diff --git a/hgeometry/kernel/src/HGeometry/HyperPlane/Class.hs b/hgeometry/kernel/src/HGeometry/HyperPlane/Class.hs
index 0596c2f88..e2b4d41b3 100644
--- a/hgeometry/kernel/src/HGeometry/HyperPlane/Class.hs
+++ b/hgeometry/kernel/src/HGeometry/HyperPlane/Class.hs
@@ -260,7 +260,7 @@ class HyperPlane_ hyperPlane d r
   -- | Construct a Hyperplane from a point and a normal. The normal points into the halfplane
   -- for which the side-test is positive.
   --
-  -- >>> myVerticalLine == fromPointAndNormal (Point2 5 30) (Vector2 (-1) 0)
+  -- >>> myVerticalLine == fromPointAndNormal (Point2 5 30) (Vector2 1 0)
   -- True
   fromPointAndNormal     :: ( Point_ point d r, Num r)
                          => point -> Vector d r -> hyperPlane
@@ -270,7 +270,7 @@ class HyperPlane_ hyperPlane d r
                                 , Has_ Metric_ d r
                                 )
                              => point -> Vector d r -> hyperPlane
-  fromPointAndNormal q n = hyperPlaneFromEquation $ cons a0 n
+  fromPointAndNormal q n = hyperPlaneFromEquation $ cons a0 (negated n)
     where
       a0 = (q^.vector) `dot` n
   {-# INLINE fromPointAndNormal #-}
diff --git a/hgeometry/kernel/src/HGeometry/HyperPlane/NonVertical.hs b/hgeometry/kernel/src/HGeometry/HyperPlane/NonVertical.hs
index 188df5856..4dd795f10 100644
--- a/hgeometry/kernel/src/HGeometry/HyperPlane/NonVertical.hs
+++ b/hgeometry/kernel/src/HGeometry/HyperPlane/NonVertical.hs
@@ -119,11 +119,10 @@ instance ( MkHyperPlaneConstraints d r
 
 
 instance ( MkHyperPlaneConstraints d r
+         , Fractional r, Eq r
          , 2 <= d
          ) => ConstructableHyperPlane_ (NonVerticalHyperPlane d r) d r where
 
-  type HyperPlaneFromEquationConstraint (NonVerticalHyperPlane d r) d r = (Fractional r, Eq r)
-
   -- | pre: the last component is not zero
   --
   --
@@ -134,17 +133,6 @@ instance ( MkHyperPlaneConstraints d r
       Nothing -> error "hyperPlaneFromEquation: Hyperplane is vertical!"
   {-# INLINE hyperPlaneFromEquation #-}
 
-  fromPointAndNormal _ n = NonVerticalHyperPlane n
-  -- see https://en.wikipedia.org/wiki/Normal_(geometry)
-  --
-  -- i.e. Alternatively, if the hyperplane is defined as the solution set of a single
-  -- linear equation a_1 x_1 + ⋯ + a_n x_n = c , then the vector n = ( a_1 , .. , a_n ) is
-  -- a normal.
-  --
-  -- FIXME: this seems fishy; don't we need the point?
-
-  --
-
 
 instance ( MkHyperPlaneConstraints d r, 1 + (d-1) ~ d
          , Num r
diff --git a/hgeometry/kernel/src/HGeometry/Line/LineEQ.hs b/hgeometry/kernel/src/HGeometry/Line/LineEQ.hs
index f8b51fbca..06b6dbe22 100644
--- a/hgeometry/kernel/src/HGeometry/Line/LineEQ.hs
+++ b/hgeometry/kernel/src/HGeometry/Line/LineEQ.hs
@@ -83,9 +83,8 @@ instance ( MkHyperPlaneConstraints 2 r
   onHyperPlane = onLine
 
 instance ( MkHyperPlaneConstraints 2 r
+         , Fractional r, Eq r
          ) => ConstructableHyperPlane_ (LineEQ r) 2 r where
-  type HyperPlaneFromEquationConstraint (LineEQ r) 2 r =
-       HyperPlaneFromEquationConstraint (NonVerticalHyperPlane 2 r) 2 r
   -- | pre: the last component is not zero
   hyperPlaneFromEquation = MkLineEQ
                          . hyperPlaneFromEquation @(NonVerticalHyperPlane 2 r)