diff --git a/brush-strokes/brush-strokes.cabal b/brush-strokes/brush-strokes.cabal
index 23cf924..e63d12b 100644
--- a/brush-strokes/brush-strokes.cabal
+++ b/brush-strokes/brush-strokes.cabal
@@ -14,6 +14,11 @@ flag use-fma
   default: True
   manual:  False
 
+flag asserts
+  description: Enable debugging assertions.
+  default: False
+  manual:  True
+
 common common
 
     build-depends:
@@ -89,6 +94,10 @@ common common
           base
             >= 4.19
 
+    if flag(asserts)
+      cpp-options:
+          -DASSERTS
+
 common extra
 
     build-depends:
@@ -133,6 +142,7 @@ library
       , Math.Root.Isolation
       , Math.Root.Isolation.Bisection
       , Math.Root.Isolation.Core
+      , Math.Root.Isolation.Degree
       , Math.Root.Isolation.GaussSeidel
       , Math.Root.Isolation.Narrowing
       , Debug.Utils
diff --git a/brush-strokes/src/cusps/bench/Bench/Types.hs b/brush-strokes/src/cusps/bench/Bench/Types.hs
index 5c559d7..bc67c07 100644
--- a/brush-strokes/src/cusps/bench/Bench/Types.hs
+++ b/brush-strokes/src/cusps/bench/Bench/Types.hs
@@ -12,8 +12,14 @@ module Bench.Types
 -- base
 import Data.Coerce
   ( coerce )
+import Data.Proxy
+  ( Proxy(..) )
+import Data.Type.Equality
+  ( (:~:)(Refl) )
 import GHC.Generics
   ( Generic )
+import GHC.TypeNats
+  ( sameNat )
 
 -- containers
 import Data.Sequence
@@ -111,10 +117,48 @@ getStrokeFunctions ( Brush brushShape brushShapeI mbRot ) sp0 crv =
         ( fmap nonDecreasing mbRot )
      )
 {-# INLINEABLE getStrokeFunctions #-}
+{-# SPECIALISE getStrokeFunctions
+    :: Brush ( ℝ 1 )
+    -> Point 1
+    -> Curve Open () ( Point 1 )
+    -> ( ℝ 1 -> Seq ( ℝ 1 -> StrokeDatum 2 () )
+       , 𝕀ℝ 1 -> Seq ( 𝕀ℝ 1 -> StrokeDatum 3 𝕀 ) )
+  #-}
+{-# SPECIALISE getStrokeFunctions
+    :: Brush ( ℝ 2 )
+    -> Point 2
+    -> Curve Open () ( Point 2 )
+    -> ( ℝ 1 -> Seq ( ℝ 1 -> StrokeDatum 2 () )
+       , 𝕀ℝ 1 -> Seq ( 𝕀ℝ 1 -> StrokeDatum 3 𝕀 ) )
+  #-}
+{-# SPECIALISE getStrokeFunctions
+    :: Brush ( ℝ 3 )
+    -> Point 3
+    -> Curve Open () ( Point 3 )
+    -> ( ℝ 1 -> Seq ( ℝ 1 -> StrokeDatum 2 () )
+       , 𝕀ℝ 1 -> Seq ( 𝕀ℝ 1 -> StrokeDatum 3 𝕀 ) )
+  #-}
+{-# SPECIALISE getStrokeFunctions
+    :: Brush ( ℝ 4 )
+    -> Point 4
+    -> Curve Open () ( Point 4 )
+    -> ( ℝ 1 -> Seq ( ℝ 1 -> StrokeDatum 2 () )
+       , 𝕀ℝ 1 -> Seq ( 𝕀ℝ 1 -> StrokeDatum 3 𝕀 ) )
+  #-}
 
 brushStrokeFunctions
   :: BrushStroke
   -> ( ℝ 1 -> Seq ( ℝ 1 -> StrokeDatum 2 () )
      , 𝕀ℝ 1 -> Seq ( 𝕀ℝ 1 -> StrokeDatum 3 𝕀 ) )
-brushStrokeFunctions ( BrushStroke { stroke = ( sp0, crv ), brush } ) =
-  getStrokeFunctions brush sp0 crv
+brushStrokeFunctions ( BrushStroke { stroke = ( sp0, crv ), brush = brush :: Brush ( ℝ nbParams ) } )
+  -- Trick for triggering specialisation... TODO improve this.
+  | Just Refl <- sameNat @nbParams @1 Proxy Proxy
+  = getStrokeFunctions @1 brush sp0 crv
+  | Just Refl <- sameNat @nbParams @2 Proxy Proxy
+  = getStrokeFunctions @2 brush sp0 crv
+  | Just Refl <- sameNat @nbParams @3 Proxy Proxy
+  = getStrokeFunctions @3 brush sp0 crv
+  | Just Refl <- sameNat @nbParams @4 Proxy Proxy
+  = getStrokeFunctions @4 brush sp0 crv
+  | otherwise
+  = getStrokeFunctions @nbParams brush sp0 crv
diff --git a/brush-strokes/src/cusps/bench/Main.hs b/brush-strokes/src/cusps/bench/Main.hs
index 06a3a34..b723f46 100644
--- a/brush-strokes/src/cusps/bench/Main.hs
+++ b/brush-strokes/src/cusps/bench/Main.hs
@@ -92,15 +92,14 @@ benchGroups :: [ ( String, NE.NonEmpty TestCase ) ]
 benchGroups =
   [ ( "ellipse"
     , NE.fromList
-      [ ellipseTestCase opts ("ε_bis=" ++ show ε_bis ++ if doBox1 then "box(1)" else "")
+      [ ellipseTestCase opts ("ε_bis=" ++ show ε_bis)
           ( 0, 1 ) pi
           ( defaultStartBoxes [ 0 .. 3 ] )
       | ε_bis <- [ 1e-6, 5e-6, 1e-5, 5e-5, 1e-4, 5e-4, 1e-4, 5e-4, 1e-3, 5e-3, 1e-2, 5e-2, 0.1, 0.2, 0.3 ]
-      , doBox1 <- [ False, True ]
       , let opts =
               RootIsolationOptions
                 { rootIsolationAlgorithms =
-                    defaultRootIsolationAlgorithms minWidth ε_bis doBox1
+                    defaultRootIsolationAlgorithms minWidth ε_bis
                 }
       ]
     )
diff --git a/brush-strokes/src/lib/Calligraphy/Brushes.hs b/brush-strokes/src/lib/Calligraphy/Brushes.hs
index a1d0b53..402eaae 100644
--- a/brush-strokes/src/lib/Calligraphy/Brushes.hs
+++ b/brush-strokes/src/lib/Calligraphy/Brushes.hs
@@ -50,9 +50,13 @@ type BrushFn i k brushParams
 -- | A brush, described as a base shape + an optional rotation.
 data Brush brushParams
   = Brush
-  { brushShape  :: BrushFn () 2 brushParams
-  , brushShapeI :: BrushFn 𝕀 3 brushParams
-  , mbRotation  :: Maybe ( brushParams -> Double )
+  { -- | Base brush shape, before applying any rotation (if any).
+    brushBaseShape  :: BrushFn () 2 brushParams
+    -- | Base brush shape, before applying any rotation (if any).
+  , brushBaseShapeI :: BrushFn 𝕀 3 brushParams
+    -- | Optional rotation angle function
+    -- (assumed to be a linear function).
+  , mbRotation      :: Maybe ( brushParams -> Double )
   }
 
 --------------------------------------------------------------------------------
@@ -74,27 +78,27 @@ type ParamsICt k i rec =
 circleBrush :: ( 1 <= RepDim params, ParamsCt params ) => Brush params
 circleBrush =
   Brush
-    { brushShape  = circleBrushFn @() @2 proxy# id
-    , brushShapeI = circleBrushFn @𝕀 @3 proxy# singleton
-    , mbRotation  = Nothing
+    { brushBaseShape  = circleBrushFn @() @2 proxy# id
+    , brushBaseShapeI = circleBrushFn @𝕀 @3 proxy# singleton
+    , mbRotation      = Nothing
     }
 
 {-# INLINEABLE ellipseBrush #-}
 ellipseBrush :: ( 3 <= RepDim params, ParamsCt params ) => Brush params
 ellipseBrush =
   Brush
-    { brushShape  = ellipseBrushFn @() @2 proxy# id
-    , brushShapeI = ellipseBrushFn @𝕀 @3 proxy# singleton
-    , mbRotation  = Just ( `index` ( Fin 3 ) )
+    { brushBaseShape  = ellipseBrushFn @() @2 proxy# id
+    , brushBaseShapeI = ellipseBrushFn @𝕀 @3 proxy# singleton
+    , mbRotation      = Just ( `index` ( Fin 3 ) )
     }
 
 {-# INLINEABLE tearDropBrush #-}
 tearDropBrush :: ( 3 <= RepDim params, ParamsCt params ) => Brush params
 tearDropBrush =
   Brush
-    { brushShape  = tearDropBrushFn @() @2 proxy# id
-    , brushShapeI = tearDropBrushFn @𝕀 @3 proxy# singleton
-    , mbRotation  = Just ( `index` ( Fin 3 ) )
+    { brushBaseShape  = tearDropBrushFn @() @2 proxy# id
+    , brushBaseShapeI = tearDropBrushFn @𝕀 @3 proxy# singleton
+    , mbRotation      = Just ( `index` ( Fin 3 ) )
     }
 
 --------------------------------------------------------------------------------
diff --git a/brush-strokes/src/lib/Math/Bezier/Stroke.hs b/brush-strokes/src/lib/Math/Bezier/Stroke.hs
index dc9b90a..dbccb1a 100644
--- a/brush-strokes/src/lib/Math/Bezier/Stroke.hs
+++ b/brush-strokes/src/lib/Math/Bezier/Stroke.hs
@@ -383,7 +383,7 @@ computeStrokeOutline rootAlgo mbCuspOptions fitParams ptParams toBrushParams bru
 
     brushShape :: ptData -> SplinePts Closed
     brushShape pt =
-      let Brush { brushShape = shapeFn, mbRotation = mbRot } = brush
+      let Brush { brushBaseShape = shapeFn, mbRotation = mbRot } = brush
           brushParams = toBrushParams $ ptParams pt
           shape = fun @Double shapeFn brushParams
       in case mbRot of
@@ -546,7 +546,7 @@ outlineFunction
   -> Curve Open crvData ptData
   -> OutlineInfo
 outlineFunction rootAlgo mbCuspOptions ptParams toBrushParams
-  ( Brush { brushShape, brushShapeI, mbRotation } ) = \ sp0 crv ->
+  ( Brush { brushBaseShape, brushBaseShapeI, mbRotation } ) = \ sp0 crv ->
   let
 
     usedParams :: C 2 ( ℝ 1 ) usedParams
@@ -561,7 +561,7 @@ outlineFunction rootAlgo mbCuspOptions ptParams toBrushParams
         coerce coerce
         path
         ( fmap toBrushParams usedParams )
-        brushShape
+        brushBaseShape
         mbRotation
 
     curvesI :: 𝕀ℝ 1 -- t
@@ -571,7 +571,7 @@ outlineFunction rootAlgo mbCuspOptions ptParams toBrushParams
         coerce coerce
         pathI
         ( fmap ( nonDecreasing toBrushParams ) usedParamsI )
-        brushShapeI
+        brushBaseShapeI
         ( fmap nonDecreasing mbRotation )
 
     usedParamsI :: C 3 ( 𝕀ℝ 1 ) ( 𝕀 usedParams )
@@ -589,9 +589,19 @@ outlineFunction rootAlgo mbCuspOptions ptParams toBrushParams
 
         D21 path_t path'_t _ = runD path t
         D21 params_t _ _     = runD usedParams t
-        brush_t              = value @Double @2 @brushParams
-                             $ runD brushShape
-                             $ toBrushParams params_t
+        brush_t              =
+          let brushParams = toBrushParams params_t
+              applyRot = case mbRotation of
+                Nothing -> id
+                Just getθ ->
+                 let θ = getθ brushParams
+                     cosθ = cos θ
+                     sinθ = sin θ
+                 in fmap ( unT . rotate cosθ sinθ . T )
+          in
+            applyRot $
+              value @Double @2 @brushParams $
+                runD brushBaseShape brushParams
 
     ( potentialCusps, definiteCusps ) =
       case mbCuspOptions of
@@ -644,6 +654,7 @@ pathAndUsedParams co toI ptParams sp0 crv =
       | let bez3 = Cubic.Bezier sp0 sp1 sp2 sp3
       -> ( bezier3 @k @i @( ℝ 2 )    co ( fmap ( toI . coords   ) bez3 )
          , bezier3 @k @i @usedParams co ( fmap ( toI . ptParams ) bez3 ) )
+{-# INLINEABLE pathAndUsedParams #-}
 
 -----------------------------------
 -- Various utility functions
@@ -1030,6 +1041,46 @@ brushStrokeData co1 co2 path params brush mbBrushRotation =
       let dbrush_t_s = dbrush_t s
           mbRotation = mbBrushRotation <&> \ getTheta -> fmap getTheta dparams_t
       in  envelopeEquation @k @i Proxy co1 dpath_t dbrush_t_s mbRotation
+{-# INLINEABLE brushStrokeData #-}
+{-# SPECIALISE brushStrokeData
+     :: ( I 𝕀 Double -> I 𝕀 ( ℝ 1 ) )
+     -> ( I 𝕀 ( ℝ 1 ) -> I 𝕀 Double )
+     -> ( C 3 ( I 𝕀 ( ℝ 1 ) ) ( I 𝕀 ( ℝ 2 ) ) )
+     -> ( C 3 ( I 𝕀 ( ℝ 1 ) ) ( I 𝕀 ( ℝ 1 ) ) )
+     -> ( C 3 ( I 𝕀 ( ℝ 1 ) ) ( Spline Closed () ( I 𝕀 ( ℝ 2 ) ) ) )
+     -> ( Maybe ( I 𝕀 ( ℝ 1 ) -> I 𝕀 Double ) )
+     -> ( I 𝕀 ( ℝ 1 ) -> Seq ( I 𝕀 ( ℝ 1 ) -> StrokeDatum 3 𝕀 ) )
+  #-}
+{-# SPECIALISE brushStrokeData
+     :: ( I 𝕀 Double -> I 𝕀 ( ℝ 1 ) )
+     -> ( I 𝕀 ( ℝ 1 ) -> I 𝕀 Double )
+     -> ( C 3 ( I 𝕀 ( ℝ 1 ) ) ( I 𝕀 ( ℝ 2 ) ) )
+     -> ( C 3 ( I 𝕀 ( ℝ 1 ) ) ( I 𝕀 ( ℝ 2 ) ) )
+     -> ( C 3 ( I 𝕀 ( ℝ 2 ) ) ( Spline Closed () ( I 𝕀 ( ℝ 2 ) ) ) )
+     -> ( Maybe ( I 𝕀 ( ℝ 2 ) -> I 𝕀 Double ) )
+     -> ( I 𝕀 ( ℝ 1 ) -> Seq ( I 𝕀 ( ℝ 1 ) -> StrokeDatum 3 𝕀 ) )
+  #-}
+{-# SPECIALISE brushStrokeData
+     :: ( I 𝕀 Double -> I 𝕀 ( ℝ 1 ) )
+     -> ( I 𝕀 ( ℝ 1 ) -> I 𝕀 Double )
+     -> ( C 3 ( I 𝕀 ( ℝ 1 ) ) ( I 𝕀 ( ℝ 2 ) ) )
+     -> ( C 3 ( I 𝕀 ( ℝ 1 ) ) ( I 𝕀 ( ℝ 3 ) ) )
+     -> ( C 3 ( I 𝕀 ( ℝ 3 ) ) ( Spline Closed () ( I 𝕀 ( ℝ 2 ) ) ) )
+     -> ( Maybe ( I 𝕀 ( ℝ 3 ) -> I 𝕀 Double ) )
+     -> ( I 𝕀 ( ℝ 1 ) -> Seq ( I 𝕀 ( ℝ 1 ) -> StrokeDatum 3 𝕀 ) )
+  #-}
+{-# SPECIALISE brushStrokeData
+     :: ( I 𝕀 Double -> I 𝕀 ( ℝ 1 ) )
+     -> ( I 𝕀 ( ℝ 1 ) -> I 𝕀 Double )
+     -> ( C 3 ( I 𝕀 ( ℝ 1 ) ) ( I 𝕀 ( ℝ 2 ) ) )
+     -> ( C 3 ( I 𝕀 ( ℝ 1 ) ) ( I 𝕀 ( ℝ 4 ) ) )
+     -> ( C 3 ( I 𝕀 ( ℝ 4 ) ) ( Spline Closed () ( I 𝕀 ( ℝ 2 ) ) ) )
+     -> ( Maybe ( I 𝕀 ( ℝ 4 ) -> I 𝕀 Double ) )
+     -> ( I 𝕀 ( ℝ 1 ) -> Seq ( I 𝕀 ( ℝ 1 ) -> StrokeDatum 3 𝕀 ) )
+  #-}
+-- TODO: these specialisations fire in the benchmarking code because
+-- we instantiate brushParams with ( ℝ nbBrushParams ), but they won't fire
+-- in the main app code because we are using types such as "Record EllipseBrushFields".
 
 --------------------------------------------------------------------------------
 -- Solving the envelolpe equation: root-finding.
diff --git a/brush-strokes/src/lib/Math/Bezier/Stroke/EnvelopeEquation.hs b/brush-strokes/src/lib/Math/Bezier/Stroke/EnvelopeEquation.hs
index 933f828..60f2710 100644
--- a/brush-strokes/src/lib/Math/Bezier/Stroke/EnvelopeEquation.hs
+++ b/brush-strokes/src/lib/Math/Bezier/Stroke/EnvelopeEquation.hs
@@ -151,22 +151,26 @@ instance HasBézier 2 () where
       D21 ( lerp @( T b ) t a b )
           ( a --> b )
           origin
+  {-# INLINEABLE line #-}
 
   bezier2 co ( bez :: Quadratic.Bezier b ) =
     D \ ( co -> t ) ->
       D21 ( Quadratic.bezier @( T b ) bez t )
           ( Quadratic.bezier'         bez t )
           ( Quadratic.bezier''        bez   )
+  {-# INLINEABLE bezier2 #-}
 
   bezier3 co ( bez :: Cubic.Bezier b ) =
     D \ ( co -> t ) ->
       D21 ( Cubic.bezier @( T b ) bez t )
           ( Cubic.bezier'         bez t )
           ( Cubic.bezier''        bez t )
+  {-# INLINEABLE bezier3 #-}
 
 instance HasEnvelopeEquation 2 where
 
   uncurryD = uncurryD2
+  {-# INLINEABLE uncurryD #-}
 
   envelopeEquation ( _ :: Proxy i ) co
     dp@( D21 ( T -> p ) p_t p_tt )
@@ -247,6 +251,7 @@ instance HasEnvelopeEquation 2 where
                , D12 ( unT u ) u_t u_s
                , D12 ( unT v ) v_t v_s
                )
+  {-# INLINEABLE envelopeEquation #-}
 
 instance HasBézier 3 () where
 
@@ -256,6 +261,7 @@ instance HasBézier 3 () where
           ( a --> b )
           origin
           origin
+  {-# INLINEABLE line #-}
 
   bezier2 co ( bez :: Quadratic.Bezier b ) =
     D \ ( co -> t ) ->
@@ -263,6 +269,7 @@ instance HasBézier 3 () where
           ( Quadratic.bezier'         bez t )
           ( Quadratic.bezier''        bez   )
           origin
+  {-# INLINEABLE bezier2 #-}
 
   bezier3 co ( bez :: Cubic.Bezier b ) =
     D \ ( co -> t ) ->
@@ -270,10 +277,12 @@ instance HasBézier 3 () where
           ( Cubic.bezier'         bez t )
           ( Cubic.bezier''        bez t )
           ( Cubic.bezier'''       bez   )
+  {-# INLINEABLE bezier3 #-}
 
 instance HasEnvelopeEquation 3 where
 
   uncurryD = uncurryD3
+  {-# INLINEABLE uncurryD #-}
 
   envelopeEquation ( _ :: Proxy i ) co
     dp@( D31 ( T -> p ) p_t p_tt p_ttt )
@@ -392,6 +401,7 @@ instance HasEnvelopeEquation 3 where
                , D22 ( unT u ) u_t u_s u_tt u_ts u_ss
                , D22 ( unT v ) v_t v_s v_tt v_ts v_ss
                )
+  {-# INLINEABLE envelopeEquation #-}
 
 instance HasBézier 3 𝕀 where
 
@@ -401,6 +411,7 @@ instance HasBézier 3 𝕀 where
           ( a --> b )
           origin
           origin
+  {-# INLINEABLE line #-}
 
   bezier2 co ( bez :: Quadratic.Bezier b ) =
     D \ ( co -> t ) ->
@@ -408,6 +419,7 @@ instance HasBézier 3 𝕀 where
           ( Quadratic.bezier'  bez t )
           ( Quadratic.bezier'' bez )
           origin
+  {-# INLINEABLE bezier2 #-}
 
   bezier3 co ( bez :: Cubic.Bezier b ) =
     D \ ( co -> t ) ->
@@ -415,6 +427,7 @@ instance HasBézier 3 𝕀 where
           ( T $ aabb ( fmap unT $ Cubic.derivative ( fmap T bez ) ) ( `evaluateQuadratic` t ) )
           ( Cubic.bezier''  bez t )
           ( Cubic.bezier''' bez )
+  {-# INLINEABLE bezier3 #-}
 
 {- Note [Computing Béziers over intervals]
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -455,6 +468,7 @@ evaluateCubic bez t =
       maxs = fmap (Cubic.bezier @( T Double ) sup_bez)
            $ inf t :| ( sup t : filter ( ∈ t ) ( Cubic.extrema sup_bez ) )
   in 𝕀 ( minimum mins ) ( maximum maxs )
+{-# INLINEABLE evaluateCubic #-}
 
 -- | Evaluate a quadratic Bézier curve, when both the coefficients and the
 -- parameter are intervals.
@@ -468,3 +482,4 @@ evaluateQuadratic bez t =
       maxs = fmap (Quadratic.bezier @( T Double ) sup_bez)
            $ inf t :| ( sup t : filter ( ∈ t ) ( Quadratic.extrema sup_bez ) )
   in 𝕀 ( minimum mins ) ( maximum maxs )
+{-# INLINEABLE evaluateQuadratic #-}
diff --git a/brush-strokes/src/lib/Math/Interval.hs b/brush-strokes/src/lib/Math/Interval.hs
index 0439ed9..a97408d 100644
--- a/brush-strokes/src/lib/Math/Interval.hs
+++ b/brush-strokes/src/lib/Math/Interval.hs
@@ -1,3 +1,5 @@
+{-# LANGUAGE CPP #-}
+
 {-# LANGUAGE ScopedTypeVariables  #-}
 {-# LANGUAGE TemplateHaskell      #-}
 {-# LANGUAGE UndecidableInstances #-}
@@ -19,8 +21,6 @@ module Math.Interval
 
 -- base
 import Prelude hiding ( Num(..), Fractional(..) )
-import Data.List
-  ( nub )
 import qualified Data.List.NonEmpty as NE
 
 -- acts
@@ -138,11 +138,23 @@ instance Torsor ( T ( 𝕀 Double ) ) ( 𝕀 Double ) where
 -------------------------------------------------------------------------------
 -- Extended division
 
--- | Extended division.
+-- | Extended division, returning either 1 or 2 intervals.
+--
+-- NB: this function can return overlapping intervals if both arguments contain 0.
+-- Otherwise, the returned intervals are guaranteed to be disjoint.
 (⊘) :: 𝕀 Double -> 𝕀 Double -> [ 𝕀 Double ]
-x ⊘ y = nub $ map ( x * ) ( extendedRecip y )
+x ⊘ y
+#ifdef ASSERTS
+  | 0 ∈ x && 0 ∈ y
+  = error $ unlines
+      [ "x ⊘ y: both arguments contain zero"
+      , "x: " ++ show x, "y: " ++ show y ]
+  | otherwise
+#endif
+  = map ( x * ) ( extendedRecip y )
 infixl 7 ⊘
 
+-- | Extended reciprocal, returning either 1 or 2 intervals.
 extendedRecip :: 𝕀 Double -> [ 𝕀 Double ]
 extendedRecip x@( 𝕀 lo hi )
   | lo == 0 && hi == 0
@@ -150,11 +162,12 @@ extendedRecip x@( 𝕀 lo hi )
   | lo >= 0 || hi <= 0
   = [ recip x ]
   | otherwise
-  = [ recip $ 𝕀 lo -0, recip $ 𝕀 0 hi ]
+  = [ 𝕀 negInf ( recip lo ), 𝕀 ( recip hi ) posInf ]
   where
-    negInf, posInf :: Double
-    negInf = -1 / 0
-    posInf =  1 / 0
+
+negInf, posInf :: Double
+negInf = -1 / 0
+posInf =  1 / 0
 
 -------------------------------------------------------------------------------
 -- Lattices.
diff --git a/brush-strokes/src/lib/Math/Linear.hs b/brush-strokes/src/lib/Math/Linear.hs
index 7948193..bd4aa1a 100644
--- a/brush-strokes/src/lib/Math/Linear.hs
+++ b/brush-strokes/src/lib/Math/Linear.hs
@@ -133,7 +133,7 @@ type role Vec nominal representational
 deriving newtype instance Show a => Show ( Vec n a )
 deriving newtype instance   Eq a =>   Eq ( Vec n a )
 deriving newtype instance  Ord a =>  Ord ( Vec n a )
-deriving newtype instance Functor  ( Vec n )
+deriving newtype instance  Functor ( Vec n )
 deriving newtype instance Foldable ( Vec n )
 deriving via ZipList
   instance Applicative ( Vec n )
diff --git a/brush-strokes/src/lib/Math/Monomial.hs b/brush-strokes/src/lib/Math/Monomial.hs
index 2356202..f027a39 100644
--- a/brush-strokes/src/lib/Math/Monomial.hs
+++ b/brush-strokes/src/lib/Math/Monomial.hs
@@ -55,25 +55,24 @@ type family Deg  f
 type family Vars f
 
 zeroMonomial :: forall k n. KnownNat n => Mon k n
-zeroMonomial = Mon $ unsafeCoerce @[ Word ] @( Vec n Word )
-                   $ replicate ( fromIntegral $ word @n ) 0
-{-# INLINEABLE zeroMonomial #-}
+zeroMonomial = Mon $ Vec $ replicate ( fromIntegral $ word @n ) 0
+{-# INLINE zeroMonomial #-}
 
 isZeroMonomial :: Mon k n -> Bool
 isZeroMonomial ( Mon ( Vec pows ) ) = all ( == 0 ) pows
+{-# INLINE isZeroMonomial #-}
 
 totalDegree :: Mon k n -> Word
 totalDegree ( Mon ( Vec pows ) ) = sum pows
 
 linearMonomial :: forall k n. ( KnownNat n, 1 <= k ) => Fin n -> Mon k n
 linearMonomial ( Fin i' ) =
-  Mon $ unsafeCoerce @[ Word ] @( Vec n Word )
-      $ replicate ( i - 1 ) 0 ++ [ 1 ] ++ replicate ( n - i ) 0
+  Mon $ Vec $ replicate ( i - 1 ) 0 ++ [ 1 ] ++ replicate ( n - i ) 0
   where
     n, i :: Int
     n = fromIntegral $ word @n
     i = fromIntegral i'
-{-# INLINEABLE linearMonomial #-}
+{-# INLINE linearMonomial #-}
 
 isLinear :: Mon k n -> Maybe ( Fin n )
 isLinear ( Mon ( Vec pows ) ) = fmap Fin $ go 1 pows
diff --git a/brush-strokes/src/lib/Math/Root/Isolation.hs b/brush-strokes/src/lib/Math/Root/Isolation.hs
index e568a39..55f8e07 100644
--- a/brush-strokes/src/lib/Math/Root/Isolation.hs
+++ b/brush-strokes/src/lib/Math/Root/Isolation.hs
@@ -102,7 +102,7 @@ newtype RootIsolationOptions n d
 defaultRootIsolationOptions :: BoxCt n d => RootIsolationOptions n d
 defaultRootIsolationOptions =
   RootIsolationOptions
-    { rootIsolationAlgorithms = defaultRootIsolationAlgorithms minWidth ε_eq True
+    { rootIsolationAlgorithms = defaultRootIsolationAlgorithms minWidth ε_eq
     }
   where
     minWidth = 1e-5
@@ -114,11 +114,10 @@ defaultRootIsolationAlgorithms
   .  BoxCt n d
   => Double -- ^ minimum width of boxes (don't bisect further)
   -> Double -- ^ threshold for progress
-  -> Bool -- ^ do box1
   -> BoxHistory n
   -> Box n
   -> Either String ( NE.NonEmpty ( RootIsolationAlgorithmWithOptions n d ) )
-defaultRootIsolationAlgorithms minWidth ε_eq doBox1 history box =
+defaultRootIsolationAlgorithms minWidth ε_eq history box =
   case history of
     lastRoundBoxes : _
       -- If, in the last round of strategies, we didn't try bisection...
@@ -132,17 +131,19 @@ defaultRootIsolationAlgorithms minWidth ε_eq doBox1 history box =
          then Left $ "widths <= " ++ show minWidth
          else Right $ NE.singleton $ AlgoWithOptions @Bisection _bisOptions
     -- Otherwise, do a normal round.
-    -- Currently: we try an interval Gauss–Seidel step followed by box(1)-consistency.
+    -- Currently: we try an interval Gauss–Seidel.
+    -- (box(1)- and box(2)-consistency don't seem to help when using
+    -- the complete interval union Gauss–Seidel step)
     _ -> Right $ AlgoWithOptions @GaussSeidel _gsOptions
-            NE.:| [ AlgoWithOptions @Box1 _box1Options
-                  | doBox1 && not verySmall ]
+            NE.:| []
 
   where
     verySmall = and $ ( \ cd -> width cd <= minWidth ) <$> coordinates box
 
     _bisOptions = defaultBisectionOptions @n @d minWidth ε_eq box
     _gsOptions = defaultGaussSeidelOptions @n @d history
-    _box1Options = defaultBox1Options @n @d minWidth ε_eq
+    _box1Options = defaultBox1Options @n @d ( minWidth * 100 ) ε_eq
+    _box2Options = ( defaultBox2Options @n @d minWidth ε_eq ) { box2LambdaMin = 0.001 }
 
     -- Did we reduce the box width by at least ε_eq
     -- in at least one of the coordinates?
@@ -249,8 +250,8 @@ isolateRootsIn ( RootIsolationOptions { rootIsolationAlgorithms } )
 -- | Execute a root isolation strategy, replacing the input box with
 -- (hopefully smaller) output boxes.
 doStrategy
-  :: BoxCt n d
-  => RootIsolationAlgorithmWithOptions n d
+  :: forall n d
+  .  RootIsolationAlgorithmWithOptions n d
   -> [ ( RootIsolationStep, Box n ) ]
   -> BoxHistory n
   -> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
@@ -260,6 +261,5 @@ doStrategy
 doStrategy algo thisRoundHist prevRoundsHist eqs cand =
   case algo of
     AlgoWithOptions @algo options -> do
-      ( step, res ) <- rootIsolationAlgorithm options thisRoundHist prevRoundsHist eqs cand
+      ( step, res ) <- rootIsolationAlgorithm @algo options thisRoundHist prevRoundsHist eqs cand
       return ( SomeRootIsolationStep @algo step, res )
-{-# INLINEABLE doStrategy #-}
diff --git a/brush-strokes/src/lib/Math/Root/Isolation/Bisection.hs b/brush-strokes/src/lib/Math/Root/Isolation/Bisection.hs
index 52f95d0..fc2db0a 100644
--- a/brush-strokes/src/lib/Math/Root/Isolation/Bisection.hs
+++ b/brush-strokes/src/lib/Math/Root/Isolation/Bisection.hs
@@ -1,5 +1,6 @@
 
-{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE ScopedTypeVariables  #-}
+{-# LANGUAGE UndecidableInstances #-}
 
 module Math.Root.Isolation.Bisection
   ( -- * The bisection method for root isolation
@@ -35,6 +36,10 @@ import GHC.TypeNats
   , type (<=)
   )
 
+-- transformers
+import Control.Monad.Trans.Writer.CPS
+  ( Writer )
+
 -- MetaBrush
 import Math.Algebra.Dual
   ( D )
@@ -52,7 +57,7 @@ import Math.Root.Isolation.Core
 
 -- | The bisection algorithm; see 'bisection'.
 data Bisection
-instance RootIsolationAlgorithm Bisection where
+instance BoxCt n d => RootIsolationAlgorithm Bisection n d where
   type instance StepDescription Bisection = ( String, Double )
   type instance RootIsolationAlgorithmOptions Bisection n d = BisectionOptions n d
   rootIsolationAlgorithm
@@ -65,6 +70,15 @@ instance RootIsolationAlgorithm Bisection where
               box
       return ( whatBis, boxes )
   {-# INLINEABLE rootIsolationAlgorithm #-}
+  {-# SPECIALISE rootIsolationAlgorithm
+        :: RootIsolationAlgorithmOptions Bisection 2 3
+        -> [ ( RootIsolationStep, Box 2 ) ]
+        -> BoxHistory 2
+        -> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
+        -> Box 2
+        -> Writer ( DoneBoxes 2 ) ( StepDescription Bisection, [ Box 2 ] ) #-}
+    -- NB: including this to be safe. The specialiser seems to sometimes
+    -- be able to generate this specialisation on its own, and sometimes not.
 
 -- | Options for the bisection method.
 type BisectionOptions :: Nat -> Nat -> Type
diff --git a/brush-strokes/src/lib/Math/Root/Isolation/Core.hs b/brush-strokes/src/lib/Math/Root/Isolation/Core.hs
index f3e2294..e6bcb29 100644
--- a/brush-strokes/src/lib/Math/Root/Isolation/Core.hs
+++ b/brush-strokes/src/lib/Math/Root/Isolation/Core.hs
@@ -1,8 +1,9 @@
 
-{-# LANGUAGE AllowAmbiguousTypes  #-}
-{-# LANGUAGE PolyKinds            #-}
-{-# LANGUAGE ScopedTypeVariables  #-}
-{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE AllowAmbiguousTypes     #-}
+{-# LANGUAGE PolyKinds               #-}
+{-# LANGUAGE ScopedTypeVariables     #-}
+{-# LANGUAGE UndecidableInstances    #-}
+{-# LANGUAGE UndecidableSuperClasses #-}
 
 -- | Core definitions and utilities common to root isolation methods.
 module Math.Root.Isolation.Core
@@ -122,9 +123,11 @@ noDoneBoxes = DoneBoxes [] []
 
 -- | Existential wrapper over any root isolation algorithm,
 -- with the options necessary to run it.
+type RootIsolationAlgorithmWithOptions :: Nat -> Nat -> Type
 data RootIsolationAlgorithmWithOptions n d where
   AlgoWithOptions
-    :: RootIsolationAlgorithm ty
+    :: forall {k :: Type} {n :: Nat} {d :: Nat} ( ty :: k )
+    .  RootIsolationAlgorithm ty n d
     => RootIsolationAlgorithmOptions ty n d
     -> RootIsolationAlgorithmWithOptions n d
 
@@ -133,15 +136,15 @@ data RootIsolationAlgorithmWithOptions n d where
 -- This design keeps the set of root isolation algorithms open-ended,
 -- while retaining the ability to inspect previous steps (using the
 -- 'IsolationStep' pattern).
-type RootIsolationAlgorithm :: forall {k}. k -> Constraint
-class ( Typeable ty, Show ( StepDescription ty ) )
-    => RootIsolationAlgorithm ty where
+type RootIsolationAlgorithm :: forall {k :: Type}. k -> Nat -> Nat -> Constraint
+class ( Typeable ty, Show ( StepDescription ty ), BoxCt n d )
+    => RootIsolationAlgorithm ty n d where
   -- | The type of additional information about an algorithm step.
   --
   -- Only really useful for debugging; gets stored in 'RootIsolationTree's.
   type StepDescription ty
   -- | Configuration options expected by this root isolation method.
-  type RootIsolationAlgorithmOptions ty (n :: Nat) (d :: Nat) = r | r -> ty n d
+  type RootIsolationAlgorithmOptions ty n d = r | r -> ty n d
   -- | Run one step of the root isolation method.
   --
   -- This gets given the equations and a box, and should attempt to
@@ -155,9 +158,7 @@ class ( Typeable ty, Show ( StepDescription ty ) )
   --    bix does not contain any solutions;
   --  - (as a writer side-effect) boxes to definitely stop processing; see 'DoneBoxes'.
   rootIsolationAlgorithm
-    :: forall (n :: Nat) (d :: Nat)
-    .  BoxCt n d
-    => RootIsolationAlgorithmOptions ty n d
+    :: RootIsolationAlgorithmOptions ty n d
         -- ^ options for this root isolation algorithm
     -> [ ( RootIsolationStep, Box n ) ]
         -- ^ history of the current round
@@ -171,20 +172,22 @@ class ( Typeable ty, Show ( StepDescription ty ) )
 
 -- | Match on an unknown root isolation algorithm step with a known algorithm.
 pattern IsolationStep
-  :: forall (ty :: Type)
-  .  RootIsolationAlgorithm ty
-  => StepDescription ty -> RootIsolationStep
+  :: forall ( ty :: Type )
+  .  Typeable ty
+  => StepDescription ty
+  -> RootIsolationStep
 pattern IsolationStep stepDescr
           <- ( rootIsolationAlgorithmStep_maybe @ty -> Just stepDescr )
-  where
-    IsolationStep stepDescr = SomeRootIsolationStep @ty stepDescr
+  -- NB: this pattern could also return @RootIsolationAlgorithm n d ty@ evidence,
+  -- but it's simpler to not do so for now.
 
 -- | Helper function used to define the 'IsolationStep' pattern.
 --
 -- Inspects whether an existential 'RootIsolationStep' packs a step for
 -- the given algorithm.
 rootIsolationAlgorithmStep_maybe
-  :: forall ty. RootIsolationAlgorithm ty
+  :: forall ty
+  .  Typeable ty
   => RootIsolationStep -> Maybe ( StepDescription ty )
 rootIsolationAlgorithmStep_maybe ( SomeRootIsolationStep @existential descr )
   | Just HRefl <- heqT @existential @ty
diff --git a/brush-strokes/src/lib/Math/Root/Isolation/Degree.hs b/brush-strokes/src/lib/Math/Root/Isolation/Degree.hs
new file mode 100644
index 0000000..8546cc5
--- /dev/null
+++ b/brush-strokes/src/lib/Math/Root/Isolation/Degree.hs
@@ -0,0 +1,126 @@
+{-# LANGUAGE ScopedTypeVariables #-}
+
+module Math.Root.Isolation.Degree
+  ( -- * Computing the topological degree
+    topologicalDegree
+  )
+  where
+
+-- base
+import Data.Kind
+  ( Type )
+import GHC.TypeNats
+  ( Nat )
+
+-- primitive
+import Data.Primitive.SmallArray
+
+-- MetaBrush
+import Math.Interval
+import Math.Linear
+
+--------------------------------------------------------------------------------
+-- Topological degree.
+
+-- | Compute the topological degree of the given interval function (2D only).
+topologicalDegree
+  :: Double
+     -- ^ minimum width when bisecting facets
+  -> ( 𝕀ℝ 2 -> 𝕀ℝ 2 )
+     -- ^ function
+  -> 𝕀ℝ 2
+     -- ^ box
+  -> Maybe Int
+topologicalDegree ε_bis f ( 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_hi y_hi ) ) =
+  topologicalDegreeFromContour $
+    smallArrayFromList $
+      concatMap ( tagFacet ε_bis f )
+        [ ( Tag ( Fin 2 ) P, 𝕀 ( ℝ2 x_hi y_lo ) ( ℝ2 x_hi y_hi ) )
+        , ( Tag ( Fin 1 ) N, 𝕀 ( ℝ2 x_lo y_hi ) ( ℝ2 x_hi y_hi ) )
+        , ( Tag ( Fin 2 ) N, 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_lo y_hi ) )
+        , ( Tag ( Fin 1 ) P, 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_hi y_lo ) ) ]
+
+-- | Tag a facet with the sign of one of the equations that is non-zero
+-- on that facet.
+--
+-- If all equations take the value 0 on that facet, bisect and recur.
+tagFacet
+  :: Double
+  -> ( 𝕀ℝ 2 -> 𝕀ℝ 2 )
+  -> ( Tag 2, 𝕀ℝ 2 )
+  -> [ Tag 2 ]
+tagFacet ε_bis f ( tag@( Tag i _ ), facet0 ) = go facet0
+  where
+    go facet
+      | f1_b_lo > 0
+      = [ Tag ( Fin 1 ) P ]
+      | f1_b_hi < 0
+      = [ Tag ( Fin 1 ) N ]
+      | f2_b_lo > 0
+      = [ Tag ( Fin 2 ) P ]
+      | f2_b_hi < 0
+      = [ Tag ( Fin 2 ) N ]
+      | width ( facet `index` i ) < ε_bis
+      = [ NoTag ] -- TODO: shortcut the whole computation;
+                  -- if we fail to tag one segment, the final answer could
+                  -- be off by { -2, -1, 0, 1, 2 } (I believe)
+      | otherwise
+      = concatMap go $ bisectFacet tag facet
+      where
+        𝕀 f1_b_lo f1_b_hi = f facet `index` Fin 1
+        𝕀 f2_b_lo f2_b_hi = f facet `index` Fin 2
+tagFacet _ _ ( NoTag, _ ) = error "impossible: input always tagged"
+
+bisectFacet :: Tag 2 -> 𝕀ℝ 2 -> [ 𝕀ℝ 2 ]
+bisectFacet ( Tag i s ) x =
+  ( if s == P then id else reverse ) $ bisectInCoord i x
+bisectFacet NoTag _ = error "impossible: input always tagged"
+
+bisectInCoord :: Fin n -> 𝕀ℝ 2 -> [ 𝕀ℝ 2 ]
+bisectInCoord ( Fin 1 ) ( 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_hi y_hi ) ) =
+  [ 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_mid y_hi ), 𝕀 ( ℝ2 x_mid y_lo ) ( ℝ2 x_hi y_hi ) ]
+    where x_mid = 0.5 * ( x_lo + x_hi )
+bisectInCoord _ ( 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_hi y_hi ) ) =
+  [ 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_hi y_mid ), 𝕀 ( ℝ2 x_lo y_mid ) ( ℝ2 x_hi y_hi ) ]
+    where y_mid = 0.5 * ( y_lo + y_hi )
+
+-- | Compute the topological degree of \( f \colon \mathbb{IR}^n \to \mathbb{IR}^n \)
+-- on a box \( D \subset \mathbb{IR}^n \) by using the signs of the components
+-- of \( f \) on a counter-clockwise polygonal contour describing the boundary of \( D \).
+topologicalDegreeFromContour :: SmallArray ( Tag 2 ) -> Maybe Int
+topologicalDegreeFromContour facetTags = go 0 0
+  where
+    n = sizeofSmallArray facetTags
+    go :: Int -> Int -> Maybe Int
+    go !d !i
+      | i >= n
+      = Just d
+      | otherwise
+      = case indexSmallArray facetTags i of
+          NoTag -> Nothing
+          Tag ( Fin 1 ) P -> go ( d + δ ) ( i + 1 )
+            where
+              δ = if indexSmallArray facetTags ( if i == n - 1 then 0 else i + 1 ) == Tag ( Fin 2 ) P
+                  then 1 else 0
+                - if indexSmallArray facetTags ( if i == 0 then n - 1 else i - 1 ) == Tag ( Fin 2 ) P
+                  then 1 else 0
+          _ -> go d ( i + 1 )
+
+--------------------------------------------------------------------------------
+-- Tagging edges with signs of certain components of the function.
+
+data Sign = P | N
+  deriving stock ( Eq, Ord )
+instance Show Sign where
+  showsPrec _ = \case { P -> showString "+"; N -> showString "-" }
+type Tag :: Nat -> Type
+newtype Tag d = MkTag Int
+  deriving newtype ( Eq, Ord )
+pattern NoTag :: Tag d
+pattern NoTag = MkTag 0
+pattern Tag :: Fin d -> Sign -> Tag d
+pattern Tag c s <- ( getTag -> (# c, s #) )
+  where Tag ( Fin c ) s = MkTag $ case s of { P -> fromIntegral c; N -> -( fromIntegral c ) }
+{-# COMPLETE Tag, NoTag #-}
+getTag :: Tag d -> (# Fin d, Sign #)
+getTag ( MkTag t ) = (# Fin ( fromIntegral $ abs t ), if t >= 0 then P else N #)
diff --git a/brush-strokes/src/lib/Math/Root/Isolation/GaussSeidel.hs b/brush-strokes/src/lib/Math/Root/Isolation/GaussSeidel.hs
index 9fad426..b1b5ff1 100644
--- a/brush-strokes/src/lib/Math/Root/Isolation/GaussSeidel.hs
+++ b/brush-strokes/src/lib/Math/Root/Isolation/GaussSeidel.hs
@@ -1,4 +1,5 @@
-{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE ScopedTypeVariables  #-}
+{-# LANGUAGE UndecidableInstances #-}
 
 module Math.Root.Isolation.GaussSeidel
   ( -- * The interval Newton method with Gauss–Seidel step
@@ -37,7 +38,7 @@ import GHC.TypeNats
 -- eigen
 import qualified Eigen.Matrix as Eigen
   ( Matrix
-  , determinant, generate, inverse, unsafeCoeff
+  , generate, inverse, unsafeCoeff
   )
 
 -- transformers
@@ -47,8 +48,6 @@ import Control.Monad.Trans.Writer.CPS
 -- MetaBrush
 import Math.Algebra.Dual
   ( D )
-import Math.Epsilon
-  ( nearZero )
 import Math.Interval
 import Math.Linear
 import Math.Module
@@ -62,13 +61,22 @@ import Math.Root.Isolation.Core
 
 -- | The interval Newton method with a Gauss–Seidel step; see 'intervalGaussSeidel'.
 data GaussSeidel
-instance RootIsolationAlgorithm GaussSeidel where
+instance BoxCt n d => RootIsolationAlgorithm GaussSeidel n d where
   type instance StepDescription GaussSeidel = ()
   type instance RootIsolationAlgorithmOptions GaussSeidel n d = GaussSeidelOptions n d
   rootIsolationAlgorithm opts _thisRoundHist _prevRoundsHist eqs box = do
-    res <- intervalGaussSeidel opts eqs box
+    res <- intervalGaussSeidel @n @d opts eqs box
     return ( (), res )
   {-# INLINEABLE rootIsolationAlgorithm #-}
+  {-# SPECIALISE rootIsolationAlgorithm
+        :: RootIsolationAlgorithmOptions GaussSeidel 2 3
+        -> [ ( RootIsolationStep, Box 2 ) ]
+        -> BoxHistory 2
+        -> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
+        -> Box 2
+        -> Writer ( DoneBoxes 2 ) ( StepDescription GaussSeidel, [ Box 2 ] ) #-}
+    -- NB: including this to be safe. The specialiser seems to sometimes
+    -- be able to generate this specialisation on its own, and sometimes not.
 
 -- | Options for the interval Gauss–Seidel method.
 type GaussSeidelOptions :: Nat -> Nat -> Type
@@ -144,13 +152,15 @@ intervalGaussSeidel
   eqs
   x
   | let x_mid = singleton $ boxMidpoint x
+        f :: 𝕀ℝ n -> 𝕀ℝ n
+        f = \ x_0 -> pickEqs $ eqs x_0 `monIndex` zeroMonomial
+
         f'_x :: Vec n ( 𝕀ℝ n )
         f'_x = fmap ( \ i -> pickEqs $ eqs x `monIndex` linearMonomial i ) ( universe @n )
-        f_x_mid = pickEqs $ eqs x_mid `monIndex` zeroMonomial
 
   = let -- Interval Newton method: take one Gauss–Seidel step
         -- for the system of equations f'(x) ( x - x_mid ) = - f(x_mid).
-        minus_f_x_mid = unT $ -1 *^ T ( boxMidpoint f_x_mid )
+        minus_f_x_mid = unT $ -1 *^ T ( boxMidpoint $ f x_mid )
 
         -- Precondition the above linear system into A ( x - x_mid ) = B.
         ( a, b ) = precondition precondMeth
@@ -160,18 +170,34 @@ intervalGaussSeidel
         -- at the origin, in order to take a Gauss–Seidel step.
         gsGuesses = map ( first ( \ x' -> unT $ x' ^+^ T x_mid ) )
                   $ gaussSeidelUpdate gsUpdate a b ( T x ^-^ T x_mid )
-    in
-         -- If the Gauss–Seidel step was a contraction, then the box
-         -- contains a unique solution (by the Banach fixed point theorem).
-         --
-         -- These boxes can thus be directly added to the solution set:
-         -- Newton's method is guaranteed to converge to the unique solution.
-         let ( done, todo ) = bimap ( map fst ) ( map fst )
-                            $ partition snd gsGuesses
-         in do tell $ noDoneBoxes { doneSolBoxes = done }
-               return todo
-  where
+        ( done, todo ) = bimap ( map fst ) ( map fst )
+                       $ partition snd gsGuesses
+    in -- If the Gauss–Seidel step was a contraction, then the box
+            -- contains a unique solution (by the Banach fixed point theorem).
+            --
+            -- These boxes can thus be directly added to the solution set:
+            -- Newton's method is guaranteed to converge to the unique solution.
+      do tell $ noDoneBoxes { doneSolBoxes = done }
+         return todo
 {-# INLINEABLE intervalGaussSeidel #-}
+{-
+
+mbDeg = topologicalDegree 0.005 f x
+det = case f'_x of
+        Vec [ c1, c2 ] ->
+          let a_11 = c1 `index` Fin 1
+              a_12 = c2 `index` Fin 1
+              a_21 = c1 `index` Fin 2
+              a_22 = c2 `index` Fin 2
+          in a_11 * a_22 - a_12 * a_21
+        _ -> error "TODO: just testing n=2 here"
+
+if | not $ 0 ∈ det
+   , mbDeg == Just 0
+   -> return []
+     -- If the Jacobian is invertible over the box, then the topological
+     -- degree tells us exactly how many solutions there are in the box.
+-}
 
 -- | A partial or complete Gauss–Seidel step for the equation \( A X = B \),
 -- refining the initial guess box for \( X \) into up to \( 2^n \) (disjoint) new boxes.
@@ -278,10 +304,7 @@ fromComponents f = do
 -- | The midpoint of a box.
 boxMidpoint :: Representable Double ( ℝ n ) => 𝕀ℝ n -> ℝ n
 boxMidpoint box =
-  tabulate $ \ i ->
-    let 𝕀 z_lo z_hi = index box i
-        z_mid = 0.5 * ( z_lo + z_hi )
-    in z_mid
+  tabulate $ \ i -> midpoint ( box `index` i )
 {-# INLINEABLE boxMidpoint #-}
 
 -- | Pre-condition the system \( AX = B \).
@@ -298,14 +321,10 @@ precondition meth as b =
       -> ( as, b )
     InverseMidpoint
       | let mat = toEigen $ fmap boxMidpoint as
-            det = Eigen.determinant mat
-      , not $ nearZero det
-      -- (TODO: a bit wasteful to compute determinant then inverse.)
       , let precond = Eigen.inverse mat
             doPrecond = matMulVec ( fromEigen precond )
+      -- TODO: avoid this when condition number is small?
       -> ( fmap doPrecond as, doPrecond b )
-      | otherwise
-      -> ( as, b )
   where
     toEigen :: Vec n ( ℝ n ) -> Eigen.Matrix n n Double
     toEigen cols =
diff --git a/brush-strokes/src/lib/Math/Root/Isolation/Narrowing.hs b/brush-strokes/src/lib/Math/Root/Isolation/Narrowing.hs
index a41ace9..05c04fb 100644
--- a/brush-strokes/src/lib/Math/Root/Isolation/Narrowing.hs
+++ b/brush-strokes/src/lib/Math/Root/Isolation/Narrowing.hs
@@ -1,4 +1,5 @@
-{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE ScopedTypeVariables  #-}
+{-# LANGUAGE UndecidableInstances #-}
 
 module Math.Root.Isolation.Narrowing
   ( -- * @box(1)@-consistency
@@ -39,6 +40,8 @@ import GHC.TypeNats
 -- transformers
 import Control.Monad.Trans.State.Strict as State
   ( State, evalState, get, put )
+import Control.Monad.Trans.Writer.CPS
+  ( Writer )
 
 -- brush-strokes
 import Math.Algebra.Dual
@@ -62,21 +65,39 @@ import Math.Root.Isolation.Core
 
 -- | A @box(1)@-consistency enforcing algorithm; see 'makeBox1Consistent'.
 data Box1
-instance RootIsolationAlgorithm Box1 where
+instance BoxCt n d => RootIsolationAlgorithm Box1 n d where
   type instance StepDescription Box1 = ()
   type instance RootIsolationAlgorithmOptions Box1 n d = Box1Options n d
   rootIsolationAlgorithm opts _thisRoundHist _prevRoundsHist eqs box =
     return $ ( (), makeBox1Consistent opts eqs box )
   {-# INLINEABLE rootIsolationAlgorithm #-}
+  {-# SPECIALISE rootIsolationAlgorithm
+        :: RootIsolationAlgorithmOptions Box1 2 3
+        -> [ ( RootIsolationStep, Box 2 ) ]
+        -> BoxHistory 2
+        -> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
+        -> Box 2
+        -> Writer ( DoneBoxes 2 ) ( StepDescription Box1, [ Box 2 ] ) #-}
+    -- NB: including this to be safe. The specialiser seems to sometimes
+    -- be able to generate this specialisation on its own, and sometimes not.
 
 -- | A @box(2)@-consistency enforcing algorithm; see 'makeBox1Consistent'.
 data Box2
-instance RootIsolationAlgorithm Box2 where
+instance BoxCt n d => RootIsolationAlgorithm Box2 n d where
   type instance StepDescription Box2 = ()
   type instance RootIsolationAlgorithmOptions Box2 n d = Box2Options n d
   rootIsolationAlgorithm opts _thisRoundHist _prevRoundsHist eqs box =
     return ( () , [ makeBox2Consistent opts eqs box ] )
   {-# INLINEABLE rootIsolationAlgorithm #-}
+  {-# SPECIALISE rootIsolationAlgorithm
+        :: RootIsolationAlgorithmOptions Box2 2 3
+        -> [ ( RootIsolationStep, Box 2 ) ]
+        -> BoxHistory 2
+        -> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
+        -> Box 2
+        -> Writer ( DoneBoxes 2 ) ( StepDescription Box2, [ Box 2 ] ) #-}
+    -- NB: including this to be safe. The specialiser seems to sometimes
+    -- be able to generate this specialisation on its own, and sometimes not.
 
 -- | Options for the @box(1)@-consistency method.
 data Box1Options n d =
diff --git a/src/app/MetaBrush/Application.hs b/src/app/MetaBrush/Application.hs
index 946b0b1..58d5e77 100644
--- a/src/app/MetaBrush/Application.hs
+++ b/src/app/MetaBrush/Application.hs
@@ -74,9 +74,7 @@ import Control.Monad.Trans.Reader
 
 -- MetaBrush
 import Math.Root.Isolation
-  ( RootIsolationOptions(..), defaultRootIsolationOptions
-  , N
-  )
+  ( RootIsolationOptions(..), defaultRootIsolationOptions )
 import Math.Bezier.Cubic.Fit
   ( FitParameters(..) )
 import Math.Bezier.Spline
@@ -208,18 +206,18 @@ runApplication application = do
   maxHistorySizeTVar   <- STM.newTVarIO @Int                                     1000
   fitParametersTVar    <- STM.newTVarIO @FitParameters $
                             FitParameters
-                              { maxSubdiv  = 5 --2 --3 -- 6
-                              , nbSegments = 3 --3 --6 -- 12
-                              , dist_tol   = 0.1 -- 5e-3
-                              , t_tol      = 0.1 -- 1e-4
-                              , maxIters   = 5   -- 100
+                              { maxSubdiv  = 0 --5 --2 --3 -- 6
+                              , nbSegments = 1 --5
+                              , dist_tol   = 5e-3
+                              , t_tol      = 1e-4
+                              , maxIters   = 20
                               }
   rootsAlgoTVar        <- STM.newTVarIO @RootSolvingAlgorithm $
                             --HalleyM2
                             NewtonRaphson
                               { maxIters = 20, precision = 8 }
-  cuspFindingOptionsTVar <- STM.newTVarIO @( Maybe ( RootIsolationOptions N 3 ) ) $
-                              Just defaultRootIsolationOptions
+  cuspFindingOptionsTVar <- STM.newTVarIO @( Maybe ( RootIsolationOptions 2 3 ) ) $
+                              Nothing --Just defaultRootIsolationOptions
 
   -- Put all these stateful variables in a record for conciseness.
   let
diff --git a/src/app/MetaBrush/Render/Document.hs b/src/app/MetaBrush/Render/Document.hs
index 4cb9f7f..c73a632 100644
--- a/src/app/MetaBrush/Render/Document.hs
+++ b/src/app/MetaBrush/Render/Document.hs
@@ -83,7 +83,7 @@ import Math.Linear
 import Math.Module
   ( Module((*^)), normalise )
 import Math.Root.Isolation
-  ( RootIsolationOptions, N )
+  ( RootIsolationOptions )
 import MetaBrush.Asset.Colours
   ( Colours, ColourRecord(..) )
 import MetaBrush.Brush
@@ -153,7 +153,7 @@ blankRender _ = pure ()
 
 getDocumentRender
   :: Colours
-  -> RootSolvingAlgorithm -> Maybe ( RootIsolationOptions N 3 ) -> FitParameters
+  -> RootSolvingAlgorithm -> Maybe ( RootIsolationOptions 2 3 ) -> FitParameters
   -> Mode -> Bool
   -> Set Modifier -> Maybe ( ℝ 2 ) -> Maybe HoldAction -> Maybe PartialPath
   -> Document
@@ -289,7 +289,7 @@ instance NFData StrokeRenderData where
 -- - Otherwise, this consists of the underlying spline path only.
 strokeRenderData
   :: RootSolvingAlgorithm
-  -> Maybe ( RootIsolationOptions N 3 )
+  -> Maybe ( RootIsolationOptions 2 3 )
   -> FitParameters
   -> Stroke
   -> Maybe ( ST RealWorld StrokeRenderData )
@@ -304,7 +304,7 @@ strokeRenderData rootAlgo mbCuspOptions fitParams
         Just ( NamedBrush { brushFunction = fn } )
           | WithParams
              { defaultParams = brush_defaults
-             , withParams    = brush@( Brush { brushShape, mbRotation = mbRot } )
+             , withParams    = brush@( Brush { brushBaseShape, mbRotation = mbRot } )
              } <- fn
           -> -- This is the key place where we need to perform impedance matching
              -- between the collection of parameters supplied along a stroke and
@@ -328,7 +328,7 @@ strokeRenderData rootAlgo mbCuspOptions fitParams
                         , strokeBrushFunction =
                             \ params ->
                             let brushParams = embedUsedParams $ toUsedParams params
-                                shape = fun @Double brushShape brushParams
+                                shape = fun @Double brushBaseShape brushParams
                                 -- TODO: remove this logic which is duplicated
                                 -- from elsewhere. The type should make it
                                 -- impossible to forget to apply the rotation.