323 lines
12 KiB
Haskell
323 lines
12 KiB
Haskell
{-# LANGUAGE BangPatterns #-}
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{-|
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Module : Geometry
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Description : Geometry helpers
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This module provides geometry functions that manipulate pairs of floats.
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Conventions:
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Seg refers to a segment, typically defined by two points, and will typically not extend beyond either of these points.
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Line refers to a line defined by two points, and extends beyond the two points.
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-}
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module Geometry
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( module Geometry
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, module Geometry.Data
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, module Geometry.Intersect
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, module Geometry.Bezier
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, module Geometry.Vector
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, module Geometry.Vector3D
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, module Geometry.LHS
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, module Geometry.Polygon
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) where
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import Geometry.Data
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import Geometry.Polygon
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import Geometry.Intersect
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import Geometry.Bezier
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import Geometry.Vector
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import Geometry.Vector3D
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import Geometry.LHS
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--import Geometry.ConvexPoly
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--import Data.Maybe
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import Data.List
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-- | Return a point a distance away from a first point towards a second point.
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-- Does not go past the second point.
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alongSegBy :: Float -> Point2 -> Point2 -> Point2
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alongSegBy !x !a !b = a +.+ y *.* normalizeV (b -.- a)
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where
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y = min x $ dist a b
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-- | Debug version of 'pointInPolygon'.
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errorPointInPolygon :: Int -> Point2 -> [Point2] -> Bool
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errorPointInPolygon !i !p xs
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| length xs == 1 = error "one point polygon"
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| length xs == 2 = error "two point polygon"
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| nub xs == xs = pointInPolygon p xs
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| otherwise = error $ "errorPointInPolygon "++ show i
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-- | Debug version of 'normalizeV'.
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errorNormalizeV :: Int -> Point2 -> Point2
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errorNormalizeV !i (V2 0 0) = error $ "problem with function: errorNormalizeV "++show i
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errorNormalizeV _ !p = normalizeV p
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-- | Debug version of 'angleVV'.
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errorAngleVV :: Int -> Point2 -> Point2 -> Float
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errorAngleVV !i (V2 0 0) _ = error $ "problem with function: errorAngleVV "++show i
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errorAngleVV !i _ (V2 0 0) = error $ "problem with function: errorAngleVV "++show i
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errorAngleVV _ !p !p' = angleVV p p'
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-- | Debug version of 'isLHS'.
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errorIsLHS :: Int -> Point2 -> Point2 -> Point2 -> Bool
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errorIsLHS !i !x !y
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| x == y = error $ "problem with function: errorIsLHS " ++show i
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| otherwise = isLHS x y
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-- | Debug version of 'closestPointOnLine'
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errorClosestPointOnLine :: Int -> Point2 -> Point2 -> Point2 -> Point2
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errorClosestPointOnLine !i !x !y
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| x == y = error $ "problem with function: errorClosestPointOnLine " ++show i
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| otherwise = closestPointOnLine x y
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-- | Debug version of 'closestPointOnLineParam'
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errorClosestPointOnLineParam :: Int -> Point2 -> Point2 -> Point2 -> Float
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errorClosestPointOnLineParam _ !x! y! z
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| x == y = dist x z
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| otherwise = closestPointOnLineParam x y z
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-- | Return midpoint between two points.
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pHalf :: Point2 -> Point2 -> Point2
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pHalf !a !b = 0.5 *.* (a +.+ b)
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-- | Test whether a circle is on a segment by intersecting a new normal segment through the
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-- center of the circle with the segment itself.
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-- Returns False if the circle center is beyond the endpoints of the
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-- segment.
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circOnSegNoEndpoints :: Point2 -> Point2 -> Point2 -> Float -> Bool
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{-# INLINE circOnSegNoEndpoints #-}
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circOnSegNoEndpoints !p1 !p2 !c !rad = intersectSegSegTest p1 p2 (c -.- thenormal) (c +.+ thenormal)
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where
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thenormal = rad *.* vNormal (normalizeV $ p1 -.- p2)
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-- | Test whether a circle is on a segment by intersecting a normal and testing
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-- the distance to the endpoints of the segment.
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circOnSeg :: Point2 -> Point2 -> Point2 -> Float -> Bool
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{-# INLINE circOnSeg #-}
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circOnSeg !p1 !p2 !c !rad = magV (p1 -.- c) <= rad
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|| magV (p2 -.- c) <= rad
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|| intersectSegSegTest p1 p2 (c -.- thenormal) (c +.+ thenormal)
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where
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thenormal = rad *.* vNormal (normalizeV $ p1 -.- p2)
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cylinderOnSeg :: Point3 -> Point3 -> Point3 -> Float -> Bool
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{-# INLINE cylinderOnSeg #-}
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cylinderOnSeg = undefined
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-- | Find the difference between two Nums.
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difference :: (Ord a, Num a) => a -> a -> a
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difference x y
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| x > y = x - y
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| otherwise = y - x
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-- | Given vector line direction and a vector movement,
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-- reflects the movement according to the line.
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reflectIn :: Point2 -> Point2 -> Point2
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reflectIn line vec = rotateV (2 * angleBetween line vec) vec
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-- | Find angle between two points.
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-- Not normalised, ranges from -2*pi to 2*pi.
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angleBetween :: Point2 -> Point2 -> Float
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angleBetween v1 v2 = argV v1 - argV v2
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-- | Return a list containing two copies of a pair.
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doublePair :: (a,a) -> [(a,a)]
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doublePair (x,y) = [(x,y),(y,x)]
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doubleV2 :: V2 a -> [V2 a]
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doubleV2 (V2 x y) = [V2 x y,V2 y x]
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-- split a list into triples, forms triangles from a polygon
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polyToTris'' :: [s] -> [s]
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polyToTris'' (a:as) = go a as
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where
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go !x (y:z:ys) = x : y : z : go x (z:ys)
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go _ _ = []
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polyToTris'' _ = []
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polyToTris :: [s] -> [s]
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{-# INLINABLE polyToTris #-}
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polyToTris (x:xs) = foldr (f x) [] $ zip xs $ tail xs
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where
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f a (b,c) ls = a:b:c:ls
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polyToTris _ = []
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polyToTris' :: [s] -> [s]
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{-# INLINE polyToTris' #-}
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polyToTris' [] = []
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polyToTris' (a:as) = prependTwo a as
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prependTwo :: a -> [a] -> [a]
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prependTwo _ [] = []
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prependTwo _ [_] = []
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prependTwo sep (x:y:xs) = sep : x : y : prependTwo sep (y:xs)
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-- | Return n equidistant points on a circle with a radius of 600.
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nRays :: Int -> [Point2]
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nRays n = nRaysRad n 600
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-- | Return n equidistant points on a circle with a radius of x.
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nRaysRad :: Int -> Float -> [Point2]
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nRaysRad n x = take n $ iterate (rotateV (2*pi/fromIntegral n)) (V2 x 0)
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-- | Test whether an angle is to the left of another angle, according to the
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-- smallest change in rotation between them.
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-- This appears to sometimes fail if the angles are not normalized.
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isLeftOfA :: Float -> Float -> Bool
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isLeftOfA angle1 angle2 = (angle1 - angle2 < pi && angle1 > angle2)
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|| (angle2 - angle1 > pi && angle2 > angle1)
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-- | Test whether a vector is to the left of another, according to the smallest
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-- change of rotation between them.
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isLeftOf :: Point2 -> Point2 -> Bool
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isLeftOf x y = isLeftOfA (argV x) (argV y)
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-- | Find the difference between two angles.
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-- Possibly not correct...
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-- TODO write tests
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diffAngles :: Float -> Float -> Float
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diffAngles x y
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| diff > pi = diffAngles (x - 2*pi) y
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| diff >= 0 = diff
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| diff > -pi = -diff
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| otherwise = diffAngles (x + 2*pi) y
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where
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diff = x-y
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mixAngles :: Float -> Float -> Float -> Float
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mixAngles frac a1 a2
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| abs (a1 - a2) <= pi = normalizeAngle $ frac * a1 + (1 - frac) * a2
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| a1 > a2 = mixAngles frac (a1 - 2*pi) a2
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| otherwise = mixAngles frac (a1 + 2*pi) a2
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-- | Return Just a point if it is inside a circle, Nothing otherwise.
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pointInCircle :: Point2 -> Float -> Point2 -> Maybe Point2
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pointInCircle p r c
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| p == c = Just p
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| magV (p -.- c) < r = Just p
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| otherwise = Nothing
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-- | Finds the height of a triangle using herons formula.
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-- The base is the line between the first two points.
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heron :: Point2 -> Point2 -> Point2 -> Float
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heron x y z
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| x == y = 0
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| otherwise =
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let a = magV $ x -.- y
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b = magV $ y -.- z
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c = magV $ z -.- x
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s = (a+b+c)/2
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area = sqrt(s*(s-a)*(s-b)*(s-c))
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in 2*area/a
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-- | Multiplies reflection in normal by factor.
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reflectInParam :: Float -> Point2 -> Point2 -> Point2
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reflectInParam x line vec =
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let angle = 2 * angleBetween line vec
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rAng = rotateV angle vec
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p = x *.* errorClosestPointOnLine 3 (V2 0 0) (vNormal line) rAng
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in rAng -.- p
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isOnSeg :: Point2 -> Point2 -> Point2 -> Bool
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isOnSeg l1 l2 p =
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errorClosestPointOnLineParam 10 l1 (l1 +.+ vNormal (l2 -.- l1)) p == 0
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&& errorClosestPointOnLineParam 11 l1 l2 p <= 1
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&& errorClosestPointOnLineParam 12 l1 l2 p >= 0
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-- | Divide a segment into a list of points with a maximal distance between
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-- them.
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-- the take 5000 here is a hack, otherwise divideLine seems to sometimes
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-- generate an infinite list, and I don't know why
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divideLine :: Float -> Point2 -> Point2 -> [Point2]
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--divideLine x a b = map (\i -> a +.+ (i / (fromIntegral numPoints)) *.* (b -.- a))
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divideLine x a b = take 5000
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$ map (\i -> a +.+ (fromIntegral i / fromIntegral numPoints *.* (b -.- a)) )
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ns
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where
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d = dist a b
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numPoints = max 1 $ ceiling $ d / x
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ns = [0 :: Int .. numPoints]
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-- | As 'divideLine', but must return an odd number of points.
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divideLineOddNumPoints :: Float -> Point2 -> Point2 -> [Point2]
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--divideLine x a b = map (\i -> a +.+ (i / (fromIntegral numPoints)) *.* (b -.- a))
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divideLineOddNumPoints x a b = take 5000
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$ map (\i -> a +.+ (fromIntegral i / fromIntegral numPoints *.* (b -.- a)) )
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ns
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where
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d = dist a b
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numPoints' = max 1 $ ceiling $ d / x
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numPoints
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| even numPoints' = numPoints'
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| otherwise = numPoints' + 1
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ns = [0 .. numPoints] :: [Int]
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divideLineExact :: Float -> Point2 -> Point2 -> [Point2]
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divideLineExact x a b = map ( (a +.+ ) . ( *.* v) ) [0 , x .. d]
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where
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d = dist a b
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v = normalizeV $ b -.- a
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-- | Given two pairs of Ints, returns a list of pairs of Ints that form
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-- a digital line between them.
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digitalLine :: (Int,Int) -> (Int,Int) -> [(Int,Int)]
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--{-# INLINE digitalLine #-}
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digitalLine (x1,y1) (x2,y2)
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| abs (x1-x2) > abs (y1-y2) =
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[ (x,( (y1-y2) * x + x1*y2 - x2*y1) `rdiv` (x1-x2) )
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| x <- intervalList x1 x2
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]
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| otherwise =
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[ ( ((x1-x2) * y + y1*x2 - y2*x1) `rdiv` (y1-y2) , y)
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| y <- intervalList y1 y2
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]
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where
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rdiv a b = round $ fromIntegral a / (fromIntegral b :: Float)
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-- | Given two pairs of 'Int's, create a list of pairs of 'Int's that form a
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-- rectangle between them.
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digitalRect :: (Int,Int) -> (Int,Int) -> [(Int,Int)]
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{-# INLINE digitalRect #-}
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digitalRect (a,b) (c,d) = [(s,t) | s <- [minx .. maxx] , t <- [miny .. maxy]]
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where
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maxx = max a c
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minx = min a c
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maxy = max b d
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miny = min b d
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-- | Given two Ints, creates the list of Ints between these.
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intervalList :: Int -> Int -> [Int]
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{-# INLINE intervalList #-}
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intervalList x y
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| y > x = [x .. y]
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| otherwise = reverse [y..x]
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-- | Create points on the circumference of a circle with maximal distance
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-- between them.
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divideCircle :: Float -> Point2 -> Float -> [Point2]
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divideCircle x cen rad = map (cen +.+) $ nRaysRad n rad
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where
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n = ceiling $ rad * 2 * pi / x
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arcStepwise
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:: Float -- ^ Maximum distance between points
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-> Float -- ^ Angle to travel
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-> Point2 -- ^ Center
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-> Point2 -- ^ Start vector from center
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-> [Point2]
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arcStepwise ssize a c v
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| a < 0 = reverse $ arcStepwisePositive ssize (negate a) c (rotateV a v)
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| otherwise = arcStepwisePositive ssize a c v
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arcStepwisePositive
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:: Float -- ^ Maximum distance between points
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-> Float -- ^ Angle to travel, assumed to be positive
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-> Point2 -- ^ Center
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-> Point2 -- ^ Start vector from center
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-> [Point2]
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arcStepwisePositive ssize a cen v = (cen +.+) . (`rotateV` v) <$> rots
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where
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rots :: [Float]
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rots = map ((a*) . (/ fromIntegral n ) . fromIntegral) [0 .. n]
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n :: Int
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n = ceiling (a * magV v / ssize)
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-- | Given a list of points, returns pairs of points linking the points into a
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-- loop.
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chainPairs :: [Point2] -> [(Point2,Point2)]
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chainPairs [] = error "tried to make chain with empty list of points"
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chainPairs [_] = error "tried to make chain with singleton list of points"
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chainPairs xs = zip xs $ tail xs
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-- | Given a list of points, returns pairs of points linking the points into a
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-- loop.
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loopPairs :: [a] -> [(a,a)]
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loopPairs [] = error "tried to make loop with empty list of elements"
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loopPairs [_] = error "tried to make loop with singleton list of elements"
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loopPairs (x:xs) = zip (x:xs) (xs ++ [x])
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-- | Test whether a point is in a cone.
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-- Note the pair is ordered.
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-- Doesn't work for obtuse angles.
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pointIsInCone
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:: Point2 -- ^ Cone point.
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-> (Point2,Point2) -- ^ Points delimiting the left and right boundaries of the cone.
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-> Point2 -- ^ Point to test.
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-> Bool
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pointIsInCone c (rightp,leftp) p = isLHS c rightp p && isLHS leftp c p
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