Typo /counterwise/ - counterclockwise.

BTW, the 'cosine's functionalty can be replaced by the outer product.
 2012-1-8 下午11:54 於 "Ray Song"  寫道：

> The 'scan' is flawed. A counterwise angle formed by the first three points
> does not guarantee p1's existence in the hull.
> 2012-1-8 下午3:32 於 "Zhi-Qiang Lei"  寫道：
> >
> > Hi,
> >
> > The Graham Scan function I wrote, looks like running well. But when I
> put it in QuickCheck, it just failed in some case. Anyone can show me some
> clues about the problem? Thanks.
> >
> > When I test it in ghci with some example, it returns the right result.
> > *Main> let xs = [Point {x = 1.0, y = 1.0},Point {x = 0.0, y = 4.0},Point
> {x = 0.0, y = 6.0},Point {x = 3.0, y = 5.0},Point {x = 4.0, y = 4.0},Point
> {x = 4.0, y = 1.0},Point {x = 3.0, y = 3.0},Point {x = 2.0, y = 2.0},Point
> {x = 5.0, y = 5.0}]
> > *Main> grahamScan xs
> > [Point {x = 1.0, y = 1.0},Point {x = 4.0, y = 1.0},Point {x = 5.0, y =
> 5.0},Point {x = 0.0, y = 6.0},Point {x = 0.0, y = 4.0}]
> > *Main> grahamScan it
> > [Point {x = 1.0, y = 1.0},Point {x = 4.0, y = 1.0},Point {x = 5.0, y =
> 5.0},Point {x = 0.0, y = 6.0},Point {x = 0.0, y = 4.0}]
> >
> > However, QuickCheck find some points which can fail it. Could it be a
> data type overflow problem?
> >
> > prop_scan_idempotent xs = not (null xs) ==> (grahamScan . grahamScan) xs
> == grahamScan xs
> >
> > *Main> quickCheck prop_scan_idempotent
> > *** Failed! Falsifiable (after 13 tests and 4 shrinks):
> > [Point {x = -6.29996952110807, y = -91.37172300100718},Point {x =
> 9.353314917365527, y = 64.35532141764591},Point {x = -23.826685687218355, y
> = 60.32049750442556},Point {x = -1.4281411275074123, y =
> 31.54197550020998},Point {x = -2.911218918860731, y = 15.564623822256719}]
> >
> > === code ===
> > module GrahamScan (grahamScan, Point(..))
> > where
> >
> > import Data.List
> > import Data.Ratio
> >
> > data Point = Point {
> >    x :: Double,
> >    y :: Double
> > } deriving (Eq, Show)
> >
> > instance Ord Point where
> >    compare (Point x1 y1) (Point x2 y2) = compare (y1, x1) (y2, x2)
> >
> > data Vector = Vector {
> >    start   :: Point,
> >    end     :: Point
> > } deriving (Eq)
> >
> > cosine :: Vector -> Double
> > cosine (Vector (Point x1 y1) (Point x2 y2)) = (x2 - x1) / ((x2 - x1) ^ 2
> + (y2 - y1) ^ 2)
> >
> > instance Ord Vector where
> >    compare a b = compare (f a) (f b) where
> >        f = negate . cosine
> >
> > sort' :: [Point] -> [Point]
> > sort' xs = pivot : fmap end sortedVectors where
> >    sortedVectors   = sort . fmap (Vector pivot) . delete pivot $ xs
> >    pivot           = minimum xs
> >
> > counterClockwise :: Point -> Point -> Point -> Bool
> > counterClockwise (Point x1 y1) (Point x2 y2) (Point x3 y3) = (x2 - x1) *
> (y3 - y1) > (y2 - y1) * (x3 - x1)
> >
> > scan :: [Point] -> [Point]
> > scan (p1 : p2 : p3 : xs)
> >    | counterClockwise p1 p2 p3 = p1 : scan (p2 : p3 : xs)
> >    | otherwise                 = scan (p1 : p3 : xs)
> > scan xs = xs
> >
> > grahamScan :: [Point] -> [Point]
> > grahamScan = scan . sort' . nub
> > === code ===
> >
> >
> > Best regards,
> > Zhi-Qiang Lei
> > zhiqiang.lei@gmail.com
> >
> >
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