Rework vector/point typesystem

Don't rely on Data.Vector.V2 and friend anymore, but use
the types we have from Diagrams already and enhance them.

LinearAlgebra/Vector.hs

@@ -2,26 +2,74 @@
 
 module LinearAlgebra.Vector where
 
-import Data.Vector.Class
+import Diagrams.TwoD.Types
+
+type Vec = R2
+type PT  = P2
+type Coord  = (Double, Double)
 
 
-type Angle = Double
-
-
--- |Checks whether the Coordinates are in a given dimension.
-inRange :: (Double, Double) -- ^ X dimension
-        -> (Double, Double) -- ^ Y dimension
-        -> (Double, Double) -- ^ Coordinates
-        -> Bool             -- ^ result
-inRange (xlD, xuD) (ylD, yuD) (x,y)
+-- |Checks whether the Point is in a given dimension.
+inRange :: Coord -- ^ X dimension
+        -> Coord -- ^ Y dimension
+        -> PT    -- ^ Coordinates
+        -> Bool  -- ^ result
+inRange (xlD, xuD) (ylD, yuD) p
   = x <= xuD && x >= xlD && y <= yuD && y >= ylD
+    where
+      (x, y) = unp2 p
 
 
 -- |Get the angle between two vectors in degrees.
-getAngle :: (Vector v) => v -> v -> Angle
-getAngle a b = (*) 180.0                    .
-                 flip (/) pi                .
-                 acos                       .
-                 flip (/) (vmag a * vmag b) .
-                 vdot a                     $
+getAngle :: Vec -> Vec -> Double
+getAngle a b = acos                                   .
+                 flip (/) (vecLength a * vecLength b) .
+                 scalarProd a                         $
                  b
+
+
+-- |Get the length of a vector.
+vecLength :: Vec -> Double
+vecLength v = sqrt (x^2 + y^2)
+  where
+    (x, y) = unr2 v
+
+
+-- |Compute the scalar product of two vectors.
+scalarProd :: Vec -> Vec -> Double
+scalarProd v1 v2 = a1 * b1 + a2 * b2
+  where
+    (a1, a2) = unr2 v1
+    (b1, b2) = unr2 v2
+
+
+-- |Construct a vector that points to a point from the origin.
+pt2Vec :: PT -> Vec
+pt2Vec = r2 . unp2
+
+
+-- |Give the point which is at the coordinates the vector
+-- points to from the origin.
+vec2Pt :: Vec -> PT
+vec2Pt = p2 . unr2
+
+
+-- |Construct a vector between two points.
+vp2 :: PT  -- ^ vector origin
+    -> PT  -- ^ vector points here
+    -> Vec
+vp2 a b = (pt2Vec b) - (pt2Vec a)
+
+
+-- |Checks if 3 points a,b,c build a counterclock wise triangle by
+-- connecting a-b-c. This is done by computing thed determinant and
+-- checking the algebraic sign.
+ccw :: PT -> PT -> PT -> Bool
+ccw a b c = (bx - ax)   *
+              (cy - ay) -
+              (by - ay) *
+              (cx - ax) >= 0
+  where
+    (ax, ay) = unp2 a
+    (bx, by) = unp2 b
+    (cx, cy) = unp2 c