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|
-- Copyright (c) 2000 Galois Connections, Inc.
-- All rights reserved. This software is distributed as
-- free software under the license in the file "LICENSE",
-- which is included in the distribution.
module Geometry
( Coords
, Ray
, Point -- abstract
, Vector -- abstract
, Matrix -- abstract
, Color -- abstract
, Box(..)
, Radian
, matrix
, coord
, color
, uncolor
, xCoord , yCoord , zCoord
, xComponent , yComponent , zComponent
, point
, vector
, nearV
, point_to_vector
, vector_to_point
, dot
, cross
, tangents
, addVV
, addPV
, subVV
, negV
, subPP
, norm
, normalize
, dist2
, sq
, distFrom0Sq
, distFrom0
, multSV
, multMM
, transposeM
, multMV
, multMP
, multMQ
, multMR
, white
, black
, addCC
, subCC
, sumCC
, multCC
, multSC
, nearC
, offsetToPoint
, epsilon
, inf
, nonZero
, eqEps
, near
, clampf
) where
import List
type Coords = (Double,Double,Double)
type Ray = (Point,Vector) -- origin of ray, and unit vector giving direction
data Point = P !Double !Double !Double -- implicit extra arg of 1
deriving (Show)
data Vector = V !Double !Double !Double -- implicit extra arg of 0
deriving (Show, Eq)
data Matrix = M !Quad !Quad !Quad !Quad
deriving (Show)
data Color = C !Double !Double !Double
deriving (Show, Eq)
data Box = B !Double !Double !Double !Double !Double !Double
deriving (Show)
data Quad = Q !Double !Double !Double !Double
deriving (Show)
type Radian = Double
type Tup4 a = (a,a,a,a)
--{-# INLINE matrix #-}
matrix :: Tup4 (Tup4 Double) -> Matrix
matrix ((m11, m12, m13, m14),
(m21, m22, m23, m24),
(m31, m32, m33, m34),
(m41, m42, m43, m44))
= M (Q m11 m12 m13 m14)
(Q m21 m22 m23 m24)
(Q m31 m32 m33 m34)
(Q m41 m42 m43 m44)
coord x y z = (x, y, z)
color r g b = C r g b
uncolor (C r g b) = (r,g,b)
{-# INLINE xCoord #-}
xCoord (P x y z) = x
{-# INLINE yCoord #-}
yCoord (P x y z) = y
{-# INLINE zCoord #-}
zCoord (P x y z) = z
{-# INLINE xComponent #-}
xComponent (V x y z) = x
{-# INLINE yComponent #-}
yComponent (V x y z) = y
{-# INLINE zComponent #-}
zComponent (V x y z) = z
point :: Double -> Double -> Double -> Point
point x y z = P x y z
vector :: Double -> Double -> Double -> Vector
vector x y z = V x y z
nearV :: Vector -> Vector -> Bool
nearV (V a b c) (V d e f) = a `near` d && b `near` e && c `near` f
point_to_vector :: Point -> Vector
point_to_vector (P x y z) = V x y z
vector_to_point :: Vector -> Point
vector_to_point (V x y z) = P x y z
{-# INLINE vector_to_quad #-}
vector_to_quad :: Vector -> Quad
vector_to_quad (V x y z) = Q x y z 0
{-# INLINE point_to_quad #-}
point_to_quad :: Point -> Quad
point_to_quad (P x y z) = Q x y z 1
{-# INLINE quad_to_point #-}
quad_to_point :: Quad -> Point
quad_to_point (Q x y z _) = P x y z
{-# INLINE quad_to_vector #-}
quad_to_vector :: Quad -> Vector
quad_to_vector (Q x y z _) = V x y z
--{-# INLINE dot #-}
dot :: Vector -> Vector -> Double
dot (V x1 y1 z1) (V x2 y2 z2) = x1 * x2 + y1 * y2 + z1 * z2
cross :: Vector -> Vector -> Vector
cross (V x1 y1 z1) (V x2 y2 z2)
= V (y1 * z2 - z1 * y2) (z1 * x2 - x1 * z2) (x1 * y2 - y1 * x2)
-- assumption: the input vector is a normal
tangents :: Vector -> (Vector, Vector)
tangents v@(V x y z)
= (v1, v `cross` v1)
where v1 | x == 0 = normalize (vector 0 z (-y))
| otherwise = normalize (vector (-y) x 0)
{-# INLINE dot4 #-}
dot4 :: Quad -> Quad -> Double
dot4 (Q x1 y1 z1 w1) (Q x2 y2 z2 w2) = x1 * x2 + y1 * y2 + z1 * z2 + w1 * w2
addVV :: Vector -> Vector -> Vector
addVV (V x1 y1 z1) (V x2 y2 z2)
= V (x1 + x2) (y1 + y2) (z1 + z2)
addPV :: Point -> Vector -> Point
addPV (P x1 y1 z1) (V x2 y2 z2)
= P (x1 + x2) (y1 + y2) (z1 + z2)
subVV :: Vector -> Vector -> Vector
subVV (V x1 y1 z1) (V x2 y2 z2)
= V (x1 - x2) (y1 - y2) (z1 - z2)
negV :: Vector -> Vector
negV (V x1 y1 z1)
= V (-x1) (-y1) (-z1)
subPP :: Point -> Point -> Vector
subPP (P x1 y1 z1) (P x2 y2 z2)
= V (x1 - x2) (y1 - y2) (z1 - z2)
--{-# INLINE norm #-}
norm :: Vector -> Double
norm (V x y z) = sqrt (sq x + sq y + sq z)
--{-# INLINE normalize #-}
-- normalize a vector to a unit vector
normalize :: Vector -> Vector
normalize v@(V x y z)
| norm /= 0 = multSV (1/norm) v
| otherwise = error "normalize empty!"
where norm = sqrt (sq x + sq y + sq z)
-- This does computes the distance *squared*
dist2 :: Point -> Point -> Double
dist2 us vs = sq x + sq y + sq z
where
(V x y z) = subPP us vs
{-# INLINE sq #-}
sq :: Double -> Double
sq d = d * d
{-# INLINE distFrom0Sq #-}
distFrom0Sq :: Point -> Double -- Distance of point from origin.
distFrom0Sq (P x y z) = sq x + sq y + sq z
{-# INLINE distFrom0 #-}
distFrom0 :: Point -> Double -- Distance of point from origin.
distFrom0 p = sqrt (distFrom0Sq p)
--{-# INLINE multSV #-}
multSV :: Double -> Vector -> Vector
multSV k (V x y z) = V (k*x) (k*y) (k*z)
--{-# INLINE multMM #-}
multMM :: Matrix -> Matrix -> Matrix
multMM m1@(M q1 q2 q3 q4) m2
= M (multMQ m2' q1)
(multMQ m2' q2)
(multMQ m2' q3)
(multMQ m2' q4)
where
m2' = transposeM m2
{-# INLINE transposeM #-}
transposeM :: Matrix -> Matrix
transposeM (M (Q e11 e12 e13 e14)
(Q e21 e22 e23 e24)
(Q e31 e32 e33 e34)
(Q e41 e42 e43 e44)) = (M (Q e11 e21 e31 e41)
(Q e12 e22 e32 e42)
(Q e13 e23 e33 e43)
(Q e14 e24 e34 e44))
--multMM m1 m2 = [map (dot4 row) (transpose m2) | row <- m1]
--{-# INLINE multMV #-}
multMV :: Matrix -> Vector -> Vector
multMV m v = quad_to_vector (multMQ m (vector_to_quad v))
--{-# INLINE multMP #-}
multMP :: Matrix -> Point -> Point
multMP m p = quad_to_point (multMQ m (point_to_quad p))
-- mat vec = map (dot4 vec) mat
{-# INLINE multMQ #-}
multMQ :: Matrix -> Quad -> Quad
multMQ (M q1 q2 q3 q4) q
= Q (dot4 q q1)
(dot4 q q2)
(dot4 q q3)
(dot4 q q4)
{-# INLINE multMR #-}
multMR :: Matrix -> Ray -> Ray
multMR m (r, v) = (multMP m r, multMV m v)
white :: Color
white = C 1 1 1
black :: Color
black = C 0 0 0
addCC :: Color -> Color -> Color
addCC (C a b c) (C d e f) = C (a+d) (b+e) (c+f)
subCC :: Color -> Color -> Color
subCC (C a b c) (C d e f) = C (a-d) (b-e) (c-f)
sumCC :: [Color] -> Color
sumCC = foldr addCC black
multCC :: Color -> Color -> Color
multCC (C a b c) (C d e f) = C (a*d) (b*e) (c*f)
multSC :: Double -> Color -> Color
multSC k (C a b c) = C (a*k) (b*k) (c*k)
nearC :: Color -> Color -> Bool
nearC (C a b c) (C d e f) = a `near` d && b `near` e && c `near` f
offsetToPoint :: Ray -> Double -> Point
offsetToPoint (r,v) i = r `addPV` (i `multSV` v)
--
epsilon, inf :: Double -- aproximate zero and infinity
epsilon = 1.0e-10
inf = 1.0e20
nonZero :: Double -> Double -- Use before a division. It makes definitions
nonZero x | x > epsilon = x -- more complete and I bet the errors that get
| x < -epsilon = x -- introduced will be undetectable if epsilon
| otherwise = epsilon -- is small enough
eqEps x y = abs (x-y) < epsilon
near = eqEps
clampf :: Double -> Double
clampf p | p < 0 = 0
| p > 1 = 1
| True = p
|
