diff --git a/hgeometry-examples/bapc2012/Gunslinger.hs b/hgeometry-examples/bapc2012/Gunslinger.hs
index fd65df2f9..c08305fa6 100644
--- a/hgeometry-examples/bapc2012/Gunslinger.hs
+++ b/hgeometry-examples/bapc2012/Gunslinger.hs
@@ -1,7 +1,6 @@
 module Main where
 
 import           Control.Lens
-import           Data.Default.Class
 import           Data.Fixed
 import qualified Data.Foldable as F
 import qualified Data.List as List
@@ -83,9 +82,6 @@ instance Show Answer where
 
 data Kind = Luke | Hatch | Dalton deriving (Show,Eq)
 
-instance Default Kind where
-  def = Dalton -- if we have to invent a value; invent a dalton.
-
 data Input = Input { _luke    :: Point 2 Int
                    , _hatch   :: Point 2 Int
                    , _daltons :: [Point 2 Int]
diff --git a/hgeometry-examples/polyLineDrawing/Main.hs b/hgeometry-examples/polyLineDrawing/Main.hs
index f26021d08..3d4918ca1 100644
--- a/hgeometry-examples/polyLineDrawing/Main.hs
+++ b/hgeometry-examples/polyLineDrawing/Main.hs
@@ -138,9 +138,6 @@ makeLenses ''Model
 instance KnownNat p => ToMisoString (RealNumber p) where
   toMisoString = toMisoString . toFixed
 
-instance Default (Point 2 R :+ Int) where
-  def = origin :+ 0
-
 ----------------------------------------
 
 initialModel :: Model
diff --git a/hgeometry-examples/polygonTriangulation/Main.hs b/hgeometry-examples/polygonTriangulation/Main.hs
index 0a089f406..09b6e5a2e 100644
--- a/hgeometry-examples/polygonTriangulation/Main.hs
+++ b/hgeometry-examples/polygonTriangulation/Main.hs
@@ -21,8 +21,6 @@ import           Miso
 import           Miso.String (MisoString,ToMisoString(..), ms)
 import           Miso.Svg hiding (height_, id_, style_, width_)
 
-import Data.Default.Class
-
 --------------------------------------------------------------------------------
 
 type R = RealNumber 5
diff --git a/hgeometry-examples/voronoiDiagram/Main.hs b/hgeometry-examples/voronoiDiagram/Main.hs
index cc6470ab6..1bc3e19d1 100644
--- a/hgeometry-examples/voronoiDiagram/Main.hs
+++ b/hgeometry-examples/voronoiDiagram/Main.hs
@@ -20,8 +20,6 @@ import           Miso
 import           Miso.String (MisoString,ToMisoString(..), ms)
 import           Miso.Svg hiding (height_, id_, style_, width_)
 
-import Data.Default.Class
-
 --------------------------------------------------------------------------------
 
 type R = RealNumber 5
@@ -37,9 +35,6 @@ makeLenses ''Model
 instance KnownNat p => ToMisoString (RealNumber p) where
   toMisoString = toMisoString . toFixed
 
-instance Default (Point 2 R :+ Int) where
-  def = origin :+ 0
-
 ----------------------------------------
 
 initialModel :: Model
diff --git a/hgeometry/bench/Util.hs b/hgeometry/bench/Util.hs
index e58473483..f00769e1e 100644
--- a/hgeometry/bench/Util.hs
+++ b/hgeometry/bench/Util.hs
@@ -19,7 +19,7 @@ import           System.Random.Stateful
 -- | Generate an infinite list random points in a sufficiently large
 -- bounding box.
 randomPoints       :: forall point r. ( UniformRange point
-                      , Point_ point 2 r
+                      , ConstructablePoint_ point 2 r
                       , Num r
                       )
                    => StdGen -- ^ generator to use
diff --git a/hgeometry/hgeometry.cabal b/hgeometry/hgeometry.cabal
index 387f3c65a..b65889da1 100644
--- a/hgeometry/hgeometry.cabal
+++ b/hgeometry/hgeometry.cabal
@@ -392,11 +392,11 @@ library
         HGeometry.LineSegment.Intersection.Naive
         HGeometry.LineSegment.Intersection.BentleyOttmann
 
-        HGeometry.LowerEnvelope
-        HGeometry.LowerEnvelope.Naive
-        HGeometry.LowerEnvelope.VertexForm
-        HGeometry.LowerEnvelope.AdjListForm
-        HGeometry.LowerEnvelope.Connected
+        HGeometry.Plane.LowerEnvelope
+        HGeometry.Plane.LowerEnvelope.Naive
+        HGeometry.Plane.LowerEnvelope.VertexForm
+        HGeometry.Plane.LowerEnvelope.AdjListForm
+        HGeometry.Plane.LowerEnvelope.Connected
 
 
         HGeometry.VoronoiDiagram
@@ -427,14 +427,14 @@ library
 
         HGeometry.LineSegment.Intersection.Types
 
-        HGeometry.LowerEnvelope.Type
-        HGeometry.LowerEnvelope.Connected.Type
-        HGeometry.LowerEnvelope.Connected.FromVertexForm
-        -- HGeometry.LowerEnvelope.Sample
+        HGeometry.Plane.LowerEnvelope.Type
+        HGeometry.Plane.LowerEnvelope.Connected.Type
+        HGeometry.Plane.LowerEnvelope.Connected.FromVertexForm
+        -- HGeometry.Plane.LowerEnvelope.Sample
 
 
-        -- HGeometry.LowerEnvelope.AtMostThree
-        -- HGeometry.LowerEnvelope.Triangulate
+        -- HGeometry.Plane.LowerEnvelope.AtMostThree
+        -- HGeometry.Plane.LowerEnvelope.Triangulate
 
         HGeometry.VoronoiDiagram.ViaLowerEnvelope
 
diff --git a/hgeometry/ipe/src/Ipe/IpeOut.hs b/hgeometry/ipe/src/Ipe/IpeOut.hs
index 5668c269c..34e729bee 100644
--- a/hgeometry/ipe/src/Ipe/IpeOut.hs
+++ b/hgeometry/ipe/src/Ipe/IpeOut.hs
@@ -20,6 +20,7 @@ import           Control.Lens hiding (Simple)
 import           Data.Foldable (toList)
 import           Data.Kind
 import           Data.List.NonEmpty (NonEmpty(..))
+import qualified Data.List.NonEmpty as NonEmpty
 import qualified Data.Sequence as Seq
 import           Data.Text (Text)
 import qualified Data.Text as Text
@@ -276,7 +277,10 @@ path = Path . Seq.singleton
 -- | Construct a PolyLine path segment
 pathSegment :: (LineSegment_ lineSegment point, Point_ point 2 r)
             => lineSegment -> PathSegment r
-pathSegment = PolyLineSegment . fmap (^.asPoint) . review _PolyLineLineSegment
+pathSegment = PolyLineSegment . fmap (^.asPoint) . lineSegmentToPolyLine
+  where
+    lineSegmentToPolyLine s = polyLineFromPoints . NonEmpty.fromList $ [s^.start, s^.end]
+
 
 -- | Draw a polygon
 ipePolygon    :: IpeOut (SimplePolygon (Point 2 r)) Path r
diff --git a/hgeometry/kernel/src/HGeometry/Ball/Class.hs b/hgeometry/kernel/src/HGeometry/Ball/Class.hs
index 38df44ec2..66cdfe5ad 100644
--- a/hgeometry/kernel/src/HGeometry/Ball/Class.hs
+++ b/hgeometry/kernel/src/HGeometry/Ball/Class.hs
@@ -49,7 +49,8 @@ class ( HasCenter ball point
   fromCenterAndSquaredRadius :: point -> NumType ball -> ball
 
   -- | A d dimensional unit ball centered at the origin.
-  unitBall :: Num (NumType ball) => ball
+  unitBall :: (r ~ NumType ball, d ~ Dimension ball
+              , Num r, ConstructablePoint_ point d r) => ball
   unitBall = fromCenterAndSquaredRadius origin 1
 
   -- | A lens to get/set the radius of a Ball
diff --git a/hgeometry/kernel/src/HGeometry/HyperPlane/Class.hs b/hgeometry/kernel/src/HGeometry/HyperPlane/Class.hs
index bf8179a56..a05662ed0 100644
--- a/hgeometry/kernel/src/HGeometry/HyperPlane/Class.hs
+++ b/hgeometry/kernel/src/HGeometry/HyperPlane/Class.hs
@@ -346,7 +346,7 @@ class HyperPlaneFromPoints hyperPlane where
                            ) => Vector d point -> hyperPlane
 
 
-instance (HyperPlane_ hyperPlane d r, Default extra)
+instance (HyperPlane_ hyperPlane d r)
          => HyperPlane_ (hyperPlane :+ extra) d r where
   evalHyperPlaneEquation h = evalHyperPlaneEquation (h^.core)
   {-# INLINE evalHyperPlaneEquation #-}
@@ -368,7 +368,7 @@ instance (ConstructableHyperPlane_ hyperPlane d r, Default extra)
   fromPointAndNormal p v = fromPointAndNormal p v :+ def
   {-# INLINE fromPointAndNormal #-}
 
-instance (NonVerticalHyperPlane_  hyperPlane d r, Default extra)
+instance (NonVerticalHyperPlane_  hyperPlane d r)
          => NonVerticalHyperPlane_ (hyperPlane :+ extra) d r where
   evalAt p = evalAt p . view core
   {-# INLINE evalAt #-}
diff --git a/hgeometry/kernel/src/HGeometry/Interval/Class.hs b/hgeometry/kernel/src/HGeometry/Interval/Class.hs
index c859fcff9..47a293359 100644
--- a/hgeometry/kernel/src/HGeometry/Interval/Class.hs
+++ b/hgeometry/kernel/src/HGeometry/Interval/Class.hs
@@ -12,11 +12,14 @@
 module HGeometry.Interval.Class
   ( Interval_, pattern Interval_
   , IntervalLike_(..)
+  , ConstructableInterval_(..)
 
   , ClosedInterval_(..), pattern ClosedInterval_
+  , ConstructableClosedInterval_(..)
   , clampTo
 
   , OpenInterval_(..), pattern OpenInterval_
+  , ConstructableOpenInterval_(..)
 
   , HasStart(..)
   , HasEnd(..)
@@ -103,6 +106,19 @@ class ( HasStart interval point, HasStartPoint interval (StartPointOf interval)
       , EndPoint_ (EndPointOf interval), IxValue (EndPointOf interval) ~ point
       , EndPoint_ (StartPointOf interval), IxValue (StartPointOf interval) ~ point
       ) => IntervalLike_ interval point | interval -> point where
+  {-# MINIMAL  #-}
+
+-- | A class for types representing Intervals
+type Interval_ :: Type -> Type -> Constraint
+class ( IntervalLike_ interval r
+      , NumType interval ~ r
+      ) => Interval_ interval r | interval -> r where
+
+--------------------------------------------------------------------------------
+
+-- | A class for constructable intervals
+type ConstructableInterval_ :: Type -> Type -> Constraint
+class  Interval_ interval point => ConstructableInterval_ interval point where
   {-# MINIMAL mkInterval #-}
 
   -- | Construct an interval given its start and end point.
@@ -124,15 +140,6 @@ class ( HasStart interval point, HasStartPoint interval (StartPointOf interval)
   {-# INLINE buildInterval #-}
 
 
---------------------------------------------------------------------------------
-
--- | A class for types representing Intervals
-type Interval_ :: Type -> Type -> Constraint
-class ( IntervalLike_ interval r
-      , NumType interval ~ r
-      ) => Interval_ interval r | interval -> r where
-
-
 --------------------------------------------------------------------------------
 -- * Closed Intervals
 
@@ -142,6 +149,12 @@ class ( Interval_ interval r
       , StartPointOf interval ~ EndPoint Closed r
       , EndPointOf interval ~ EndPoint Closed r
       ) => ClosedInterval_ interval r where
+
+-- | A class representing constructable closed intervals
+class ( ClosedInterval_ interval r
+      , ConstructableInterval_ interval r
+      ) => ConstructableClosedInterval_ interval r where
+
   -- | Construct an interval given its start and end point.
   mkClosedInterval     :: r -> r -> interval
   mkClosedInterval s e = mkInterval (ClosedE s) (ClosedE e)
@@ -157,10 +170,10 @@ class ( Interval_ interval r
 -- | Pattern matching on an arbitrary closed interval
 pattern ClosedInterval_     :: ClosedInterval_ interval r => r -> r -> interval
 pattern ClosedInterval_ l u <- (startAndEnd -> (l,u))
-  where
-    ClosedInterval_ l u = mkClosedInterval l u
 {-# COMPLETE ClosedInterval_ #-}
-{-# INLINE ClosedInterval_ #-}
+--   where
+--     ClosedInterval_ l u = mkClosedInterval l u
+-- {-# INLINE ClosedInterval_ #-}
 
 -- | Clamps a value to an interval. I.e. if the value lies outside the range we
 -- report the closest value "in the range".
@@ -182,6 +195,11 @@ class ( Interval_ interval r
       , StartPointOf interval ~ EndPoint Open r
       , EndPointOf interval ~ EndPoint Open r
       ) => OpenInterval_ interval r | interval -> r where
+
+-- | Constructable Open intervals
+class ( OpenInterval_ interval r
+      , ConstructableInterval_ interval r
+      ) => ConstructableOpenInterval_ interval r | interval -> r where
   -- | Construct an interval given its start s and end point t.
   --
   -- pre: s < t
@@ -192,19 +210,19 @@ class ( Interval_ interval r
 -- | Pattern matching on an arbitrary open interval
 pattern OpenInterval_     :: OpenInterval_ interval r => r -> r -> interval
 pattern OpenInterval_ l u <- (startAndEnd -> (l,u))
-  where
-    OpenInterval_ l u = mkOpenInterval l u
+  -- where
+  --   OpenInterval_ l u = mkOpenInterval l u
 {-# COMPLETE OpenInterval_ #-}
 
 --------------------------------------------------------------------------------
 
--- | Pattern to match on intervals or construct them.
+-- | Pattern to match on intervals
 pattern Interval_     :: Interval_ interval r
                       => StartPointOf interval -> EndPointOf interval -> interval
 pattern Interval_ s t <- (startAndEndPoint -> (s,t))
-  where
-    Interval_ s t = mkInterval s t
 {-# COMPLETE Interval_ #-}
+  -- where
+  --   Interval_ s t = mkInterval s t
 
 
 -- | Compute where the given query value is with respect to the interval.
@@ -320,20 +338,28 @@ type instance StartPointOf (interval :+ extra) = StartPointOf interval
 type instance EndPointOf (interval :+ extra) = EndPointOf interval
 
 instance ( IntervalLike_ interval point
-         , Default extra
          ) => IntervalLike_ (interval :+ extra) point where
-  mkInterval s t = mkInterval s t :+ def
 
 instance ( Interval_ interval r
-         , Default extra
          ) => Interval_ (interval :+ extra) r
 
 instance ( ClosedInterval_ interval r
+         ) => ClosedInterval_ (interval :+ extra) r
+
+instance ( OpenInterval_ interval r
+         ) => OpenInterval_ (interval :+ extra) r
+
+instance ( ConstructableInterval_ interval point
          , Default extra
-         ) => ClosedInterval_ (interval :+ extra) r where
+         ) => ConstructableInterval_ (interval :+ extra) point where
+  mkInterval s t = mkInterval s t :+ def
+
+instance ( ConstructableClosedInterval_ interval r
+         , Default extra
+         ) => ConstructableClosedInterval_ (interval :+ extra) r where
   mkClosedInterval s t = mkClosedInterval s t :+ def
 
-instance ( OpenInterval_ interval r
+instance ( ConstructableOpenInterval_ interval r
          , Default extra
-         ) => OpenInterval_ (interval :+ extra) r where
+         ) => ConstructableOpenInterval_ (interval :+ extra) r where
   mkOpenInterval s t = mkOpenInterval s t :+ def
diff --git a/hgeometry/kernel/src/HGeometry/Interval/HalfOpen.hs b/hgeometry/kernel/src/HGeometry/Interval/HalfOpen.hs
index df64fc82e..0407dbd31 100644
--- a/hgeometry/kernel/src/HGeometry/Interval/HalfOpen.hs
+++ b/hgeometry/kernel/src/HGeometry/Interval/HalfOpen.hs
@@ -52,10 +52,12 @@ instance HasEndPoint (HalfOpenInterval r) (EndPoint Closed r) where
   endPoint = lens _right (\ai x -> ai { _right = x})
 
 instance IntervalLike_ (HalfOpenInterval r) r where
-  mkInterval = HalfOpenInterval
 
 instance Interval_ (HalfOpenInterval r) r where
 
+instance ConstructableInterval_ (HalfOpenInterval r) r where
+  mkInterval = HalfOpenInterval
+
 instance (Ord r) => Point 1 r `HasIntersectionWith` HalfOpenInterval r where
   (Point1 q) `intersects` i = q `stabsInterval` i
 
diff --git a/hgeometry/kernel/src/HGeometry/Interval/Internal.hs b/hgeometry/kernel/src/HGeometry/Interval/Internal.hs
index 4dea2b4e1..1b491b3d9 100644
--- a/hgeometry/kernel/src/HGeometry/Interval/Internal.hs
+++ b/hgeometry/kernel/src/HGeometry/Interval/Internal.hs
@@ -83,15 +83,19 @@ type instance EndPointOf   (Interval endPoint r) = endPoint r
 
 instance ( EndPoint_ (endPoint r), IxValue (endPoint r) ~ r
          ) => IntervalLike_ (Interval endPoint r) r where
-  mkInterval = Interval
 
 instance ( EndPoint_ (endPoint r), IxValue (endPoint r) ~ r
          ) => Interval_ (Interval endPoint r) r where
 
+instance ( EndPoint_ (endPoint r), IxValue (endPoint r) ~ r
+         ) => ConstructableInterval_ (Interval endPoint r) r where
+  mkInterval = Interval
 
 instance ClosedInterval_ (ClosedInterval r) r
+instance ConstructableClosedInterval_ (ClosedInterval r) r
 
 instance OpenInterval_ (OpenInterval r) r
+instance ConstructableOpenInterval_ (OpenInterval r) r
 
 --------------------------------------------------------------------------------
 
diff --git a/hgeometry/kernel/src/HGeometry/LineSegment/Class.hs b/hgeometry/kernel/src/HGeometry/LineSegment/Class.hs
index 04d9070fd..d20f3db3c 100644
--- a/hgeometry/kernel/src/HGeometry/LineSegment/Class.hs
+++ b/hgeometry/kernel/src/HGeometry/LineSegment/Class.hs
@@ -11,6 +11,7 @@
 --------------------------------------------------------------------------------
 module HGeometry.LineSegment.Class
   ( LineSegment_(..), pattern LineSegment_
+  , ConstructableLineSegment_(..)
   , ClosedLineSegment_
   , OpenLineSegment_
 
@@ -57,14 +58,18 @@ class HasOnSegment lineSegment d | lineSegment -> d where
 class ( IntervalLike_ lineSegment point
       , Point_ point (Dimension lineSegment) (NumType lineSegment)
       ) => LineSegment_ lineSegment point | lineSegment -> point where
-  {-# MINIMAL uncheckedLineSegment #-}
+  {-# MINIMAL  #-}
 
+-- | A class representing line segments
+class ( LineSegment_ lineSegment point
+      ) => ConstructableLineSegment_ lineSegment point where
+  {-# MINIMAL uncheckedLineSegment #-}
 
   -- | Create a segment
   --
   -- pre: the points are disjoint
   uncheckedLineSegment     :: point -> point -> lineSegment
-  uncheckedLineSegment s t = mkInterval (mkEndPoint s) (mkEndPoint t)
+  -- uncheckedLineSegment s t = mkInterval (mkEndPoint s) (mkEndPoint t)
 
   -- | smart constructor that creates a valid segment, i.e. it
   -- validates that the endpoints are disjoint.
@@ -74,6 +79,9 @@ class ( IntervalLike_ lineSegment point
     | s^.vector /= t^.vector = Just $ uncheckedLineSegment s t
     | otherwise              = Nothing
 
+
+
+
 -- | A class representing Closed Linesegments
 class ( LineSegment_ lineSegment point
       , StartPointOf lineSegment ~ EndPoint Closed point
@@ -87,12 +95,10 @@ class ( LineSegment_ lineSegment point
       ) => OpenLineSegment_ lineSegment point where
 
 
--- | Constructs a line segment from the start and end point
+-- | Deconstructs a line segment from the start and end point
 pattern LineSegment_     :: forall lineSegment point. LineSegment_ lineSegment point
                          => point -> point -> lineSegment
 pattern LineSegment_ s t <- (startAndEnd -> (s,t))
-  where
-    LineSegment_ s t = uncheckedLineSegment s t
 {-# COMPLETE LineSegment_ #-}
 
 --------------------------------------------------------------------------------
@@ -108,7 +114,9 @@ pattern LineSegment_ s t <- (startAndEnd -> (s,t))
 -- >>> interpolate 1 $ ClosedLineSegment origin (Point2 10.0 10.0)
 -- Point2 10.0 10.0
 interpolate       :: forall lineSegment d point r
-                     . (Fractional r, LineSegment_ lineSegment point, Point_ point d r)
+                     . ( Fractional r, LineSegment_ lineSegment point
+                       , ConstructablePoint_ point d r
+                       )
                   => r -> lineSegment -> point
 interpolate lam (LineSegment_ s t) =
   fromVector $ (s^.vector ^* (1-lam)) ^+^ (t^.vector ^* lam)
@@ -221,16 +229,17 @@ yCoordAt x (LineSegment_ (Point2_ px py) (Point2_ qx qy))
 --------------------------------------------------------------------------------
 
 instance ( LineSegment_ segment point
-         , Default extra
          ) => LineSegment_ (segment :+ extra) point where
+
+instance ( ConstructableLineSegment_ segment point
+         , Default extra
+         ) => ConstructableLineSegment_ (segment :+ extra) point where
   uncheckedLineSegment p q = uncheckedLineSegment p q :+ def
 
 instance ( ClosedLineSegment_ segment point
-         , Default extra
          ) => ClosedLineSegment_ (segment :+ extra) point where
 
 instance ( OpenLineSegment_ segment point
-         , Default extra
          ) => OpenLineSegment_ (segment :+ extra) point where
 
 instance HasOnSegment lineSegment d =>  HasOnSegment (lineSegment :+ extra) d where
diff --git a/hgeometry/kernel/src/HGeometry/LineSegment/Internal.hs b/hgeometry/kernel/src/HGeometry/LineSegment/Internal.hs
index c1c98a90f..32458564b 100644
--- a/hgeometry/kernel/src/HGeometry/LineSegment/Internal.hs
+++ b/hgeometry/kernel/src/HGeometry/LineSegment/Internal.hs
@@ -63,10 +63,10 @@ import Text.Read
 type LineSegment :: (Type -> Type) -> Type -> Type
 newtype LineSegment endPoint point = MkLineSegment (Interval endPoint point)
 
--- | Default implementation of Closed LineSegments
+-- | A type representing Closed LineSegments
 type ClosedLineSegment point = LineSegment (EndPoint Closed) point
 
--- | Default implementation of Open LineSegments
+-- | A type representing Open LineSegments
 type OpenLineSegment point   = LineSegment (EndPoint Open) point
 
 -- | Construct a line Segment
@@ -133,12 +133,16 @@ type instance EndPointOf   (LineSegment endPoint point) = endPoint point
 instance ( IxValue (endPoint point) ~ point
          , EndPoint_ (endPoint point)
          ) => IntervalLike_ (LineSegment endPoint point) point where
-  mkInterval = LineSegment
 
 instance ( IxValue (endPoint point) ~ point
          , EndPoint_ (endPoint point)
          , Point_ point (Dimension point) (NumType point)
          ) => LineSegment_ (LineSegment endPoint point) point where
+
+instance ( IxValue (endPoint point) ~ point
+         , EndPoint_ (endPoint point)
+         , Point_ point (Dimension point) (NumType point)
+         ) => ConstructableLineSegment_ (LineSegment endPoint point) point where
   uncheckedLineSegment s t = LineSegment (mkEndPoint s) (mkEndPoint t)
 
 instance ( Point_ point (Dimension point) (NumType point)
diff --git a/hgeometry/kernel/src/HGeometry/Viewport.hs b/hgeometry/kernel/src/HGeometry/Viewport.hs
index 67a37055e..06e8a5bb8 100644
--- a/hgeometry/kernel/src/HGeometry/Viewport.hs
+++ b/hgeometry/kernel/src/HGeometry/Viewport.hs
@@ -117,7 +117,7 @@ centeredOrigin       :: ( Fractional r
                         , Point_ point 2 r
                         ) => rectangle -> Viewport r
 centeredOrigin rect' = Viewport rect
-                                (translation $ centerPoint rect' .-. origin)
+                                (translation $ (centerPoint rect')^.vector)
   where rect = Box (rect'^.minPoint.asPoint) (rect'^.maxPoint.asPoint)
 
 
diff --git a/hgeometry/point/src/HGeometry/Point.hs b/hgeometry/point/src/HGeometry/Point.hs
index 56d717665..4f769facd 100644
--- a/hgeometry/point/src/HGeometry/Point.hs
+++ b/hgeometry/point/src/HGeometry/Point.hs
@@ -20,6 +20,7 @@ module HGeometry.Point
   -- , {- | construct a 3-dimensional point -} pattern Point3
   -- , {- | construct a 4-dimensional point -} pattern Point4
   , Point_(..), pattern Point1_, pattern Point2_, pattern Point3_, pattern Point4_
+  , ConstructablePoint_(..)
   , HasVector(..)
   , HasCoordinates(..)
   , asPoint
diff --git a/hgeometry/point/src/HGeometry/Point/Class.hs b/hgeometry/point/src/HGeometry/Point/Class.hs
index 1d34eaa56..4f482236a 100644
--- a/hgeometry/point/src/HGeometry/Point/Class.hs
+++ b/hgeometry/point/src/HGeometry/Point/Class.hs
@@ -1,6 +1,7 @@
 {-# LANGUAGE UndecidableInstances #-}
 {-# LANGUAGE AllowAmbiguousTypes #-}
 {-# LANGUAGE DefaultSignatures #-}
+{-# OPTIONS_GHC -Wno-orphans #-}
 --------------------------------------------------------------------------------
 -- |
 -- Module      :  HGeometry.Point.Class
@@ -16,6 +17,7 @@ module HGeometry.Point.Class
   , HasCoordinates(..)
   , Affine_(..)
   , Point_(..), pattern Point1_, pattern Point2_, pattern Point3_, pattern Point4_
+  , ConstructablePoint_(..)
   , origin
   , pointFromList
   , coord
@@ -24,7 +26,6 @@ module HGeometry.Point.Class
   -- , projectPoint
   -- , PointFor
   , HasPoints(..), HasPoints'
-  , NoDefault(..)
   ) where
 
 import           Control.Lens
@@ -32,7 +33,6 @@ import           Data.Default.Class
 import           Data.Function (on)
 import qualified Data.List.NonEmpty as NonEmpty
 import           Data.Proxy (Proxy(..))
-import           GHC.Generics (Generic)
 import           GHC.TypeNats
 import           HGeometry.Ext
 import           HGeometry.Properties
@@ -161,13 +161,7 @@ instance ( d ~ Dimension (v r)
 class ( Affine_ point d r
       , HasVector point point
       ) => Point_ point d r where
-  {-# MINIMAL fromVector #-}
-
-  -- | Construct a point from a vector
-  --
-  -- >>> fromVector (Vector4 1 2 3 4) :: Point 4 Int
-  -- Point4 1 2 3 4
-  fromVector :: Vector d r -> point
+  {-# MINIMAL #-}
 
   -- | Get the coordinate in a given dimension. This operation is unsafe in the
   -- sense that no bounds are checked. Consider using `coord` instead.
@@ -183,6 +177,16 @@ class ( Affine_ point d r
   {-# INLINE coord' #-}
 
 
+-- | Type class for constructable points
+class Point_ point d r => ConstructablePoint_ point d r where
+  {-# MINIMAL fromVector #-}
+
+  -- | Construct a point from a vector
+  --
+  -- >>> fromVector (Vector4 1 2 3 4) :: Point 4 Int
+  -- Point4 1 2 3 4
+  fromVector :: Vector d r -> point
+
 
 -- | Get the coordinate in a given dimension
 --
@@ -202,7 +206,13 @@ instance ( d ~ Dimension (v r)
          , r ~ IxValue (v r)
          , Vector_ (v r) d r
          , Additive_ (Vector d r) d r
-         ) => Point_ (Linear.Point v r) d r where
+         ) => Point_ (Linear.Point v r) d r
+
+instance ( d ~ Dimension (v r)
+         , r ~ IxValue (v r)
+         , Vector_ (v r) d r
+         , Additive_ (Vector d r) d r
+         ) => ConstructablePoint_ (Linear.Point v r) d r where
   fromVector = Linear.P . review _Vector
   {-# INLINE fromVector #-}
 
@@ -210,46 +220,44 @@ instance ( d ~ Dimension (v r)
 --
 -- >>> origin :: Point 4 Int
 -- Point4 0 0 0 0
-origin :: forall point d r. (Num r, Point_ point d r) => point
+origin :: forall point d r. (Num r, ConstructablePoint_ point d r) => point
 origin = fromVector zero
 {-# INLINE origin #-}
 
--- | A bidirectional pattern synonym for 1 dimensional points.
+-- | A pattern synonym for 1 dimensional points.
 pattern Point1_   :: Point_ point 1 r => r -> point
 pattern Point1_ x <- (view xCoord -> x)
-  where
-    Point1_ x = fromVector $ Vector1 x
-{-# INLINE Point1_ #-}
+--   where
+--     Point1_ x = fromVector $ Vector1 x
+-- {-# INLINE Point1_ #-}
 {-# COMPLETE Point1_ #-}
 
--- | A bidirectional pattern synonym for 2 dimensional points.
+-- | A pattern synonym for 2 dimensional points.
 pattern Point2_     :: ( Point_ point 2 r
                        ) => r -> r -> point
 pattern Point2_ x y <- (view vector -> Vector2 x y)
- where
-   Point2_ x y = fromVector $ Vector2 x y
-{-# INLINE Point2_ #-}
+--  where
+--    Point2_ x y = fromVector $ Vector2 x y
+-- {-# INLINE Point2_ #-}
 {-# COMPLETE Point2_ #-}
 
 
--- | A bidirectional pattern synonym for 3 dimensional points.
+-- | A pattern synonym for 3 dimensional points.
 pattern Point3_       :: ( Point_ point 3 r
-                         -- , ConstructableVector_ (Vector.VectorFamily 3 r) 3 r
                          ) => r -> r -> r -> point
 pattern Point3_ x y z <- (view vector -> Vector3 x y z)
-  where
-    Point3_ x y z = fromVector $ Vector3 x y z
-{-# INLINE Point3_ #-}
+--   where
+--     Point3_ x y z = fromVector $ Vector3 x y z
+-- {-# INLINE Point3_ #-}
 {-# COMPLETE Point3_ #-}
 
 -- | A bidirectional pattern synonym for 4 dimensional points.
 pattern Point4_         :: ( Point_ point 4 r
-                           -- , ConstructableVector_ (Vector.VectorFamily 4 r) 4 r
                            ) => r -> r -> r -> r -> point
 pattern Point4_ x y z w <- (view vector -> Vector4 x y z w)
-  where
-    Point4_ x y z w = fromVector $ Vector4 x y z w
-{-# INLINE Point4_ #-}
+--   where
+--     Point4_ x y z w = fromVector $ Vector4 x y z w
+-- {-# INLINE Point4_ #-}
 {-# COMPLETE Point4_ #-}
 
 
@@ -262,7 +270,7 @@ pattern Point4_ x y z w <- (view vector -> Vector4 x y z w)
 -- Nothing
 -- >>> pointFromList [1,2,3,4] :: Maybe (Point 3 Int)
 -- Nothing
-pointFromList :: ( Point_ point d r
+pointFromList :: ( ConstructablePoint_ point d r
                  , Vector_ (Vector d r) d r
                  ) => [r] -> Maybe point
 pointFromList = fmap fromVector . vectorFromList
@@ -356,15 +364,11 @@ instance Affine_ point d r => Affine_ (point :+ extra) d r where
   p .+^ v = p&core %~ (.+^ v)
   {-# INLINE (.+^) #-}
 
-instance (Point_ point d r, Default extra) => Point_ (point :+ extra) d r where
+instance (Point_ point d r) => Point_ (point :+ extra) d r where
   {-# SPECIALIZE instance Point_ point d r => Point_ (point :+ ()) d r #-}
-  fromVector v = fromVector v :+ def
 
-
--- | A newtype that can discharge the Default constraint in an unsafe way, if you really
--- sure that you'll never actually need the default
-newtype NoDefault extra = NoDefault extra
-  deriving newtype (Show,Read,Eq,Ord,Enum,Num,Bounded,Real,Fractional,RealFrac,Generic)
-
-instance Default (NoDefault extra) where
-  def = error "NoDefault does not have an actual default. So something went wrong"
+instance (ConstructablePoint_ point d r, Default extra)
+          => ConstructablePoint_ (point :+ extra) d r where
+  {-# SPECIALIZE instance ConstructablePoint_ point d r
+                          => ConstructablePoint_ (point :+ ()) d r #-}
+  fromVector v = fromVector v :+ def
diff --git a/hgeometry/point/src/HGeometry/Point/PointF.hs b/hgeometry/point/src/HGeometry/Point/PointF.hs
index e14ca5a0a..0bdd6921f 100644
--- a/hgeometry/point/src/HGeometry/Point/PointF.hs
+++ b/hgeometry/point/src/HGeometry/Point/PointF.hs
@@ -108,6 +108,10 @@ instance ( Additive_ vector       d r
 instance ( Additive_ vector       d r
          , Additive_ (Vector d r) d r
          ) => Point_ (PointF vector) d r where
+
+instance ( Additive_ vector       d r
+         , Additive_ (Vector d r) d r
+         ) => ConstructablePoint_ (PointF vector) d r where
   fromVector = Point . review _Vector
   {-# INLINE fromVector #-}
 
diff --git a/hgeometry/point/src/HGeometry/Point/Quadrants.hs b/hgeometry/point/src/HGeometry/Point/Quadrants.hs
index 6315c1f1f..535d7319b 100644
--- a/hgeometry/point/src/HGeometry/Point/Quadrants.hs
+++ b/hgeometry/point/src/HGeometry/Point/Quadrants.hs
@@ -35,7 +35,7 @@ quadrantWith c p = case ( (c^.xCoord) `compare` (p^.xCoord)
                                    (LT, GT) -> BottomRight
 
 -- | Quadrants with respect to the origin
-quadrant :: (Ord r, Num r, Point_ point 2 r) => point -> Quadrant
+quadrant :: (Ord r, Num r, ConstructablePoint_ point 2 r) => point -> Quadrant
 quadrant = quadrantWith origin
 
 -- | Given a center point c, and a set of points, partition the points into
diff --git a/hgeometry/point/test/Point/CmpAround.hs b/hgeometry/point/test/Point/CmpAround.hs
index fa076a67c..949d3e336 100644
--- a/hgeometry/point/test/Point/CmpAround.hs
+++ b/hgeometry/point/test/Point/CmpAround.hs
@@ -1,14 +1,13 @@
 module Point.CmpAround where
 
 import Control.Lens ((^.))
-import Data.Default.Class
 import HGeometry.Ext
 import HGeometry.Point
 
 
 -- | Counter clockwise ordering of the points around c. Points are ordered with
 -- respect to the positive x-axis.
-ccwCmpAroundByQuadrant       :: (Num r, Ord r, Default p)
+ccwCmpAroundByQuadrant       :: (Num r, Ord r)
                              => Point 2 r :+ p -> Point 2 r :+ p -> Point 2 r :+ p -> Ordering
 ccwCmpAroundByQuadrant c q r = case (quadrantWith c q `compare` quadrantWith c r) of
                        EQ -> case ccw (c^.core) (q^.core) (r^.core) of
@@ -20,7 +19,7 @@ ccwCmpAroundByQuadrant c q r = case (quadrantWith c q `compare` quadrantWith c r
 
 
 
-cwCmpAroundByQuadrant :: (Num r, Ord r, Default p)
+cwCmpAroundByQuadrant :: (Num r, Ord r)
             => Point 2 r :+ p -> Point 2 r :+ p -> Point 2 r :+ p -> Ordering
 cwCmpAroundByQuadrant c q r = case (quadrantWith c q `compare` quadrantWith c r) of
                        EQ -> case ccw (c^.core) (q^.core) (r^.core) of
@@ -32,7 +31,7 @@ cwCmpAroundByQuadrant c q r = case (quadrantWith c q `compare` quadrantWith c r)
                                 -- specified by the quadrant.
 
 -- | Original implementation of cw with distance check
-cwCmpAroundByQuadrantWithDist       :: (Num r, Ord r, Default p)
+cwCmpAroundByQuadrantWithDist       :: (Num r, Ord r)
                                     => Point 2 r :+ p -> Point 2 r :+ p -> Point 2 r :+ p
                                     -> Ordering
 cwCmpAroundByQuadrantWithDist c q r = case (quadrantWith c q `compare` quadrantWith c r) of
diff --git a/hgeometry/src/HGeometry/HalfPlane/CommonIntersection.hs b/hgeometry/src/HGeometry/HalfPlane/CommonIntersection.hs
index 54af40d60..1f0c6a813 100644
--- a/hgeometry/src/HGeometry/HalfPlane/CommonIntersection.hs
+++ b/hgeometry/src/HGeometry/HalfPlane/CommonIntersection.hs
@@ -8,11 +8,13 @@ module HGeometry.HalfPlane.CommonIntersection
   -- , LowerBoundary(..)
   ) where
 
+
+import           Control.Lens
 import           Control.Lens hiding (Empty)
 import           Data.Bifunctor (first)
-import           Data.Default.Class
 import           Data.Foldable1
 import           Data.List.NonEmpty (NonEmpty(..))
+import qualified Data.List.NonEmpty as NonEmpty
 import           Data.Ord (comparing)
 import           Data.Sequence (Seq(..))
 import qualified Data.Sequence as Seq
@@ -67,7 +69,6 @@ commonIntersection     :: forall f halfPlane r.
                           , HalfPlane_ halfPlane r
                           , Fractional r, Ord r
 
-                          , Default (LineEQ r :+ halfPlane), Default halfPlane -- FIXME
                           , Show halfPlane, Show r
                           )
                        => f halfPlane -> CommonIntersection halfPlane r
@@ -273,7 +274,6 @@ boundaries :: ( HalfPlane_ halfPlane r
               , Ord r, Fractional r
 
 
-              , Default (LineEQ r :+ halfPlane), Default halfPlane -- FIXME
               , Show r, Show halfPlane
               ) => NonVerticals halfPlane r -> These2 (Chain Seq (LineEQ r :+ halfPlane) r)
 boundaries = bimap upperBoundary lowerBoundary
@@ -431,7 +431,6 @@ upperEnvelope :: forall g f line r.
                     , IsIntersectableWith line line
                     , Intersection line line ~ Maybe (LineLineIntersection line)
                     , HasFromFoldable g, Functor g
-                    , Default line -- TODO hack
                     )
                  => f line -> UpperEnvelopeF g (Point 2 r) line
 upperEnvelope = bimap (over yCoord negate) flipY . lowerEnvelope . fmap flipY
diff --git a/hgeometry/src/HGeometry/Line/LowerEnvelope.hs b/hgeometry/src/HGeometry/Line/LowerEnvelope.hs
index b6ae4ab3b..74d1cba07 100644
--- a/hgeometry/src/HGeometry/Line/LowerEnvelope.hs
+++ b/hgeometry/src/HGeometry/Line/LowerEnvelope.hs
@@ -21,7 +21,6 @@ module HGeometry.Line.LowerEnvelope
   ) where
 
 import           Control.Lens
-import           Data.Default.Class
 import           Data.Foldable1
 import qualified Data.List as List
 import           Data.List.NonEmpty (NonEmpty(..))
@@ -91,7 +90,6 @@ lowerEnvelope    :: forall g f line r.
                     , IsIntersectableWith line line
                     , Intersection line line ~ Maybe (LineLineIntersection line)
                     , HasFromFoldable g
-                    , Default line -- TODO hack
                     )
                  => f line -> LowerEnvelopeF g (Point 2 r) line
 lowerEnvelope = construct . fmap (view extra) . NonEmpty.reverse
diff --git a/hgeometry/src/HGeometry/LineSegment/Intersection/BentleyOttmann.hs b/hgeometry/src/HGeometry/LineSegment/Intersection/BentleyOttmann.hs
index d4af6f33c..27871b374 100644
--- a/hgeometry/src/HGeometry/LineSegment/Intersection/BentleyOttmann.hs
+++ b/hgeometry/src/HGeometry/LineSegment/Intersection/BentleyOttmann.hs
@@ -468,11 +468,12 @@ instance ( HasEndPoint lineSegment endPoint
 type instance StartPointOf (Flipped lineSegment) = StartPointOf lineSegment
 type instance EndPointOf   (Flipped lineSegment) = EndPointOf   lineSegment
 
-instance IntervalLike_ lineSegment point => IntervalLike_ (Flipped lineSegment) point where
-  mkInterval s t = NotFlipped $ mkInterval s t
-instance LineSegment_ lineSegment point => LineSegment_ (Flipped lineSegment) point where
-  uncheckedLineSegment s t = NotFlipped $ uncheckedLineSegment s t
+instance IntervalLike_ lineSegment point => IntervalLike_ (Flipped lineSegment) point
+instance LineSegment_ lineSegment point => LineSegment_ (Flipped lineSegment) point
 
+instance ConstructableLineSegment_ lineSegment point
+         => ConstructableLineSegment_ (Flipped lineSegment) point where
+  uncheckedLineSegment s t = NotFlipped $ uncheckedLineSegment s t
 
 instance segment `HasIntersectionWith` segment
           => (Flipped segment) `HasIntersectionWith` (Flipped segment) where
diff --git a/hgeometry/src/HGeometry/LowerEnvelope.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope.hs
similarity index 57%
rename from hgeometry/src/HGeometry/LowerEnvelope.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope.hs
index 6907f8f75..a155c14ef 100644
--- a/hgeometry/src/HGeometry/LowerEnvelope.hs
+++ b/hgeometry/src/HGeometry/Plane/LowerEnvelope.hs
@@ -1,6 +1,6 @@
 --------------------------------------------------------------------------------
 -- |
--- Module      :  HGeometry.LowerEnvelope
+-- Module      :  HGeometry.Plane.LowerEnvelope
 -- Copyright   :  (C) Frank Staals
 -- License     :  see the LICENSE file
 -- Maintainer  :  Frank Staals
@@ -8,14 +8,14 @@
 -- Computing the lower envelope of a set of planes
 --
 --------------------------------------------------------------------------------
-module HGeometry.LowerEnvelope
-  (  module HGeometry.LowerEnvelope.Naive
-  ,  module HGeometry.LowerEnvelope.Type
-  -- , module HGeometry.LowerEnvelope.Naive
+module HGeometry.Plane.LowerEnvelope
+  (  module HGeometry.Plane.LowerEnvelope.Naive
+  ,  module HGeometry.Plane.LowerEnvelope.Type
+  -- , module HGeometry.Plane.LowerEnvelope.Naive
   ) where
 
-import HGeometry.LowerEnvelope.Naive
-import HGeometry.LowerEnvelope.Type
+import HGeometry.Plane.LowerEnvelope.Naive
+import HGeometry.Plane.LowerEnvelope.Type
 
 
 --------------------------------------------------------------------------------
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/AdjListForm.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/AdjListForm.hs
similarity index 95%
rename from hgeometry/src/HGeometry/LowerEnvelope/AdjListForm.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/AdjListForm.hs
index 6d2d1f61e..d1e66ea7c 100644
--- a/hgeometry/src/HGeometry/LowerEnvelope/AdjListForm.hs
+++ b/hgeometry/src/HGeometry/Plane/LowerEnvelope/AdjListForm.hs
@@ -1,7 +1,7 @@
 {-# LANGUAGE UndecidableInstances #-}
 --------------------------------------------------------------------------------
 -- |
--- Module      :  HGeometry.LowerEnvelope.AdjListForm
+-- Module      :  HGeometry.Plane.LowerEnvelope.AdjListForm
 -- Copyright   :  (C) Frank Staals
 -- License     :  see the LICENSE file
 -- Maintainer  :  Frank Staals
@@ -10,7 +10,7 @@
 -- form.
 --
 --------------------------------------------------------------------------------
-module HGeometry.LowerEnvelope.AdjListForm
+module HGeometry.Plane.LowerEnvelope.AdjListForm
   ( LowerEnvelope(..)
   , LowerEnvelope'(LowerEnvelope)
   , ParallelPlane
@@ -45,9 +45,9 @@ import           HGeometry.Algorithms.DivideAndConquer
 import           HGeometry.HyperPlane.Class
 import           HGeometry.HyperPlane.NonVertical
 import           HGeometry.Line.LineEQ
-import           HGeometry.LowerEnvelope.Connected
-import           HGeometry.LowerEnvelope.VertexForm (IntersectionLine(..), intersectionLine)
-import qualified HGeometry.LowerEnvelope.VertexForm as VertexForm
+import           HGeometry.Plane.LowerEnvelope.Connected
+import           HGeometry.Plane.LowerEnvelope.VertexForm (IntersectionLine(..), intersectionLine)
+import qualified HGeometry.Plane.LowerEnvelope.VertexForm as VertexForm
 import           HGeometry.Point
 import           HGeometry.Properties
 import           HGeometry.Vector.NonEmpty.Util ()
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/AtMostThree.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/AtMostThree.hs
similarity index 100%
rename from hgeometry/src/HGeometry/LowerEnvelope/AtMostThree.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/AtMostThree.hs
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/BatchedPointLoc.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/BatchedPointLoc.hs
similarity index 100%
rename from hgeometry/src/HGeometry/LowerEnvelope/BatchedPointLoc.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/BatchedPointLoc.hs
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/ConflictLists.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/ConflictLists.hs
similarity index 100%
rename from hgeometry/src/HGeometry/LowerEnvelope/ConflictLists.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/ConflictLists.hs
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/Connected.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Connected.hs
similarity index 79%
rename from hgeometry/src/HGeometry/LowerEnvelope/Connected.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/Connected.hs
index 9455fdba0..2214cb9ff 100644
--- a/hgeometry/src/HGeometry/LowerEnvelope/Connected.hs
+++ b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Connected.hs
@@ -10,7 +10,7 @@
 -- form.
 --
 --------------------------------------------------------------------------------
-module HGeometry.LowerEnvelope.Connected
+module HGeometry.Plane.LowerEnvelope.Connected
   ( LowerEnvelope'(LowerEnvelope)
   , theUnboundedVertex, boundedVertices
 
@@ -28,9 +28,9 @@ module HGeometry.LowerEnvelope.Connected
   , projectedEdgeGeometries, projectedEdgeGeometry
   ) where
 
-import           HGeometry.LowerEnvelope.Type
-import           HGeometry.LowerEnvelope.Connected.Type
-import           HGeometry.LowerEnvelope.Connected.FromVertexForm
+import           HGeometry.Plane.LowerEnvelope.Type
+import           HGeometry.Plane.LowerEnvelope.Connected.Type
+import           HGeometry.Plane.LowerEnvelope.Connected.FromVertexForm
 import           HGeometry.Vector.NonEmpty.Util ()
 
 
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/Connected/FromVertexForm.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Connected/FromVertexForm.hs
similarity index 98%
rename from hgeometry/src/HGeometry/LowerEnvelope/Connected/FromVertexForm.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/Connected/FromVertexForm.hs
index 0d8df9938..e3ee52041 100644
--- a/hgeometry/src/HGeometry/LowerEnvelope/Connected/FromVertexForm.hs
+++ b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Connected/FromVertexForm.hs
@@ -1,7 +1,7 @@
 {-# LANGUAGE UndecidableInstances #-}
 --------------------------------------------------------------------------------
 -- |
--- Module      :  HGeometry.LowerEnvelope.FromVertexForm
+-- Module      :  HGeometry.Plane.LowerEnvelope.FromVertexForm
 -- Copyright   :  (C) Frank Staals
 -- License     :  see the LICENSE file
 -- Maintainer  :  Frank Staals
@@ -10,7 +10,7 @@
 -- form.
 --
 --------------------------------------------------------------------------------
-module HGeometry.LowerEnvelope.Connected.FromVertexForm
+module HGeometry.Plane.LowerEnvelope.Connected.FromVertexForm
   ( fromVertexForm'
   ) where
 
@@ -33,9 +33,9 @@ import           HGeometry.Foldable.Sort
 import           HGeometry.Foldable.Util
 import           HGeometry.HyperPlane.NonVertical
 import           HGeometry.Line
-import           HGeometry.LowerEnvelope.Connected.Type
-import           HGeometry.LowerEnvelope.Type
-import qualified HGeometry.LowerEnvelope.VertexForm as VertexForm
+import           HGeometry.Plane.LowerEnvelope.Connected.Type
+import           HGeometry.Plane.LowerEnvelope.Type
+import qualified HGeometry.Plane.LowerEnvelope.VertexForm as VertexForm
 import           HGeometry.Point
 import           HGeometry.Properties
 import           HGeometry.Vector
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/Connected/Type.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Connected/Type.hs
similarity index 98%
rename from hgeometry/src/HGeometry/LowerEnvelope/Connected/Type.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/Connected/Type.hs
index 1693b42de..47312b5db 100644
--- a/hgeometry/src/HGeometry/LowerEnvelope/Connected/Type.hs
+++ b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Connected/Type.hs
@@ -1,7 +1,7 @@
 {-# LANGUAGE UndecidableInstances #-}
 --------------------------------------------------------------------------------
 -- |
--- Module      :  HGeometry.LowerEnvelope.Connected.Type
+-- Module      :  HGeometry.Plane.LowerEnvelope.Connected.Type
 -- Copyright   :  (C) Frank Staals
 -- License     :  see the LICENSE file
 -- Maintainer  :  Frank Staals
@@ -10,7 +10,7 @@
 -- form.
 --
 --------------------------------------------------------------------------------
-module HGeometry.LowerEnvelope.Connected.Type
+module HGeometry.Plane.LowerEnvelope.Connected.Type
   ( LowerEnvelope'(LowerEnvelope)
   , theUnboundedVertex, boundedVertices
   , traverseLowerEnvelope
@@ -57,9 +57,9 @@ import           HGeometry.HyperPlane.Class
 import           HGeometry.HyperPlane.NonVertical
 import           HGeometry.Line
 import           HGeometry.LineSegment
-import           HGeometry.LowerEnvelope.Type
-import           HGeometry.LowerEnvelope.VertexForm (IntersectionLine(..),intersectionLine)
-import qualified HGeometry.LowerEnvelope.VertexForm as VertexForm
+import           HGeometry.Plane.LowerEnvelope.Type
+import           HGeometry.Plane.LowerEnvelope.VertexForm (IntersectionLine(..),intersectionLine)
+import qualified HGeometry.Plane.LowerEnvelope.VertexForm as VertexForm
 import           HGeometry.Point
 import           HGeometry.Properties
 import           HGeometry.Vector
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/DivideAndConquer.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/DivideAndConquer.hs
similarity index 100%
rename from hgeometry/src/HGeometry/LowerEnvelope/DivideAndConquer.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/DivideAndConquer.hs
diff --git a/hgeometry/src/HGeometry/Plane/LowerEnvelope/EpsApproximation.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/EpsApproximation.hs
new file mode 100644
index 000000000..666b5d9df
--- /dev/null
+++ b/hgeometry/src/HGeometry/Plane/LowerEnvelope/EpsApproximation.hs
@@ -0,0 +1,32 @@
+module HGeometry.LowerEnvelope.EpsApproximation
+  (
+
+  ) where
+
+import Control.Monad.State.Class
+import HGeometry.HyperPlane.Class
+import HGeometry.HyperPlane.NonVertical
+import System.Random.Stateful
+import Witherable
+
+--------------------------------------------------------------------------------
+
+-- | Given a value r, and a set of planes H, we construct a C/r-approximation A of H w.r.t
+-- the shallow downward ranges.
+epsApproximation :: ( Plane_ plane r
+                    , Ord r, Fractional r, Foldable f, Functor f, Ord plane
+                    )
+                 => Int -- ^ the parameter r
+                 -> f plane -> f plane
+epsApproximation = undefined
+
+-- | Given a value r, and a set of planes H, we construct a C/r-approximation A of H w.r.t
+-- the shallow downward ranges.
+epsApproximation'      :: ( Plane_ plane r
+                          , Ord r, Fractional r, Foldable f, Functor f, Ord plane
+                          , RandomGen gen, MonadState gen m
+                          , Show plane, Show r
+                          )
+                       => Int -- ^ the parameter r
+                       -> f plane -> m (f plane)
+epsApproximation' r hs = undefined
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/Naive.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Naive.hs
similarity index 95%
rename from hgeometry/src/HGeometry/LowerEnvelope/Naive.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/Naive.hs
index 96e9fe874..5ebd50ced 100644
--- a/hgeometry/src/HGeometry/LowerEnvelope/Naive.hs
+++ b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Naive.hs
@@ -1,5 +1,5 @@
 {-# LANGUAGE UndecidableInstances #-}
-module HGeometry.LowerEnvelope.Naive
+module HGeometry.Plane.LowerEnvelope.Naive
   ( lowerEnvelope
   , lowerEnvelopeVertexForm
   -- , triangulatedLowerEnvelope
@@ -16,8 +16,8 @@ import qualified Data.Set as Set
 import           HGeometry.Combinatorial.Util
 import           HGeometry.HyperPlane.Class
 import           HGeometry.HyperPlane.NonVertical
-import           HGeometry.LowerEnvelope.AdjListForm (LowerEnvelope, fromVertexForm)
-import           HGeometry.LowerEnvelope.VertexForm
+import           HGeometry.Plane.LowerEnvelope.AdjListForm (LowerEnvelope, fromVertexForm)
+import           HGeometry.Plane.LowerEnvelope.VertexForm
 import           HGeometry.Point
 
 --------------------------------------------------------------------------------
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/Sample.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Sample.hs
similarity index 100%
rename from hgeometry/src/HGeometry/LowerEnvelope/Sample.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/Sample.hs
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/Triangulate.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Triangulate.hs
similarity index 100%
rename from hgeometry/src/HGeometry/LowerEnvelope/Triangulate.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/Triangulate.hs
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/Type.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Type.hs
similarity index 99%
rename from hgeometry/src/HGeometry/LowerEnvelope/Type.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/Type.hs
index 8368bde61..f3a7c8194 100644
--- a/hgeometry/src/HGeometry/LowerEnvelope/Type.hs
+++ b/hgeometry/src/HGeometry/Plane/LowerEnvelope/Type.hs
@@ -1,5 +1,5 @@
 {-# LANGUAGE UndecidableInstances #-}
-module HGeometry.LowerEnvelope.Type
+module HGeometry.Plane.LowerEnvelope.Type
   ( VertexID
   , BoundedVertexF(Vertex)
   , location, definers, location2, incidentEdgesB
diff --git a/hgeometry/src/HGeometry/LowerEnvelope/VertexForm.hs b/hgeometry/src/HGeometry/Plane/LowerEnvelope/VertexForm.hs
similarity index 97%
rename from hgeometry/src/HGeometry/LowerEnvelope/VertexForm.hs
rename to hgeometry/src/HGeometry/Plane/LowerEnvelope/VertexForm.hs
index 6de37fcdb..e6be0926c 100644
--- a/hgeometry/src/HGeometry/LowerEnvelope/VertexForm.hs
+++ b/hgeometry/src/HGeometry/Plane/LowerEnvelope/VertexForm.hs
@@ -1,7 +1,7 @@
 {-# LANGUAGE UndecidableInstances #-}
 --------------------------------------------------------------------------------
 -- |
--- Module      :  HGeometry.LowerEnvelope.VertexForm
+-- Module      :  HGeometry.Plane.LowerEnvelope.VertexForm
 -- Copyright   :  (C) Frank Staals
 -- License     :  see the LICENSE file
 -- Maintainer  :  Frank Staals
@@ -10,7 +10,7 @@
 -- i.e. storing only the vertices.
 --
 --------------------------------------------------------------------------------
-module HGeometry.LowerEnvelope.VertexForm
+module HGeometry.Plane.LowerEnvelope.VertexForm
   ( VertexForm(VertexForm)
   , hasVertices, vertices'
   , singleton
@@ -31,7 +31,7 @@ import           HGeometry.Combinatorial.Util
 import           HGeometry.HyperPlane.NonVertical
 import           HGeometry.Intersection
 import           HGeometry.Line
-import           HGeometry.LowerEnvelope.Type
+import           HGeometry.Plane.LowerEnvelope.Type
 import           HGeometry.Point
 import           HGeometry.Properties
 import           Hiraffe.Graph
diff --git a/hgeometry/src/HGeometry/PlaneGraph/Class.hs b/hgeometry/src/HGeometry/PlaneGraph/Class.hs
index 313008613..6ca4b61e1 100644
--- a/hgeometry/src/HGeometry/PlaneGraph/Class.hs
+++ b/hgeometry/src/HGeometry/PlaneGraph/Class.hs
@@ -28,7 +28,6 @@ module HGeometry.PlaneGraph.Class
 
 import Control.Lens
 import Data.Coerce
-import Data.Default.Class
 import Data.Foldable1
 import Data.Functor.Apply
 import Data.Maybe (fromMaybe)
@@ -234,9 +233,9 @@ polygonFromFace      :: forall planeGraph vertex r.( PlaneGraph_ planeGraph vert
                      -> SimplePolygon (vertex :+ VertexIx planeGraph)
 polygonFromFace gr fi = poly'&vertices.extra %~ coerce
   where
-    poly' :: SimplePolygon (vertex :+ NoDefault (VertexIx planeGraph))
+    poly' :: SimplePolygon (vertex :+ VertexIx planeGraph)
     poly' = uncheckedFromCCWPoints
-         . fmap (\vi -> gr^?!vertexAt vi :+ NoDefault vi)
+         . fmap (\vi -> gr^?!vertexAt vi :+ vi)
          $ boundaryVertices fi gr
         -- note that this is safe, since boundaryVerticesOf guarantees that for
         -- interior faces, the vertices are returned in CCW order.
@@ -264,13 +263,6 @@ interiorFacePolygonAt fi = theFold
         draw _ = let poly =  polygonFromFace gr fi
                  in poly >$ indexed pPolyFPoly fi poly
 
-
-newtype NoDefault e = NoDefault e
-
-instance Default (NoDefault e) where
-  def = undefined
-
-
 --------------------------------------------------------------------------------
 
 -- | get the minimum of the elements the lens points to using the given comparison function
diff --git a/hgeometry/src/HGeometry/PolyLine.hs b/hgeometry/src/HGeometry/PolyLine.hs
index eff6e9f93..0cb11e2db 100644
--- a/hgeometry/src/HGeometry/PolyLine.hs
+++ b/hgeometry/src/HGeometry/PolyLine.hs
@@ -18,25 +18,21 @@ module HGeometry.PolyLine
   ) where
 
 
-import           Control.DeepSeq (NFData)
-import           Control.Lens
--- import qualified Data.Foldable as F
-import           Data.Functor.Classes
--- import qualified Data.List.NonEmpty as NonEmpty
-import           Data.Semigroup.Foldable
-import           Data.Vector.NonEmpty.Internal (NonEmptyVector(..))
--- import qualified Data.Vector.NonEmpty as NV
-import           GHC.Generics
-import           HGeometry.Box
-import           HGeometry.Point
-import           HGeometry.Foldable.Util
-import           HGeometry.PolyLine.Class
-import           HGeometry.Interval.Class
-import           HGeometry.Properties
-import           HGeometry.Transformation
--- import           HGeometry.Vector
-import           HGeometry.Vector.NonEmpty.Util ()
-import           Hiraffe.Graph
+import Control.DeepSeq (NFData)
+import Control.Lens
+import Data.Functor.Classes
+import Data.Semigroup.Foldable
+import Data.Vector.NonEmpty.Internal (NonEmptyVector(..))
+import GHC.Generics
+import HGeometry.Box
+import HGeometry.Foldable.Util
+import HGeometry.Interval.Class
+import HGeometry.Point
+import HGeometry.PolyLine.Class
+import HGeometry.Properties
+import HGeometry.Transformation
+import HGeometry.Vector.NonEmpty.Util ()
+import Hiraffe.Graph
 
 --------------------------------------------------------------------------------
 
@@ -124,4 +120,12 @@ instance ( Traversable1 f
          , Point_ point d r
          , TraversableWithIndex Int f
          ) => PolyLine_ (PolyLineF f point) point where
+
+instance ( Traversable1 f
+         , IxValue (f point) ~ point, Index (f point) ~ Int
+         , Ixed (f point)
+         , HasFromFoldable1 f
+         , Point_ point d r
+         , TraversableWithIndex Int f
+         ) => ConstructablePolyLine_ (PolyLineF f point) point where
   polyLineFromPoints = PolyLine . fromFoldable1
diff --git a/hgeometry/src/HGeometry/PolyLine/Class.hs b/hgeometry/src/HGeometry/PolyLine/Class.hs
index d53731c9e..dcb7c6013 100644
--- a/hgeometry/src/HGeometry/PolyLine/Class.hs
+++ b/hgeometry/src/HGeometry/PolyLine/Class.hs
@@ -10,23 +10,17 @@
 --------------------------------------------------------------------------------
 module HGeometry.PolyLine.Class
   ( PolyLine_(..)
+  , ConstructablePolyLine_(..)
   , _PolyLineLineSegment
   ) where
 
 import           Control.Lens
--- import Control.Lens.Internal.Fold (NonEmptyDList(..))
--- import Data.Functor.Apply (Apply)
--- import Data.Functor.Contravariant (phantom)
 import qualified Data.List.NonEmpty as NonEmpty
--- import Data.Maybe (fromMaybe)
 import           Data.Semigroup.Foldable
 import           HGeometry.LineSegment.Class
 import           HGeometry.Point.Class
 import           HGeometry.Properties
--- import HGeometry.Vector.Class
 import           Hiraffe.Graph
--- import Data.Function (on)
--- import Data.Semigroup (First(..))
 
 --------------------------------------------------------------------------------
 
@@ -41,6 +35,9 @@ class ( HasVertices polyLine polyLine
       , Dimension polyLine ~ Dimension point
       ) => PolyLine_ polyLine point | polyLine -> point where
 
+-- | Class for constructable polylglines
+class PolyLine_ polyLine point => ConstructablePolyLine_ polyLine point where
+
   -- | Constructs a polyline from a given sequence of points.
   --
   -- pre: there should be at least two distinct points
@@ -50,12 +47,13 @@ class ( HasVertices polyLine polyLine
 -- maybe make these two functions into a prism instead
 
 -- | Prism between a polyline and a line segment
-_PolyLineLineSegment :: ( LineSegment_ lineSegment point
-                        , PolyLine_ polyLine point
+_PolyLineLineSegment :: ( ConstructableLineSegment_ lineSegment point
+                        , ConstructablePolyLine_ polyLine point
                         ) => Prism' polyLine lineSegment
 _PolyLineLineSegment = prism' lineSegmentToPolyLine polyLineToLineSegment
   where
     lineSegmentToPolyLine s = polyLineFromPoints . NonEmpty.fromList $ [s^.start, s^.end]
 
     polyLineToLineSegment pl
-      | lengthOf vertices pl == 2 = Just $ uncheckedLineSegment (pl^.start) (pl^.end)                     | otherwise                 = Nothing
+      | lengthOf vertices pl == 2 = Just $ uncheckedLineSegment (pl^.start) (pl^.end)
+      | otherwise                 = Nothing
diff --git a/hgeometry/src/HGeometry/Polygon/Convex/MinkowskiSum.hs b/hgeometry/src/HGeometry/Polygon/Convex/MinkowskiSum.hs
index 00d3789bd..884ea3d2e 100644
--- a/hgeometry/src/HGeometry/Polygon/Convex/MinkowskiSum.hs
+++ b/hgeometry/src/HGeometry/Polygon/Convex/MinkowskiSum.hs
@@ -15,7 +15,6 @@ module HGeometry.Polygon.Convex.MinkowskiSum
   ) where
 
 import Control.Lens
-import Data.Default.Class
 import Data.List.NonEmpty (NonEmpty(..))
 import Data.Ord (comparing)
 import HGeometry.Ext
@@ -38,7 +37,6 @@ import HGeometry.Polygon.Simple.Class
 minkowskiSum     :: ( Ord r, Num r
                     , ConvexPolygon_ convexPolygon  point r
                     , ConvexPolygon_ convexPolygon' point' r
-                    , Default point'
                     )
                  => convexPolygon -> convexPolygon'
                  -> ConvexPolygon (point :+ point')
diff --git a/hgeometry/src/HGeometry/Polygon/Simple/Class.hs b/hgeometry/src/HGeometry/Polygon/Simple/Class.hs
index fa2695d0a..7fafe3ae2 100644
--- a/hgeometry/src/HGeometry/Polygon/Simple/Class.hs
+++ b/hgeometry/src/HGeometry/Polygon/Simple/Class.hs
@@ -55,7 +55,7 @@ class ( Polygon_ simplePolygon point r
   -- | Compute the centroid of a simple polygon.
   --
   -- running time: \(O(n)\)
-  centroid      :: Fractional r => simplePolygon -> point
+  centroid      :: (Fractional r, ConstructablePoint_ point' 2 r) => simplePolygon -> point'
   centroid poly = fromVector $ sum' xs ^/ (6 * signedArea poly)
     where
       xs = [ (p^.vector ^+^ q^.vector) ^* (p^.xCoord * q^.yCoord - q^.xCoord * p^.yCoord)
diff --git a/hgeometry/src/HGeometry/Polygon/Triangulation/MakeMonotone.hs b/hgeometry/src/HGeometry/Polygon/Triangulation/MakeMonotone.hs
index ed0133bc9..58c0e8ba9 100644
--- a/hgeometry/src/HGeometry/Polygon/Triangulation/MakeMonotone.hs
+++ b/hgeometry/src/HGeometry/Polygon/Triangulation/MakeMonotone.hs
@@ -15,7 +15,6 @@ module HGeometry.Polygon.Triangulation.MakeMonotone
   ) where
 
 import           Control.Lens
-import           Data.Default.Class
 import qualified Data.Map as Map
 import           Data.Ord (Down (..), comparing)
 import qualified Data.Set as Set
@@ -51,7 +50,7 @@ makeMonotone pg = constructGraph pg (computeDiagonals pg)
 --
 -- running time: \(O(n\log n)\)
 computeDiagonals    :: ( Polygon_ polygon point r, Point_ point 2 r
-                       , Ord r, Num r, Ord (VertexIx polygon), Default (VertexIx polygon)
+                       , Ord r, Num r, Ord (VertexIx polygon)
                        )
                     => polygon -> [Diagonal polygon]
 computeDiagonals pg =
@@ -91,7 +90,6 @@ p `cmpSweep` q = comparing (^.yCoord) p q <> comparing (Down . (^.xCoord)) p q
 -- | Handle an event
 handle :: forall polygon point r. ( Polygon_ polygon point r
                                   , Point_ point 2 r, Num r, Ord r
-                                  , Default (VertexIx polygon)
                                   , Ord (VertexIx polygon)
                                   )
        => polygon
diff --git a/hgeometry/src/HGeometry/Polygon/Triangulation/Types.hs b/hgeometry/src/HGeometry/Polygon/Triangulation/Types.hs
index 47f5ecffa..1a6ba0cb3 100644
--- a/hgeometry/src/HGeometry/Polygon/Triangulation/Types.hs
+++ b/hgeometry/src/HGeometry/Polygon/Triangulation/Types.hs
@@ -18,7 +18,6 @@ import           HGeometry.Point
 import           HGeometry.Polygon.Class
 import           HGeometry.Polygon.Simple.Class
 import           HGeometry.Vector
-import           Hiraffe.Graph.Class
 import           Hiraffe.PlanarGraph as PlanarGraph
 
 -- import qualified Data.Foldable as F
diff --git a/hgeometry/src/HGeometry/VoronoiDiagram/ViaLowerEnvelope.hs b/hgeometry/src/HGeometry/VoronoiDiagram/ViaLowerEnvelope.hs
index db9129099..2744c1003 100644
--- a/hgeometry/src/HGeometry/VoronoiDiagram/ViaLowerEnvelope.hs
+++ b/hgeometry/src/HGeometry/VoronoiDiagram/ViaLowerEnvelope.hs
@@ -20,7 +20,6 @@ module HGeometry.VoronoiDiagram.ViaLowerEnvelope
   ) where
 
 import           Control.Lens
-import           Data.Default.Class
 import           Data.Foldable1
 import qualified Data.Set as Set
 import           HGeometry.Box
@@ -28,9 +27,9 @@ import           HGeometry.Duality
 import           HGeometry.Ext
 import           HGeometry.HyperPlane.Class
 import           HGeometry.HyperPlane.NonVertical
-import           HGeometry.LowerEnvelope.AdjListForm
-import           HGeometry.LowerEnvelope.Naive (lowerEnvelopeVertexForm)
-import           HGeometry.LowerEnvelope.VertexForm (VertexForm, vertices')
+import           HGeometry.Plane.LowerEnvelope.AdjListForm
+import           HGeometry.Plane.LowerEnvelope.Naive (lowerEnvelopeVertexForm)
+import           HGeometry.Plane.LowerEnvelope.VertexForm (VertexForm, vertices')
 import           HGeometry.Point
 import           HGeometry.Properties
 
@@ -86,7 +85,7 @@ instance (Ord (NumType point), Num (NumType point)) => IsBoxable (VoronoiDiagram
 -- the lower envelope of these planes.
 --
 -- \(O(n\log n)\)
-voronoiDiagram     :: ( Point_ point 2 r, Functor f, Default point, Ord point
+voronoiDiagram     :: ( Point_ point 2 r, Functor f, Ord point
                       , Ord r, Fractional r, Foldable1 f
                       , Show point, Show r
                       ) => f point -> VoronoiDiagram point
@@ -98,14 +97,13 @@ voronoiDiagram pts = case lowerEnvelope' . fmap (\p -> liftPointToPlane p :+ p)
     getPoint = view (_Wrapped'.extra.to ColinearPoint)
 
 -- | Computes all voronoi vertices
-voronoiVertices :: ( Point_ point 2 r, Functor f, Default point, Ord point
+voronoiVertices :: ( Point_ point 2 r, Functor f, Ord point
                    , Ord r, Fractional r, Foldable f
                    ) => f point -> [Point 2 r]
 voronoiVertices = map (projectPoint . fst)
                 . itoListOf vertices'
                 . upperEnvelopeVertexForm
                 . fmap (\p -> liftPointToPlane p :+ p)
--- FIXME: get rid of the default point constraint
 -- FIXME: get rid of the ord point constraint
 
 -- | Computes the vertex form of the upper envelope. The z-coordinates are still flipped.
@@ -120,7 +118,7 @@ upperEnvelopeVertexForm = lowerEnvelopeVertexForm . fmap flipZ
 --------------------------------------------------------------------------------
 
 -- | Get the halflines and line segments representing the VoronoiDiagram
-edgeGeometries :: (Point_ point 2 r, Ord r, Fractional r, Default point
+edgeGeometries :: (Point_ point 2 r, Ord r, Fractional r
 
                   , Show point, Show r
                   )
diff --git a/hgeometry/test-with-ipe/test/HalfPlane/CommonIntersectionSpec.hs b/hgeometry/test-with-ipe/test/HalfPlane/CommonIntersectionSpec.hs
index a67133e28..d85bc3a04 100644
--- a/hgeometry/test-with-ipe/test/HalfPlane/CommonIntersectionSpec.hs
+++ b/hgeometry/test-with-ipe/test/HalfPlane/CommonIntersectionSpec.hs
@@ -3,7 +3,6 @@ module HalfPlane.CommonIntersectionSpec(spec) where
 
 import           Control.Lens hiding (below)
 import           Control.Monad ((>=>))
-import           Data.Default.Class
 import qualified Data.Foldable as F
 import           Data.List.NonEmpty (NonEmpty(..))
 import qualified Data.List.NonEmpty as NonEmpty
@@ -132,11 +131,6 @@ rightOf = HalfSpace Positive . VerticalLineThrough
 
 --------------------------------------------------------------------------------
 
--- FIXME: this instance does not really make sene I think, but whatever
-instance Num r => Default (HalfPlane r) where
-  def = HalfSpace Negative (VerticalLineThrough 0)
-instance Num r => Default (LineEQ r) where
-  def = LineEQ 1 0
 
 --------------------------------------------------------------------------------
 
diff --git a/hgeometry/test-with-ipe/test/Line/LowerEnvelopeSpec.hs b/hgeometry/test-with-ipe/test/Line/LowerEnvelopeSpec.hs
index 3a5aad88b..0c2120ea3 100644
--- a/hgeometry/test-with-ipe/test/Line/LowerEnvelopeSpec.hs
+++ b/hgeometry/test-with-ipe/test/Line/LowerEnvelopeSpec.hs
@@ -3,7 +3,6 @@ module Line.LowerEnvelopeSpec(spec) where
 
 import           Control.Lens hiding (below)
 import           Control.Monad ((>=>))
-import           Data.Default.Class
 import           Data.Foldable (minimumBy)
 import           Data.List.NonEmpty (NonEmpty(..))
 import qualified Data.List.NonEmpty as NonEmpty
@@ -88,7 +87,3 @@ theAnswer2 = LowerEnvelope
                          )
 
 --------------------------------------------------------------------------------
-
--- FIXME: hack
-instance Num r => Default (LineEQ r) where
-  def = LineEQ 1 0
diff --git a/hgeometry/test-with-ipe/test/PlaneGraph/RenderSpec.hs b/hgeometry/test-with-ipe/test/PlaneGraph/RenderSpec.hs
index caa710ff2..af3f67767 100644
--- a/hgeometry/test-with-ipe/test/PlaneGraph/RenderSpec.hs
+++ b/hgeometry/test-with-ipe/test/PlaneGraph/RenderSpec.hs
@@ -98,7 +98,7 @@ spec = describe "render planegraph tests" $ do
 
 drawGraph    :: ( PlaneGraph_ planeGraph vertex
                 , IsTransformable vertex
-                , Point_ vertex 2 r, Ord r, Real r
+                , ConstructablePoint_ vertex 2 r, Ord r, Real r
                 , Fractional r, Show r, Eq (FaceIx planeGraph)
                 , Show (Vertex planeGraph), Show (Dart planeGraph), Show (Face planeGraph)
                 , Show (EdgeIx planeGraph)
@@ -117,7 +117,7 @@ drawVertex _ v = [ iO $ ipeDiskMark (v^.asPoint) ! attr SLayer "vertex"
                                                          -- ! attr SStroke Ipe.red
                  ]
 
-drawEdge        :: ( PlaneGraph_ planeGraph vertex, Point_ vertex 2 r, IsTransformable vertex
+drawEdge        :: ( PlaneGraph_ planeGraph vertex, ConstructablePoint_ vertex 2 r, IsTransformable vertex
                    , Show (EdgeIx planeGraph), Fractional r, Real r)
                 => planeGraph -> EdgeIx planeGraph -> ClosedLineSegment vertex -> [IpeObject r]
 drawEdge gr d s = [ iO $ ipeLineSegment s ! attr SLayer "edges"
@@ -127,7 +127,7 @@ drawEdge gr d s = [ iO $ ipeLineSegment s ! attr SLayer "edges"
     c = interpolate 0.5 s ^. asPoint
 
 
-drawDart        :: ( PlaneGraph_ planeGraph vertex, Point_ vertex 2 r, IsTransformable vertex
+drawDart        :: ( PlaneGraph_ planeGraph vertex, ConstructablePoint_ vertex 2 r, IsTransformable vertex
                    , Show (Dart planeGraph), Fractional r, Real r)
                 => planeGraph -> DartIx planeGraph -> ClosedLineSegment vertex -> [IpeObject r]
 drawDart gr d s = [ iO $ ipeLineSegment (offset s)
diff --git a/hgeometry/test-with-ipe/test/Polygon/Convex/ConvexSpec.hs b/hgeometry/test-with-ipe/test/Polygon/Convex/ConvexSpec.hs
index 54fa6cc18..dc2b316cf 100644
--- a/hgeometry/test-with-ipe/test/Polygon/Convex/ConvexSpec.hs
+++ b/hgeometry/test-with-ipe/test/Polygon/Convex/ConvexSpec.hs
@@ -6,7 +6,6 @@ module Polygon.Convex.ConvexSpec
 
 import           Control.Arrow ((&&&))
 import           Control.Lens
-import           Data.Default.Class
 import qualified Data.List.NonEmpty as NonEmpty
 import           Golden
 import           HGeometry.ConvexHull.GrahamScan (convexHull)
@@ -29,12 +28,6 @@ import           Test.Hspec.WithTempFile
 
 type R = RealNumber 10
 
-instance Default (Point 2 R) where
-  def = origin
-
-instance Default (Point 2 Rational) where
-  def = origin
-
 --------------------------------------------------------------------------------
 
 spec :: Spec
@@ -104,15 +97,16 @@ toSpec (TestCase poly) = do
 --------------------------------------------------------------------------------
 
 -- | Center the given polygon at the origin. I.e. places the centroid at the origin.
-centerAtOrigin    :: ( SimplePolygon_ polygon  point r
+centerAtOrigin    :: forall polygon point r.
+                     ( SimplePolygon_ polygon  point r
                      , Fractional r
                      , IsTransformable polygon
                      ) => polygon -> polygon
-centerAtOrigin pg = translateBy (origin .-. centroid pg) pg
+centerAtOrigin pg = translateBy (origin .-. (centroid pg :: Point 2 r)) pg
 
 --------------------------------------------------------------------------------
 
-minkowskiTests       ::  (Fractional r, Ord r, Show r, Default (Point 2 r)
+minkowskiTests       ::  (Fractional r, Ord r, Show r
                          , IpeWriteText r
                          )
                      => String -> [ConvexPolygon (Point 2 r)] -> Spec
@@ -121,7 +115,7 @@ minkowskiTests s pgs = describe ("Minkowskisums on " ++ s) $
       zip [0..]
           [ (p,centerAtOrigin q) | p <- pgs, q <- pgs ]
 
-minkowskiTest       ::  ( Fractional r, Ord r, Show r, Default (Point 2 r)
+minkowskiTest       ::  ( Fractional r, Ord r, Show r
                         , IpeWriteText r
                         )
                     => Int -> ConvexPolygon (Point 2 r) -> ConvexPolygon (Point 2 r) -> Spec
@@ -160,7 +154,6 @@ instance (ShiftedEq b, Eq (ElemCyclic b)) => Eq (F a b) where
 naiveMinkowski     :: ( Ord r, Num r
                       , ConvexPolygon_ convexPolygon  point r
                       , ConvexPolygon_ convexPolygon' point' r
-                      , Default point'
                       )
                    => convexPolygon -> convexPolygon'
                    -> ConvexPolygon (point :+ point')
diff --git a/hgeometry/test-with-ipe/test/Polygon/Triangulation/TriangulateSpec.hs b/hgeometry/test-with-ipe/test/Polygon/Triangulation/TriangulateSpec.hs
index b6b803d96..568db5728 100644
--- a/hgeometry/test-with-ipe/test/Polygon/Triangulation/TriangulateSpec.hs
+++ b/hgeometry/test-with-ipe/test/Polygon/Triangulation/TriangulateSpec.hs
@@ -58,7 +58,7 @@ spec = describe "triangulateSpec" $ do
 
 _drawGraph    :: ( PlaneGraph_ planeGraph vertex
                  , IsTransformable vertex
-                 , Point_ vertex 2 r, Ord r, Real r, Fractional r, Show r, Eq (FaceIx planeGraph)
+                 , ConstructablePoint_ vertex 2 r, Ord r, Real r, Fractional r, Show r, Eq (FaceIx planeGraph)
                  , Show (Vertex planeGraph), Show (Dart planeGraph), Show (Face planeGraph)
                  , Show (EdgeIx planeGraph)
                  ) => planeGraph -> [IpeObject r]
diff --git a/hgeometry/test-with-ipe/test/VoronoiDiagram/VoronoiSpec.hs b/hgeometry/test-with-ipe/test/VoronoiDiagram/VoronoiSpec.hs
index a2b85109d..95d46591b 100644
--- a/hgeometry/test-with-ipe/test/VoronoiDiagram/VoronoiSpec.hs
+++ b/hgeometry/test-with-ipe/test/VoronoiDiagram/VoronoiSpec.hs
@@ -6,15 +6,14 @@ module VoronoiDiagram.VoronoiSpec
   ) where
 
 import           Control.Lens
-import           Data.Default.Class
 import           Golden
 -- import HGeometry.Combinatorial.Util
 import           HGeometry.Duality
 import           HGeometry.Ext
 import           HGeometry.HyperPlane.Class
 import           HGeometry.HyperPlane.NonVertical
-import           HGeometry.LowerEnvelope
-import           HGeometry.LowerEnvelope.AdjListForm
+import           HGeometry.Plane.LowerEnvelope
+import           HGeometry.Plane.LowerEnvelope.AdjListForm
 import qualified Data.Set as Set
 import qualified Data.List.NonEmpty as NonEmpty
 import qualified Data.Sequence as Seq
@@ -104,7 +103,7 @@ grow d (Box p q) = Box (p&coordinates %~ subtract d)
                        (q&coordinates %~ (+d))
 
 
-instance (HasDefaultIpeOut point, Point_ point 2 r, Fractional r, Ord r, Default point
+instance (HasDefaultIpeOut point, Point_ point 2 r, Fractional r, Ord r
          , Show r, Show point
          )
          => HasDefaultIpeOut (VoronoiDiagram point) where
@@ -114,7 +113,7 @@ instance (HasDefaultIpeOut point, Point_ point 2 r, Fractional r, Ord r, Default
     ConnectedVD vd  -> defIO vd
 
 
-instance (HasDefaultIpeOut point, Point_ point 2 r, Fractional r, Ord r, Default point
+instance (HasDefaultIpeOut point, Point_ point 2 r, Fractional r, Ord r
          , Show r, Show point
          )
          => HasDefaultIpeOut (VoronoiDiagram' point) where
@@ -126,10 +125,6 @@ instance (HasDefaultIpeOut point, Point_ point 2 r, Fractional r, Ord r, Default
         Left hl   -> iO $ ipeHalfLineIn bRect hl
         Right seg -> iO' seg
 
-
-instance Default (Point 2 R) where
-  def = error "not def"
-
 inputs :: [Point 2 R]
 inputs = [origin, Point2 10 10, Point2 10 0]
 
diff --git a/hgeometry/test/LowerEnvelope/NaiveSpec.hs b/hgeometry/test/LowerEnvelope/NaiveSpec.hs
index 535e25bc9..647e408e5 100644
--- a/hgeometry/test/LowerEnvelope/NaiveSpec.hs
+++ b/hgeometry/test/LowerEnvelope/NaiveSpec.hs
@@ -8,8 +8,8 @@ import           HGeometry.Combinatorial.Util
 import           HGeometry.HyperPlane.Class
 import           HGeometry.HyperPlane.NonVertical
 import           HGeometry.Instances ()
-import           HGeometry.LowerEnvelope
-import           HGeometry.LowerEnvelope.VertexForm
+import           HGeometry.Plane.LowerEnvelope
+import           HGeometry.Plane.LowerEnvelope.VertexForm
 import           HGeometry.Number.Real.Rational
 import           HGeometry.Point
 import           HGeometry.Vector
diff --git a/hgeometry/test/Polygon/Convex/ConvexSpec.hs b/hgeometry/test/Polygon/Convex/ConvexSpec.hs
index ca9b8fddc..33582febf 100644
--- a/hgeometry/test/Polygon/Convex/ConvexSpec.hs
+++ b/hgeometry/test/Polygon/Convex/ConvexSpec.hs
@@ -4,7 +4,6 @@ module Polygon.Convex.ConvexSpec
 
 import           Control.Lens hiding (elements)
 import           Control.Monad.State
-import           Data.Default.Class
 import qualified Data.List.NonEmpty as NonEmpty
 import           HGeometry.Boundary
 import           HGeometry.Box
@@ -34,8 +33,6 @@ import           Test.QuickCheck.Instances ()
 
 -- type R = RealNumber 10
 
-instance Default (Point 2 Rational) where
-  def = origin
 instance Arbitrary (ConvexPolygon (Point 2 Rational)) where
   arbitrary = let granularity = 1000000 in
     sized $ \n -> do
@@ -148,17 +145,17 @@ spec = describe "Convex Polygon tests" $ do
 --------------------------------------------------------------------------------
 
 -- | Center the given polygon at the origin. I.e. places the centroid at the origin.
-centerAtOrigin    :: ( SimplePolygon_ polygon  point r
+centerAtOrigin    :: forall polygon point r.
+                     ( SimplePolygon_ polygon  point r
                      , Fractional r
                      , IsTransformable polygon
                      ) => polygon -> polygon
-centerAtOrigin pg = translateBy (origin .-. centroid pg) pg
+centerAtOrigin pg = translateBy (origin .-. (centroid pg :: Point 2 r)) pg
 
 
 naiveMinkowski     :: ( Ord r, Num r
                       , ConvexPolygon_ convexPolygon  point r
                       , ConvexPolygon_ convexPolygon' point' r
-                      , Default point'
                       )
                    => convexPolygon -> convexPolygon'
                    -> ConvexPolygon (point :+ point')
diff --git a/hgeometry/test/Polygon/Triangulation/MakeMonotoneSpec.hs b/hgeometry/test/Polygon/Triangulation/MakeMonotoneSpec.hs
index e53555dc3..a022c5685 100644
--- a/hgeometry/test/Polygon/Triangulation/MakeMonotoneSpec.hs
+++ b/hgeometry/test/Polygon/Triangulation/MakeMonotoneSpec.hs
@@ -1,7 +1,6 @@
 module Polygon.Triangulation.MakeMonotoneSpec where
 
 import           Control.Lens
-import           Data.Default.Class
 import           Data.Maybe (fromJust)
 import qualified Data.Set as Set
 import           HGeometry.Ext
@@ -40,10 +39,6 @@ spec = describe "GeomBook Example" $ do
                seg2 = ClosedLineSegment (Point2 16 25) (Point2 13 23)
            cmpX q seg2 `shouldBe` EQ
 
-
-instance Default VertexType where
-  def = Regular
-
 sort'               :: Ord a => Vector 2 a-> Vector 2 a
 sort' (Vector2 x y) = Vector2 (min x y) (max x y)