四元数旋转有一个奇怪的行为（Haskell OpenGL）
Quaternions rotation has a weird behaviour (Haskell OpenGL)
我一直在关注 Haskell OpenGL tutorial。
3D 中的旋转 space 引起了我的兴趣，所以我开始学习欧拉角，最后学习四元数。
我想使用四元数实现我自己的函数来执行旋转（在立方体上），我基于这两篇论文：mostly this one and this one.
当我只在一个轴上执行旋转时，我的函数工作正常，但是当我在 X 和 Y 上执行旋转时，立方体开始随机前进并在旋转时被“阻挡”。
Video of the cube performing rotation on XY.
当我设置三个轴（X、Y、Z）时，它会放大得更多（但没有那个奇怪的阻挡物）：video。
这是我的程序代码：
这里是创建一个window的主文件，设置idle函数并在屏幕上输出旋转角度A的结果，其中A在每帧增加0.05。
module Main (main) where
import Core
import Utils
import Data.IORef
import Graphics.UI.GLUT
import Graphics.Rendering.OpenGL
main :: IO ()
main = do
createAWindow "177013"
mainLoop
createAWindow :: [Char] -> IO ()
createAWindow windowName = do
(procName, _args) <- getArgsAndInitialize
createWindow windowName
initialDisplayMode $= [DoubleBuffered]
angle <- newIORef 0.0
delta <- newIORef 0.05
displayCallback $= (start angle)
reshapeCallback $= Just reshape
keyboardMouseCallback $= Just keyboardMouse
idleCallback $= Just (idle angle delta)
reshape :: ReshapeCallback
reshape size = do
viewport $= (Position 0 0, size)
postRedisplay Nothing
keyboardMouse :: KeyboardMouseCallback
keyboardMouse _ _ _ _ = return ()
idle :: IORef GLfloat -> IORef GLfloat -> IdleCallback
idle angle delta = do
d <- get delta
a <- get angle
angle $~! (+d)
postRedisplay Nothing
start :: IORef GLfloat -> DisplayCallback
start angle = do
clear [ColorBuffer]
loadIdentity
a <- get angle
let c = rotate3f (0, 0, 0) [X,Y,Z] a $ cube3f 0.2 -- here I'm rotating on X, Y and Z axis
draw3f Quads c CCyan
flush
swapBuffers
where
这是定义旋转函数的核心文件（还有一些其他的）。我添加了一些评论，因为它可能是一些低质量的 haskell 代码。
module Core (draw3f, vertex3f, rotate3f, translate3f, rotate3d, Colors(..), Axes(..)) where
import Control.Lens
import Graphics.Rendering.OpenGL
data Axes = X | Y | Z
deriving Eq
data Colors = CRed | CGreen | CBlue | CYellow | CWhite | CMagenta | CCyan | CBlack | CNone | CPreset
deriving Eq
rotate3f :: (GLfloat, GLfloat, GLfloat) -> [Axes] -> GLfloat -> [(GLfloat, GLfloat, GLfloat)] -> [(GLfloat, GLfloat, GLfloat)]
rotate3f _ _ _ [] = []
rotate3f _ [] _ _ = []
rotate3f o axes a p = let p' = translate3f p u -- translation if I don't want to rotate it by the origin
q = cos a' : ((\x -> if x `elem` axes then sin a' else 0) <$> [X,Y,Z]) -- if the axe is set then its related component is equal to sin theta/2, otherwise it will be 0
q' = q !! 0 : (negate <$> (tail q)) -- quaternion inversion
in translate3f ((rotate q q') <$> p') [(0,0,0),o] -- rotate and translate again to put the object where it belongs
where
a' = (a * (pi / 180)) / 2 -- convert to radians and divide by 2 as all q components takes theta/2
u :: [(GLfloat, GLfloat, GLfloat)]
u = [o,(0,0,0)]
rotate :: [GLfloat] -> [GLfloat] -> (GLfloat, GLfloat, GLfloat) -> (GLfloat, GLfloat, GLfloat)
rotate q q' (x,y,z) = let p = [0,x,y,z]
qmul q1 q2 = [(q1 !! 0) * (q2 !! 0) - (q1 !! 1) * (q2 !! 1) - (q1 !! 2) * (q2 !! 2) - (q1 !! 3) * (q2 !! 3),
(q1 !! 0) * (q2 !! 1) + (q1 !! 1) * (q2 !! 0) + (q1 !! 2) * (q2 !! 3) - (q1 !! 3) * (q2 !! 2),
(q1 !! 0) * (q2 !! 2) - (q1 !! 1) * (q2 !! 3) + (q1 !! 2) * (q2 !! 0) + (q1 !! 3) * (q2 !! 1),
(q1 !! 0) * (q2 !! 3) + (q1 !! 1) * (q2 !! 2) - (q1 !! 2) * (q2 !! 1) + (q1 !! 3) * (q2 !! 0)]
p' = qmul (qmul q p) q'
in (p' !! 1, p' !! 2, p' !! 3)
translate3f :: [(GLfloat, GLfloat, GLfloat)] -> [(GLfloat, GLfloat, GLfloat)] -> [(GLfloat, GLfloat, GLfloat)]
translate3f p [(ax,ay,az),(bx,by,bz)] = map (\(x,y,z) -> (x + (bx - ax), y + (by - ay), z + (bz - az))) p
draw3f :: PrimitiveMode -> [(GLfloat, GLfloat, GLfloat)] -> Colors -> IO()
draw3f shape points color = renderPrimitive shape $ mapM_ (\(x,y,z) -> vertex3f x y z color) points
vertex3f :: GLfloat -> GLfloat -> GLfloat -> Colors -> IO()
vertex3f x y z c = do
if c /= CPreset
then color $ Color3 (c' ^. _1) (c' ^. _2) ((c' ^. _3) :: GLfloat)
else return ()
vertex $ Vertex3 x y z
where
c' :: (GLfloat, GLfloat, GLfloat)
c' = case c of CRed -> (1,0,0)
CGreen -> (0,1,0)
CBlue -> (0,0,1)
CYellow -> (1,1,0)
CMagenta -> (1,0,1)
CCyan -> (0,1,1)
CBlack -> (0,0,0)
_ -> (1,1,1)
这是 utils 文件，其中只有立方体的定义，来自 Haskell OpenGL 教程
module Utils (cube3f) where
import Core
import Graphics.UI.GLUT
import Graphics.Rendering.OpenGL
cube3f :: GLfloat -> [(GLfloat, GLfloat, GLfloat)]
cube3f w = [( w, w, w), ( w, w,-w), ( w,-w,-w), ( w,-w, w),
( w, w, w), ( w, w,-w), (-w, w,-w), (-w, w, w),
( w, w, w), ( w,-w, w), (-w,-w, w), (-w, w, w),
(-w, w, w), (-w, w,-w), (-w,-w,-w), (-w,-w, w),
( w,-w, w), ( w,-w,-w), (-w,-w,-w), (-w,-w, w),
( w, w,-w), ( w,-w,-w), (-w,-w,-w), (-w, w,-w)]
最后，如果它可以帮助人们查看我的算法是否存在问题，这里有一些使用我的函数的旋转示例：
点 (1, 2, 3) 在 X 轴上围绕点 (0, 0, 0)（原点）旋转 90° 给出：(0.99999994,-3.0,2.0)
相同的旋转但在 X 轴和 Y 轴上给出：(5.4999995,-0.99999994,-0.49999988)
再次旋转相同但在 X、Y 和 Z 轴上给出：(5.9999995,1.9999999,3.9999995)
你指向的关于四元数旋转的第二篇论文有这句话：
“(x̂, ŷ, ẑ) 是定义旋转轴的单位向量。”.
所以四元数必须归一化，分量的平方和等于1
因此，例如，如果您涉及所有 3 个轴，则它必须是 (cos θ/2, r3sin θ/2, r3sin θ/2, r3* sin θ/2) 其中 r3 是 3 的平方根的倒数。这就是我如何解释你在 post 末尾提到的旋转结果在多个轴时无法保持向量的长度参与其中。
因此，关键部分是函数中的这一行 rotate3f:
q = cos a' : ((\x -> if x `elem` axes then sin a' else 0) <$> [X,Y,Z])
缺少归一化因子。
您的代码提供了许多提高可读性的机会。您可以考虑使用 CodeReview 了解更多详情。
一个主要问题是源代码行太宽。如果 reader 必须使用水平滑块，那么理解代码和查找错误就会困难得多。下面，我会尽量避免超过 80 个字符的宽度。
首先，我们需要一些四元数基础设施：
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE ExplicitForAll #-}
type GLfloat = Float
type GLfloatV3 = (GLfloat, GLfloat, GLfloat)
type QuatFloat = [GLfloat]
data Axes = X | Y | Z deriving Eq
qmul :: QuatFloat -> QuatFloat -> QuatFloat
qmul [qa0, qa1, qa2, qa3] [qb0, qb1, qb2, qb3] =
[
qa0*qb0 - qa1*qb1 - qa2*qb2 - qa3*qb3 ,
qa0*qb1 + qa1*qb0 + qa2*qb3 - qa3*qb2 ,
qa0*qb2 - qa1*qb3 + qa2*qb0 + qa3*qb1 ,
qa0*qb3 + qa1*qb2 - qa2*qb1 + qa3*qb0
]
qmul _ _ = error "Quaternion length differs from 4"
qconj :: QuatFloat -> QuatFloat
qconj q = (head q) : (map negate (tail q)) -- q-conjugation
rotate :: [GLfloat] -> [GLfloat] -> GLfloatV3 -> GLfloatV3
rotate q q' (x,y,z) = let p = [0, x,y,z]
[q0,q1,q2,q3] = qmul (qmul q p) q'
in (q1, q2, q3)
请注意，定义特殊类型的想法不仅可以减少代码宽度，还可以提供额外的灵活性。如果有一天您决定用比普通列表更有效的其他数据结构来表示四元数，则可以在保持客户端代码不变的情况下完成。
接下来，旋转代码正确。函数 rotQuat0 是您的初始算法，它再现了您问题末尾提到的数值结果。函数 rotQuat1 是给出 1-归一化四元数的修改版本。
-- original code:
rotQuat0 :: [Axes] -> GLfloat -> QuatFloat
rotQuat0 axes angle = let fn x = if (x `elem` axes) then (sin angle) else 0
in (cos angle) : (map fn [X,Y,Z])
-- modified code:
rotQuat1 :: [Axes] -> GLfloat -> QuatFloat
rotQuat1 axes angle = let corr = 1.0 / sqrt (fromIntegral (length axes))
fn x = if (x `elem` axes) then corr*(sin angle) else 0
in (cos angle) : (map fn [X,Y,Z])
代码使用 rotQuat1:
rotate3f :: GLfloatV3 -> [Axes] -> GLfloat -> [GLfloatV3] -> [GLfloatV3]
rotate3f _ _ _ [] = []
rotate3f _ [] _ _ = []
rotate3f org axes degθ pts =
let -- convert to radians and divide by 2, as all q components take θ/2
a' = (degθ * (pi / 180)) / 2
u :: [GLfloatV3]
u = [org, (0,0,0)]
-- translation if I don't want to rotate it by the origin
p' = translate3f pts u
-- if the axis is set, then its related component is
-- equal to sin θ/2, otherwise it will be zero
---- q = cos a' : ((\x -> if x `elem` axes then sin a' else 0) <$> [X,Y,Z])
q = rotQuat1 axes a' -- modified version
q' = qconj q
-- rotate and translate again to put the object where it belongs
in translate3f ((rotate q q') <$> p') [(0,0,0), org]
translate3f :: [GLfloatV3] -> [GLfloatV3] -> [GLfloatV3]
translate3f pts [(ax,ay,az), (bx,by,bz)] =
let dx = bx - ax
dy = by - ay
dz = bz - az
in map (\(x,y,z) -> (x + dx, y + dy, z + dz)) pts
测试代码：
sqNorm3 :: GLfloatV3 -> GLfloat
sqNorm3 (x,y,z) = x*x + y*y +z*z
printAsLines :: Show α => [α] -> IO ()
printAsLines xs = mapM_ (putStrLn . show) xs
main = do
let pt = (1,2,3) :: GLfloatV3
pt1 = rotate3f (0,0,0) [X] 90 [pt]
pt2 = rotate3f (0,0,0) [X,Y] 90 [pt]
pt3 = rotate3f (0,0,0) [X,Y,Z] 90 [pt]
pts = map head [pt1, pt2, pt3]
ptN = map sqNorm3 pts
printAsLines pts
putStrLn " "
printAsLines ptN
让我们检查函数 rotQuat1，初始 (1,2,3) 输入向量（即 1+4+9=13）的平方范数保持不变，适合适当的旋转:
$ ghc opengl00.hs -o ./opengl00.x && ./opengl00.x
[1 of 1] Compiling Main ( opengl00.hs, opengl00.o )
Linking ./opengl00.x ...
(0.99999994,-3.0,2.0)
(3.6213198,-0.62132025,0.70710695)
(2.5773501,0.84529924,2.5773501)
14.0
13.999995
13.999998
$
不幸的是，我没有足够的时间来安装 OpenGL 基础结构和重现动画。请让我们知道这是否解决了整个问题。
我一直在关注 Haskell OpenGL tutorial。 3D 中的旋转 space 引起了我的兴趣，所以我开始学习欧拉角，最后学习四元数。
我想使用四元数实现我自己的函数来执行旋转（在立方体上），我基于这两篇论文：mostly this one and this one.
当我只在一个轴上执行旋转时，我的函数工作正常，但是当我在 X 和 Y 上执行旋转时，立方体开始随机前进并在旋转时被“阻挡”。
Video of the cube performing rotation on XY.
当我设置三个轴（X、Y、Z）时，它会放大得更多（但没有那个奇怪的阻挡物）：video。
这是我的程序代码：
这里是创建一个window的主文件，设置idle函数并在屏幕上输出旋转角度A的结果，其中A在每帧增加0.05。
module Main (main) where
import Core
import Utils
import Data.IORef
import Graphics.UI.GLUT
import Graphics.Rendering.OpenGL
main :: IO ()
main = do
createAWindow "177013"
mainLoop
createAWindow :: [Char] -> IO ()
createAWindow windowName = do
(procName, _args) <- getArgsAndInitialize
createWindow windowName
initialDisplayMode $= [DoubleBuffered]
angle <- newIORef 0.0
delta <- newIORef 0.05
displayCallback $= (start angle)
reshapeCallback $= Just reshape
keyboardMouseCallback $= Just keyboardMouse
idleCallback $= Just (idle angle delta)
reshape :: ReshapeCallback
reshape size = do
viewport $= (Position 0 0, size)
postRedisplay Nothing
keyboardMouse :: KeyboardMouseCallback
keyboardMouse _ _ _ _ = return ()
idle :: IORef GLfloat -> IORef GLfloat -> IdleCallback
idle angle delta = do
d <- get delta
a <- get angle
angle $~! (+d)
postRedisplay Nothing
start :: IORef GLfloat -> DisplayCallback
start angle = do
clear [ColorBuffer]
loadIdentity
a <- get angle
let c = rotate3f (0, 0, 0) [X,Y,Z] a $ cube3f 0.2 -- here I'm rotating on X, Y and Z axis
draw3f Quads c CCyan
flush
swapBuffers
where
这是定义旋转函数的核心文件（还有一些其他的）。我添加了一些评论，因为它可能是一些低质量的 haskell 代码。
module Core (draw3f, vertex3f, rotate3f, translate3f, rotate3d, Colors(..), Axes(..)) where
import Control.Lens
import Graphics.Rendering.OpenGL
data Axes = X | Y | Z
deriving Eq
data Colors = CRed | CGreen | CBlue | CYellow | CWhite | CMagenta | CCyan | CBlack | CNone | CPreset
deriving Eq
rotate3f :: (GLfloat, GLfloat, GLfloat) -> [Axes] -> GLfloat -> [(GLfloat, GLfloat, GLfloat)] -> [(GLfloat, GLfloat, GLfloat)]
rotate3f _ _ _ [] = []
rotate3f _ [] _ _ = []
rotate3f o axes a p = let p' = translate3f p u -- translation if I don't want to rotate it by the origin
q = cos a' : ((\x -> if x `elem` axes then sin a' else 0) <$> [X,Y,Z]) -- if the axe is set then its related component is equal to sin theta/2, otherwise it will be 0
q' = q !! 0 : (negate <$> (tail q)) -- quaternion inversion
in translate3f ((rotate q q') <$> p') [(0,0,0),o] -- rotate and translate again to put the object where it belongs
where
a' = (a * (pi / 180)) / 2 -- convert to radians and divide by 2 as all q components takes theta/2
u :: [(GLfloat, GLfloat, GLfloat)]
u = [o,(0,0,0)]
rotate :: [GLfloat] -> [GLfloat] -> (GLfloat, GLfloat, GLfloat) -> (GLfloat, GLfloat, GLfloat)
rotate q q' (x,y,z) = let p = [0,x,y,z]
qmul q1 q2 = [(q1 !! 0) * (q2 !! 0) - (q1 !! 1) * (q2 !! 1) - (q1 !! 2) * (q2 !! 2) - (q1 !! 3) * (q2 !! 3),
(q1 !! 0) * (q2 !! 1) + (q1 !! 1) * (q2 !! 0) + (q1 !! 2) * (q2 !! 3) - (q1 !! 3) * (q2 !! 2),
(q1 !! 0) * (q2 !! 2) - (q1 !! 1) * (q2 !! 3) + (q1 !! 2) * (q2 !! 0) + (q1 !! 3) * (q2 !! 1),
(q1 !! 0) * (q2 !! 3) + (q1 !! 1) * (q2 !! 2) - (q1 !! 2) * (q2 !! 1) + (q1 !! 3) * (q2 !! 0)]
p' = qmul (qmul q p) q'
in (p' !! 1, p' !! 2, p' !! 3)
translate3f :: [(GLfloat, GLfloat, GLfloat)] -> [(GLfloat, GLfloat, GLfloat)] -> [(GLfloat, GLfloat, GLfloat)]
translate3f p [(ax,ay,az),(bx,by,bz)] = map (\(x,y,z) -> (x + (bx - ax), y + (by - ay), z + (bz - az))) p
draw3f :: PrimitiveMode -> [(GLfloat, GLfloat, GLfloat)] -> Colors -> IO()
draw3f shape points color = renderPrimitive shape $ mapM_ (\(x,y,z) -> vertex3f x y z color) points
vertex3f :: GLfloat -> GLfloat -> GLfloat -> Colors -> IO()
vertex3f x y z c = do
if c /= CPreset
then color $ Color3 (c' ^. _1) (c' ^. _2) ((c' ^. _3) :: GLfloat)
else return ()
vertex $ Vertex3 x y z
where
c' :: (GLfloat, GLfloat, GLfloat)
c' = case c of CRed -> (1,0,0)
CGreen -> (0,1,0)
CBlue -> (0,0,1)
CYellow -> (1,1,0)
CMagenta -> (1,0,1)
CCyan -> (0,1,1)
CBlack -> (0,0,0)
_ -> (1,1,1)
这是 utils 文件，其中只有立方体的定义，来自 Haskell OpenGL 教程
module Utils (cube3f) where
import Core
import Graphics.UI.GLUT
import Graphics.Rendering.OpenGL
cube3f :: GLfloat -> [(GLfloat, GLfloat, GLfloat)]
cube3f w = [( w, w, w), ( w, w,-w), ( w,-w,-w), ( w,-w, w),
( w, w, w), ( w, w,-w), (-w, w,-w), (-w, w, w),
( w, w, w), ( w,-w, w), (-w,-w, w), (-w, w, w),
(-w, w, w), (-w, w,-w), (-w,-w,-w), (-w,-w, w),
( w,-w, w), ( w,-w,-w), (-w,-w,-w), (-w,-w, w),
( w, w,-w), ( w,-w,-w), (-w,-w,-w), (-w, w,-w)]
最后，如果它可以帮助人们查看我的算法是否存在问题，这里有一些使用我的函数的旋转示例：
点 (1, 2, 3) 在 X 轴上围绕点 (0, 0, 0)（原点）旋转 90° 给出：(0.99999994,-3.0,2.0)
相同的旋转但在 X 轴和 Y 轴上给出：(5.4999995,-0.99999994,-0.49999988)
再次旋转相同但在 X、Y 和 Z 轴上给出：(5.9999995,1.9999999,3.9999995)
你指向的关于四元数旋转的第二篇论文有这句话：
“(x̂, ŷ, ẑ) 是定义旋转轴的单位向量。”.
所以四元数必须归一化，分量的平方和等于1
因此，例如，如果您涉及所有 3 个轴，则它必须是 (cos θ/2, r3sin θ/2, r3sin θ/2, r3* sin θ/2) 其中 r3 是 3 的平方根的倒数。这就是我如何解释你在 post 末尾提到的旋转结果在多个轴时无法保持向量的长度参与其中。
因此，关键部分是函数中的这一行 rotate3f:
q = cos a' : ((\x -> if x `elem` axes then sin a' else 0) <$> [X,Y,Z])
缺少归一化因子。
您的代码提供了许多提高可读性的机会。您可以考虑使用 CodeReview 了解更多详情。
一个主要问题是源代码行太宽。如果 reader 必须使用水平滑块，那么理解代码和查找错误就会困难得多。下面，我会尽量避免超过 80 个字符的宽度。
首先，我们需要一些四元数基础设施：
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE ExplicitForAll #-}
type GLfloat = Float
type GLfloatV3 = (GLfloat, GLfloat, GLfloat)
type QuatFloat = [GLfloat]
data Axes = X | Y | Z deriving Eq
qmul :: QuatFloat -> QuatFloat -> QuatFloat
qmul [qa0, qa1, qa2, qa3] [qb0, qb1, qb2, qb3] =
[
qa0*qb0 - qa1*qb1 - qa2*qb2 - qa3*qb3 ,
qa0*qb1 + qa1*qb0 + qa2*qb3 - qa3*qb2 ,
qa0*qb2 - qa1*qb3 + qa2*qb0 + qa3*qb1 ,
qa0*qb3 + qa1*qb2 - qa2*qb1 + qa3*qb0
]
qmul _ _ = error "Quaternion length differs from 4"
qconj :: QuatFloat -> QuatFloat
qconj q = (head q) : (map negate (tail q)) -- q-conjugation
rotate :: [GLfloat] -> [GLfloat] -> GLfloatV3 -> GLfloatV3
rotate q q' (x,y,z) = let p = [0, x,y,z]
[q0,q1,q2,q3] = qmul (qmul q p) q'
in (q1, q2, q3)
请注意，定义特殊类型的想法不仅可以减少代码宽度，还可以提供额外的灵活性。如果有一天您决定用比普通列表更有效的其他数据结构来表示四元数，则可以在保持客户端代码不变的情况下完成。
接下来，旋转代码正确。函数 rotQuat0 是您的初始算法，它再现了您问题末尾提到的数值结果。函数 rotQuat1 是给出 1-归一化四元数的修改版本。
-- original code:
rotQuat0 :: [Axes] -> GLfloat -> QuatFloat
rotQuat0 axes angle = let fn x = if (x `elem` axes) then (sin angle) else 0
in (cos angle) : (map fn [X,Y,Z])
-- modified code:
rotQuat1 :: [Axes] -> GLfloat -> QuatFloat
rotQuat1 axes angle = let corr = 1.0 / sqrt (fromIntegral (length axes))
fn x = if (x `elem` axes) then corr*(sin angle) else 0
in (cos angle) : (map fn [X,Y,Z])
代码使用 rotQuat1:
rotate3f :: GLfloatV3 -> [Axes] -> GLfloat -> [GLfloatV3] -> [GLfloatV3]
rotate3f _ _ _ [] = []
rotate3f _ [] _ _ = []
rotate3f org axes degθ pts =
let -- convert to radians and divide by 2, as all q components take θ/2
a' = (degθ * (pi / 180)) / 2
u :: [GLfloatV3]
u = [org, (0,0,0)]
-- translation if I don't want to rotate it by the origin
p' = translate3f pts u
-- if the axis is set, then its related component is
-- equal to sin θ/2, otherwise it will be zero
---- q = cos a' : ((\x -> if x `elem` axes then sin a' else 0) <$> [X,Y,Z])
q = rotQuat1 axes a' -- modified version
q' = qconj q
-- rotate and translate again to put the object where it belongs
in translate3f ((rotate q q') <$> p') [(0,0,0), org]
translate3f :: [GLfloatV3] -> [GLfloatV3] -> [GLfloatV3]
translate3f pts [(ax,ay,az), (bx,by,bz)] =
let dx = bx - ax
dy = by - ay
dz = bz - az
in map (\(x,y,z) -> (x + dx, y + dy, z + dz)) pts
测试代码：
sqNorm3 :: GLfloatV3 -> GLfloat
sqNorm3 (x,y,z) = x*x + y*y +z*z
printAsLines :: Show α => [α] -> IO ()
printAsLines xs = mapM_ (putStrLn . show) xs
main = do
let pt = (1,2,3) :: GLfloatV3
pt1 = rotate3f (0,0,0) [X] 90 [pt]
pt2 = rotate3f (0,0,0) [X,Y] 90 [pt]
pt3 = rotate3f (0,0,0) [X,Y,Z] 90 [pt]
pts = map head [pt1, pt2, pt3]
ptN = map sqNorm3 pts
printAsLines pts
putStrLn " "
printAsLines ptN
让我们检查函数 rotQuat1，初始 (1,2,3) 输入向量（即 1+4+9=13）的平方范数保持不变，适合适当的旋转:
$ ghc opengl00.hs -o ./opengl00.x && ./opengl00.x
[1 of 1] Compiling Main ( opengl00.hs, opengl00.o )
Linking ./opengl00.x ...
(0.99999994,-3.0,2.0)
(3.6213198,-0.62132025,0.70710695)
(2.5773501,0.84529924,2.5773501)
14.0
13.999995
13.999998
$
不幸的是，我没有足够的时间来安装 OpenGL 基础结构和重现动画。请让我们知道这是否解决了整个问题。