pre><code class="no-highlight">Constant Function -> Constant List of Triangles -> Display
|
||
</code></pre>
|
||
|
||
<p>Now we have </p>
|
||
|
||
<pre><code class="no-highlight">Main loop -> World -> Function -> List of Objects -> Atoms -> Display
|
||
</code></pre>
|
||
|
||
<p>The World state could change.
|
||
The compiler can no more optimize the computation for us.
|
||
We have to manually explain when to redraw the shape.</p>
|
||
|
||
<p>To optimize we must do some things in a lower level.
|
||
Mostly the program remains the same,
|
||
but it will provide the list of atoms directly.</p>
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><code class="haskell">main :: IO ()
|
||
main = yMainLoop inputActionMap idleAction initialWorld
|
||
</code></pre>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<p>Our initial world state is slightly changed:</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>The use of <code>eps</code> is a hint to make a better zoom by computing with the right bounds.</p>
|
||
|
||
<p>We use the <code>YGL.getObject3DFromShapeFunction</code> function directly.
|
||
This way instead of providing <code>XYFunc</code>, we provide directly a list of Atoms.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>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.</p>
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
<p>All the rest is exactly the same.</p>
|
||
|
||
<div style="display:none">
|
||
|
||
<div class="codehighlight">
|
||
|
||
|
||
<pre><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></pre>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
|
||
<p>And you can also consider minor changes in the <code>YGL.hs</code> source file.</p>
|
||
|
||
<ul>
|
||
<li><a href="code/06_Mandelbulb/YGL.hs"><code>YGL.hs</code></a>, the 3D rendering framework</li>
|
||
<li><a href="code/06_Mandelbulb/Mandel.hs"><code>Mandel</code></a>, the mandel function</li>
|
||
<li><a href="code/06_Mandelbulb/ExtComplex.hs"><code>ExtComplex</code></a>, the extended complexes</li>
|
||
</ul>
|
||
|
||
<p><a href="code/06_Mandelbulb/Mandelbulb.lhs" class="cut">Download the source code of this section → 06_Mandelbulb/<strong>Mandelbulb.lhs</strong> </a></p>
|
||
|
||
<h2 id="conclusion">Conclusion</h2>
|
||
|
||
<p>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.</p>
|
||
|
||
<p>Once you are used to pure functional style,
|
||
it is hard not to see all advantages it offers.</p>
|
||
|
||
<p>The code in the two last sections is completely pure and functional.
|
||
Furthermore I don’t use <code>GLfloat</code>, <code>Color3</code> 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.</p>
|
||
|
||
<p>The <code>YGL</code> 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.</p>
|
||
|
||
<p>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.</p>
|
||
|
||
<p>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).</p>
|
||
<hr/><div class="footnotes">
|
||
<ol>
|
||
<li id="fn:001">
|
||
<p>Unfortunately, I couldn’t make this program to work on my Mac. More precisely, I couldn’t make the <a href="http://openil.sourceforge.net/">DevIL</a> library work on Mac to output the image. Yes I have done a <code>brew install libdevil</code>. But even a minimal program who simply write some <code>jpg</code> didn’t worked. I tried both with <code>Haskell</code> and <code>C</code>.<a href="#fnref:001" rel="reference">↩</a></p>
|
||
</li>
|
||
<li id="fn:011">
|
||
<p>Generally in Haskell you need to declare a lot of import lines.
|
||
This is something I find annoying.
|
||
In particular, it should be possible to create a special file, Import.hs
|
||
which make all the necessary import for you, as you generally need them all.
|
||
I understand why this is cleaner to force the programmer not to do so,
|
||
but, each time I do a copy/paste, I feel something is wrong.
|
||
I believe this concern can be generalized to the lack of namespace in Haskell.<a href="#fnref:011" rel="reference">↩</a></p>
|
||
</li>
|
||
</ol>
|
||
</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
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