From 7bd85bc2977af8cbe8b67fc995534afa7c79da57 Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Wed, 15 Jul 2015 19:21:49 -0400
Subject: [PATCH 01/12] Added plane and solid angles as base dimensions.

---
 src/Numeric/Units/Dimensional/DK.hs           |  24 +--
 .../Dimensional/DK/Dimensions/TermLevel.hs    |  45 +++---
 .../Dimensional/DK/Dimensions/TypeLevel.hs    |  49 +++---
 src/Numeric/Units/Dimensional/DK/NonSI.hs     |   4 +-
 .../Units/Dimensional/DK/Quantities.hs        | 144 +++++++++---------
 src/Numeric/Units/Dimensional/DK/SIUnits.hs   |   4 +-
 6 files changed, 141 insertions(+), 129 deletions(-)

diff --git a/src/Numeric/Units/Dimensional/DK.hs b/src/Numeric/Units/Dimensional/DK.hs
index 303a0f55..b558c0d5 100644
--- a/src/Numeric/Units/Dimensional/DK.hs
+++ b/src/Numeric/Units/Dimensional/DK.hs
@@ -210,10 +210,10 @@ module Numeric.Units.Dimensional.DK
     (*~~), (/~~), sum, mean, dimensionlessLength, nFromTo,
     -- * Dimension Synonyms
     -- $dimension-synonyms
-    DOne, DLength, DMass, DTime, DElectricCurrent, DThermodynamicTemperature, DAmountOfSubstance, DLuminousIntensity,
+    DOne, DLength, DMass, DTime, DElectricCurrent, DThermodynamicTemperature, DAmountOfSubstance, DLuminousIntensity, DPlaneAngle, DSolidAngle,
     -- * Quantity Synonyms
     -- $quantity-synonyms
-    Dimensionless, Length, Mass, Time, ElectricCurrent, ThermodynamicTemperature, AmountOfSubstance, LuminousIntensity,
+    Dimensionless, Length, Mass, Time, ElectricCurrent, ThermodynamicTemperature, AmountOfSubstance, LuminousIntensity, PlaneAngle, SolidAngle,
     -- * Constants
     -- $constants
     _0, _1, _2, _3, _4, _5, _6, _7, _8, _9, pi, tau,
@@ -372,6 +372,8 @@ type ElectricCurrent          = Quantity DElectricCurrent
 type ThermodynamicTemperature = Quantity DThermodynamicTemperature
 type AmountOfSubstance        = Quantity DAmountOfSubstance
 type LuminousIntensity        = Quantity DLuminousIntensity
+type PlaneAngle               = Quantity DPlaneAngle
+type SolidAngle               = Quantity DSolidAngle
 
 {- $dimension-arithmetic
 When performing arithmetic on units and quantities the arithmetics
@@ -568,16 +570,10 @@ We continue by defining elementary functions on 'Dimensionless'
 that may be obviously useful.
 -}
 
-exp, log, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh
+exp, log, sinh, cosh, tanh, asinh, acosh, atanh
   :: Floating a => Dimensionless a -> Dimensionless a
 exp   = fmap Prelude.exp
 log   = fmap Prelude.log
-sin   = fmap Prelude.sin
-cos   = fmap Prelude.cos
-tan   = fmap Prelude.tan
-asin  = fmap Prelude.asin
-acos  = fmap Prelude.acos
-atan  = fmap Prelude.atan
 sinh  = fmap Prelude.sinh
 cosh  = fmap Prelude.cosh
 tanh  = fmap Prelude.tanh
@@ -585,6 +581,16 @@ asinh = fmap Prelude.asinh
 acosh = fmap Prelude.acosh
 atanh = fmap Prelude.atanh
 
+sin, cos, tan :: Floating a => PlaneAngle a -> Dimensionless a
+sin   = (*~ one) . Prelude.sin . (/~ siUnit)
+cos   = (*~ one) . Prelude.cos . (/~ siUnit)
+tan   = (*~ one) . Prelude.tan . (/~ siUnit)
+
+asin, acos, atan :: Floating a => Dimensionless a -> PlaneAngle a
+asin  = (*~ siUnit) . Prelude.asin . (/~ one)
+acos  = (*~ siUnit) . Prelude.acos . (/~ one)
+atan  = (*~ siUnit) . Prelude.atan . (/~ one)
+
 (**) :: Floating a => Dimensionless a -> Dimensionless a -> Dimensionless a
 Dimensional x ** Dimensional y = Dimensional (x Prelude.** y)
 
diff --git a/src/Numeric/Units/Dimensional/DK/Dimensions/TermLevel.hs b/src/Numeric/Units/Dimensional/DK/Dimensions/TermLevel.hs
index 2cf7e968..9608ebea 100644
--- a/src/Numeric/Units/Dimensional/DK/Dimensions/TermLevel.hs
+++ b/src/Numeric/Units/Dimensional/DK/Dimensions/TermLevel.hs
@@ -9,7 +9,7 @@
    Portability: GHC only
 
 This module defines physical dimensions expressed in terms of
-the SI base dimensions, including arithmetic.
+the SI base dimensions along with plane and solid angles, including arithmetic.
 
 -}
 module Numeric.Units.Dimensional.DK.Dimensions.TermLevel
@@ -22,7 +22,7 @@ module Numeric.Units.Dimensional.DK.Dimensions.TermLevel
   (*), (/), (^), recip,
   -- * Synonyms for Base Dimensions
   dOne,
-  dLength, dMass, dTime, dElectricCurrent, dThermodynamicTemperature, dAmountOfSubstance, dLuminousIntensity,
+  dLength, dMass, dTime, dElectricCurrent, dThermodynamicTemperature, dAmountOfSubstance, dLuminousIntensity, dPlaneAngle, dSolidAngle,
   -- * Deconstruction
   asList
 )
@@ -32,10 +32,11 @@ import Data.Monoid (Monoid(..))
 import Prelude (id, (+), (-), Int, Show, Eq, Ord)
 import qualified Prelude as P
 
--- | A physical dimension, encoded as 7 integers, representing a factorization of the dimension into the
--- 7 SI base dimensions. By convention they are stored in the same order as 
+-- | A physical dimension, encoded as 9 integers, representing a factorization of the dimension into the
+-- 7 SI base dimensions, along with the dimensions of plane and solid angles.
+-- By convention they are stored in the same order as 
 -- in the 'Numeric.Units.Dimensional.DK.Dimensions.TypeLevel.Dimension' data kind.
-data Dimension' = Dim' !Int !Int !Int !Int !Int !Int !Int 
+data Dimension' = Dim' !Int !Int !Int !Int !Int !Int !Int !Int !Int
   deriving (Show, Eq, Ord)
 
 -- | The monoid of dimensions under multiplication.
@@ -53,16 +54,18 @@ instance HasDimension Dimension' where
 
 -- | The dimension of dimensionless values.
 dOne :: Dimension'
-dOne = Dim' 0 0 0 0 0 0 0
-
-dLength, dMass, dTime, dElectricCurrent, dThermodynamicTemperature, dAmountOfSubstance, dLuminousIntensity :: Dimension'
-dLength                   = Dim' 1 0 0 0 0 0 0
-dMass                     = Dim' 0 1 0 0 0 0 0
-dTime                     = Dim' 0 0 1 0 0 0 0
-dElectricCurrent          = Dim' 0 0 0 1 0 0 0
-dThermodynamicTemperature = Dim' 0 0 0 0 1 0 0
-dAmountOfSubstance        = Dim' 0 0 0 0 0 1 0
-dLuminousIntensity        = Dim' 0 0 0 0 0 0 1
+dOne = Dim' 0 0 0 0 0 0 0 0 0
+
+dLength, dMass, dTime, dElectricCurrent, dThermodynamicTemperature, dAmountOfSubstance, dLuminousIntensity, dPlaneAngle, dSolidAngle :: Dimension'
+dLength                   = Dim' 1 0 0 0 0 0 0 0 0
+dMass                     = Dim' 0 1 0 0 0 0 0 0 0
+dTime                     = Dim' 0 0 1 0 0 0 0 0 0
+dElectricCurrent          = Dim' 0 0 0 1 0 0 0 0 0
+dThermodynamicTemperature = Dim' 0 0 0 0 1 0 0 0 0
+dAmountOfSubstance        = Dim' 0 0 0 0 0 1 0 0 0
+dLuminousIntensity        = Dim' 0 0 0 0 0 0 1 0 0
+dPlaneAngle               = Dim' 0 0 0 0 0 0 0 1 0
+dSolidAngle               = Dim' 0 0 0 0 0 0 0 0 1
 
 {-
 We will reuse the operators and function names from the Prelude.
@@ -75,21 +78,21 @@ infixl 7  *, /
 
 -- | Forms the product of two dimensions.
 (*) :: Dimension' -> Dimension' -> Dimension'
-(Dim' l m t i th n j) * (Dim' l' m' t' i' th' n' j') = Dim' (l + l') (m + m') (t + t') (i + i') (th + th') (n + n') (j + j')
+(Dim' l m t i th n j pa sa) * (Dim' l' m' t' i' th' n' j' pa' sa') = Dim' (l + l') (m + m') (t + t') (i + i') (th + th') (n + n') (j + j') (pa + pa') (sa + sa')
 
 -- | Forms the quotient of two dimensions.
 (/) :: Dimension' -> Dimension' -> Dimension'
-(Dim' l m t i th n j) / (Dim' l' m' t' i' th' n' j') = Dim' (l - l') (m - m') (t - t') (i - i') (th - th') (n - n') (j - j')
+(Dim' l m t i th n j pa sa) / (Dim' l' m' t' i' th' n' j' pa' sa') = Dim' (l - l') (m - m') (t - t') (i - i') (th - th') (n - n') (j - j') (pa - pa') (sa - sa')
 
 -- | Raises a dimension to an integer power.
 (^) :: Dimension' -> Int -> Dimension'
-(Dim' l m t i th n j) ^ x = Dim' (x P.* l) (x P.* m) (x P.* t) (x P.* i) (x P.* th) (x P.* n) (x P.* j)
+(Dim' l m t i th n j pa sa) ^ x = Dim' (x P.* l) (x P.* m) (x P.* t) (x P.* i) (x P.* th) (x P.* n) (x P.* j) (x P.* pa) (x P.* sa)
 
 -- | Forms the reciprocal of a dimension.
 recip :: Dimension' -> Dimension'
 recip = (dOne /)
 
--- | Converts a dimension to a list of 7 integers, representing the exponent associated with each
--- of the 7 SI base dimensions in the standard order.
+-- | Converts a dimension to a list of 9 integers, representing the exponent associated with each
+-- of the 7 SI base dimensions in the standard order, followed by the dimensions of plane and spherical angles.
 asList :: Dimension' -> [Int]
-asList (Dim' l m t i th n j) = [l, m, t, i, th, n, j]
+asList (Dim' l m t i th n j pa sa) = [l, m, t, i, th, n, j, pa, sa]
diff --git a/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs b/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs
index 8f257835..25ab7038 100644
--- a/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs
+++ b/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs
@@ -18,7 +18,7 @@
    Portability: GHC only
 
 This module defines type-level physical dimensions expressed in terms of
-the SI base dimensions using 'Numeric.NumType.DK.NumType' for type-level integers.
+the SI base dimensions along with plane and solid angles using 'Numeric.NumType.DK.NumType' for type-level integers.
 
 Type-level arithmetic, synonyms for the base dimensions, and conversion to the term-level are included.
 -}
@@ -30,7 +30,7 @@ module Numeric.Units.Dimensional.DK.Dimensions.TypeLevel
   type (*), type (/), type (^), type Recip, type Root,
   -- * Synonyms for Base Dimensions
   DOne,
-  DLength, DMass, DTime, DElectricCurrent, DThermodynamicTemperature, DAmountOfSubstance, DLuminousIntensity,
+  DLength, DMass, DTime, DElectricCurrent, DThermodynamicTemperature, DAmountOfSubstance, DLuminousIntensity, DPlaneAngle, DSolidAngle,
   -- * Conversion to Term Level
   type KnownDimension
 )
@@ -45,6 +45,7 @@ import qualified Numeric.NumType.DK.Integers as N
 import Numeric.Units.Dimensional.DK.Dimensions.TermLevel
 
 -- | Represents a physical dimension in the basis of the 7 SI base dimensions, 
+-- along with the dimensions of plane and solid angles,
 -- where the respective dimensions are represented by type variables
 -- using the following convention.
 --
@@ -55,19 +56,23 @@ import Numeric.Units.Dimensional.DK.Dimensions.TermLevel
 --  * th: Thermodynamic temperature
 --  * n: Amount of substance
 --  * j: Luminous intensity
+--  * pa: Plane angle
+--  * sa: Solid angle
 --
 -- For the equivalent term-level representation, see 'Dimension''
-data Dimension = Dim TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt
+data Dimension = Dim TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt
 
 -- | The type-level dimensions of dimensionless values.
-type DOne                      = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
-type DLength                   = 'Dim 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
-type DMass                     = 'Dim 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
-type DTime                     = 'Dim 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
-type DElectricCurrent          = 'Dim 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero
-type DThermodynamicTemperature = 'Dim 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero
-type DAmountOfSubstance        = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
-type DLuminousIntensity        = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1
+type DOne                      = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DLength                   = 'Dim 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DMass                     = 'Dim 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DTime                     = 'Dim 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DElectricCurrent          = 'Dim 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DThermodynamicTemperature = 'Dim 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
+type DAmountOfSubstance        = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero
+type DLuminousIntensity        = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero
+type DPlaneAngle               = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
+type DSolidAngle               = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1
 
 {-
 We will reuse the operators and function names from the Prelude.
@@ -83,16 +88,16 @@ infixl 7  *, /
 type family (a::Dimension) * (b::Dimension) where
   DOne * d = d
   d * DOne = d
-  ('Dim l  m  t  i  th  n  j) * ('Dim l' m' t' i' th' n' j')
-    = 'Dim (l + l') (m + m') (t + t') (i + i') (th + th') (n + n') (j + j')
+  ('Dim l  m  t  i  th  n  j pa sa) * ('Dim l' m' t' i' th' n' j' pa' sa')
+    = 'Dim (l + l') (m + m') (t + t') (i + i') (th + th') (n + n') (j + j') (pa + pa') (sa + sa')
 
 -- | Division of dimensions corresponds to subtraction of the base
 -- dimensions' exponents.
 type family (a::Dimension) / (d::Dimension) where
   d / DOne = d
   d / d = DOne
-  ('Dim l  m  t  i  th  n  j) / ('Dim l' m' t' i' th' n' j')
-    = 'Dim (l - l') (m - m') (t - t') (i - i') (th - th') (n - n') (j - j')
+  ('Dim l  m  t  i  th  n  j pa sa) / ('Dim l' m' t' i' th' n' j' pa' sa')
+    = 'Dim (l - l') (m - m') (t - t') (i - i') (th - th') (n - n') (j - j') (pa - pa') (sa - sa')
 
 -- | The reciprocal of a dimension is defined as the result of dividing 'DOne' by it,
 -- or of negating each of the base dimensions' exponents.
@@ -107,8 +112,8 @@ type family (d::Dimension) ^ (x::TypeInt) where
   DOne ^ x = DOne
   d ^ 'Zero = DOne
   d ^ 'Pos1 = d
-  ('Dim l  m  t  i  th  n  j) ^ x
-    = 'Dim (l N.* x) (m N.* x) (t N.* x) (i N.* x) (th N.* x) (n N.* x) (j N.* x)
+  ('Dim l  m  t  i  th  n  j pa sa) ^ x
+    = 'Dim (l N.* x) (m N.* x) (t N.* x) (i N.* x) (th N.* x) (n N.* x) (j N.* x) (pa N.* x) (sa N.* x)
 
 -- | Roots of dimensions corresponds to division of the base dimensions'
 -- exponents by the order(?) of the root.
@@ -117,8 +122,8 @@ type family (d::Dimension) ^ (x::TypeInt) where
 type family Root (d::Dimension) (x::TypeInt) where
   Root DOne x = DOne
   Root d 'Pos1 = d
-  Root ('Dim l  m  t  i  th  n  j) x
-    = 'Dim (l N./ x) (m N./ x) (t N./ x) (i N./ x) (th N./ x) (n N./ x) (j N./ x)
+  Root ('Dim l  m  t  i  th  n  j pa sa) x
+    = 'Dim (l N./ x) (m N./ x) (t N./ x) (i N./ x) (th N./ x) (n N./ x) (j N./ x) (pa N./ x) (sa N./ x)
 
 -- | A KnownDimension is one for which we can construct a term-level representation.
 -- Each validly constructed type of kind 'Dimension' has a 'KnownDimension' instance.
@@ -134,7 +139,9 @@ instance ( KnownTypeInt l
          , KnownTypeInt th
          , KnownTypeInt n
          , KnownTypeInt j
-         ) => HasDimension (Proxy ('Dim l m t i th n j))
+         , KnownTypeInt pa
+         , KnownTypeInt sa
+         ) => HasDimension (Proxy ('Dim l m t i th n j pa sa))
   where 
     dimension _ = Dim'
                 (toNum (Proxy :: Proxy l))
@@ -144,3 +151,5 @@ instance ( KnownTypeInt l
                 (toNum (Proxy :: Proxy th))
                 (toNum (Proxy :: Proxy n))
                 (toNum (Proxy :: Proxy j))
+                (toNum (Proxy :: Proxy pa))
+                (toNum (Proxy :: Proxy sa))
diff --git a/src/Numeric/Units/Dimensional/DK/NonSI.hs b/src/Numeric/Units/Dimensional/DK/NonSI.hs
index bd5c50e0..917576ff 100644
--- a/src/Numeric/Units/Dimensional/DK/NonSI.hs
+++ b/src/Numeric/Units/Dimensional/DK/NonSI.hs
@@ -142,9 +142,9 @@ mile = prefix 1760 yard
 nauticalMile = prefix 1852 meter
 knot :: (Fractional a) => Unit DVelocity a
 knot = nauticalMile / hour
-revolution :: (Floating a) => Unit DOne a
+revolution :: (Floating a) => Unit DPlaneAngle a
 revolution = prefix (2 Prelude.* Prelude.pi) radian
-solid :: (Floating a) => Unit DOne a
+solid :: (Floating a) => Unit DSolidAngle a
 solid = prefix (4 Prelude.* Prelude.pi) steradian
 teaspoon :: (Fractional a) => Unit DVolume a
 teaspoon = prefix 5 (milli liter)
diff --git a/src/Numeric/Units/Dimensional/DK/Quantities.hs b/src/Numeric/Units/Dimensional/DK/Quantities.hs
index 164cddbc..95d6dce4 100644
--- a/src/Numeric/Units/Dimensional/DK/Quantities.hs
+++ b/src/Numeric/Units/Dimensional/DK/Quantities.hs
@@ -35,7 +35,7 @@ module Numeric.Units.Dimensional.DK.Quantities
   Area, Volume, Velocity, Acceleration, WaveNumber, MassDensity, Density, SpecificVolume, CurrentDensity,
   MagneticFieldStrength, AmountOfSubstanceConcentration, Concentration, Luminance,
   -- $table3
-  PlaneAngle, SolidAngle, Frequency, Force, Pressure, Stress, Energy, Work, QuantityOfHeat, Power, RadiantFlux,
+  Frequency, Force, Pressure, Stress, Energy, Work, QuantityOfHeat, Power, RadiantFlux,
   ElectricCharge, QuantityOfElectricity, ElectricPotential, PotentialDifference, ElectromotiveForce,
   Capacitance, ElectricResistance, ElectricConductance, MagneticFlux, MagneticFluxDensity,
   Inductance, LuminousFlux, Illuminance, CelsiusTemperature,
@@ -57,7 +57,7 @@ module Numeric.Units.Dimensional.DK.Quantities
   -- $dimension-aliases
   DArea, DVolume, DVelocity, DAcceleration, DWaveNumber, DMassDensity, DDensity, DSpecificVolume, DCurrentDensity,
   DMagneticFieldStrength, DAmountOfSubstanceConcentration, DConcentration, DLuminance,
-  DPlaneAngle, DSolidAngle, DFrequency, DForce, DPressure, DStress, DEnergy, DWork, DQuantityOfHeat, DPower, DRadiantFlux,
+  DFrequency, DForce, DPressure, DStress, DEnergy, DWork, DQuantityOfHeat, DPower, DRadiantFlux,
   DElectricCharge, DQuantityOfElectricity, DElectricPotential, DPotentialDifference, DElectromotiveForce,
   DCapacitance, DElectricResistance, DElectricConductance, DMagneticFlux, DMagneticFluxDensity,
   DInductance, DLuminousFlux, DIlluminance, DCelsiusTemperature,
@@ -73,8 +73,8 @@ module Numeric.Units.Dimensional.DK.Quantities
 where
 
 import Numeric.Units.Dimensional.DK
-  ( Dimension (Dim), Quantity, Dimensionless
-  , DOne, DLuminousIntensity, DThermodynamicTemperature
+  ( Dimension (Dim), Quantity
+  , DThermodynamicTemperature, DPlaneAngle
   , Unit, DLength, (^)  -- Used only for 'square' and 'cubic'.
   )
 import Numeric.NumType.DK.Integers
@@ -104,41 +104,41 @@ For each 'Quantity' alias supplied above, we also supply a corresponding 'Dimens
 These dimension aliases may be convenient for supplying type signatures for 'Unit's or for other type-level dimensional programming.
 -}
 
-type DArea = 'Dim 'Pos2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DArea = 'Dim 'Pos2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type Area  = Quantity DArea
 
-type DVolume = 'Dim 'Pos3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DVolume = 'Dim 'Pos3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type Volume  = Quantity DVolume
 
-type DVelocity = 'Dim 'Pos1 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DVelocity = 'Dim 'Pos1 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type Velocity  = Quantity DVelocity
 
-type DAcceleration = 'Dim 'Pos1 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero
+type DAcceleration = 'Dim 'Pos1 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type Acceleration  = Quantity DAcceleration
 
-type DWaveNumber = 'Dim 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DWaveNumber = 'Dim 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type WaveNumber  = Quantity DWaveNumber
 
-type DMassDensity = 'Dim 'Neg3 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DMassDensity = 'Dim 'Neg3 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type DDensity     = DMassDensity
 type MassDensity  = Quantity DMassDensity
 type Density      = MassDensity -- Short name.
 
-type DSpecificVolume = 'Dim 'Pos3 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DSpecificVolume = 'Dim 'Pos3 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type SpecificVolume  = Quantity DSpecificVolume
 
-type DCurrentDensity = 'Dim 'Neg2 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero
+type DCurrentDensity = 'Dim 'Neg2 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
 type CurrentDensity  = Quantity DCurrentDensity
 
-type DMagneticFieldStrength = 'Dim 'Neg1 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero
+type DMagneticFieldStrength = 'Dim 'Neg1 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
 type MagneticFieldStrength  = Quantity DMagneticFieldStrength
 
-type DAmountOfSubstanceConcentration = 'Dim 'Neg3 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
+type DAmountOfSubstanceConcentration = 'Dim 'Neg3 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero
 type DConcentration                  = DAmountOfSubstanceConcentration
 type AmountOfSubstanceConcentration  = Quantity DAmountOfSubstanceConcentration
 type Concentration                   = AmountOfSubstanceConcentration -- Short name.
 
-type DLuminance = 'Dim 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1
+type DLuminance = 'Dim 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero
 type Luminance  = Quantity DLuminance
 
 
@@ -149,69 +149,63 @@ SI coherent derived units with special names and symbols.
 
 -}
 
-type DPlaneAngle = DOne
-type PlaneAngle  = Dimensionless
-
-type DSolidAngle = DOne
-type SolidAngle  = Dimensionless
-
-type DFrequency = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DFrequency = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type Frequency  = Quantity DFrequency
 
-type DForce = 'Dim 'Pos1 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero
+type DForce = 'Dim 'Pos1 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type Force  = Quantity DForce
 
-type DPressure = 'Dim 'Neg1 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero
+type DPressure = 'Dim 'Neg1 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type DStress   = DPressure
 type Pressure  = Quantity DPressure
 type Stress    = Quantity DStress
 
-type DEnergy         = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero
+type DEnergy         = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type DWork           = DEnergy
 type DQuantityOfHeat = DEnergy
 type Energy          = Quantity DEnergy
 type Work            = Quantity DWork
 type QuantityOfHeat  = Quantity DQuantityOfHeat
 
-type DPower       = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero
+type DPower       = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type DRadiantFlux = DPower
 type Power        = Quantity DPower
 type RadiantFlux  = Quantity DRadiantFlux
 
-type DElectricCharge        = 'Dim 'Zero 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero
+type DElectricCharge        = 'Dim 'Zero 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
 type DQuantityOfElectricity = DElectricCharge
 type ElectricCharge         = Quantity DElectricCharge
 type QuantityOfElectricity  = Quantity DQuantityOfElectricity
 
-type DElectricPotential   = 'Dim 'Pos2 'Pos1 'Neg3 'Neg1 'Zero 'Zero 'Zero
+type DElectricPotential   = 'Dim 'Pos2 'Pos1 'Neg3 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type DPotentialDifference = DElectricPotential
 type DElectromotiveForce  = DElectricPotential
 type ElectricPotential    = Quantity DElectricPotential
 type PotentialDifference  = Quantity DPotentialDifference
 type ElectromotiveForce   = Quantity DElectromotiveForce
 
-type DCapacitance = 'Dim 'Neg2 'Neg1 'Pos4 'Pos2 'Zero 'Zero 'Zero
+type DCapacitance = 'Dim 'Neg2 'Neg1 'Pos4 'Pos2 'Zero 'Zero 'Zero 'Zero 'Zero
 type Capacitance  = Quantity DCapacitance
 
-type DElectricResistance = 'Dim 'Pos2 'Pos1 'Neg3 'Neg2 'Zero 'Zero 'Zero
+type DElectricResistance = 'Dim 'Pos2 'Pos1 'Neg3 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
 type ElectricResistance  = Quantity DElectricResistance
 
-type DElectricConductance = 'Dim 'Neg2 'Neg1 'Pos3 'Pos2 'Zero 'Zero 'Zero
+type DElectricConductance = 'Dim 'Neg2 'Neg1 'Pos3 'Pos2 'Zero 'Zero 'Zero 'Zero 'Zero
 type ElectricConductance  = Quantity DElectricConductance
 
-type DMagneticFlux = 'Dim 'Pos2 'Pos1 'Neg2 'Neg1 'Zero 'Zero 'Zero
+type DMagneticFlux = 'Dim 'Pos2 'Pos1 'Neg2 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type MagneticFlux  = Quantity DMagneticFlux
 
-type DMagneticFluxDensity = 'Dim 'Zero 'Pos1 'Neg2 'Neg1 'Zero 'Zero 'Zero
+type DMagneticFluxDensity = 'Dim 'Zero 'Pos1 'Neg2 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type MagneticFluxDensity  = Quantity DMagneticFluxDensity
 
-type DInductance = 'Dim 'Pos2 'Pos1 'Neg2 'Neg2 'Zero 'Zero 'Zero
+type DInductance = 'Dim 'Pos2 'Pos1 'Neg2 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
 type Inductance  = Quantity DInductance
 
-type DLuminousFlux = DLuminousIntensity
+type DLuminousFlux = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Pos1
 type LuminousFlux  = Quantity DLuminousFlux
 
-type DIlluminance = 'Dim 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1
+type DIlluminance = 'Dim 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Pos1
 type Illuminance  = Quantity DIlluminance
 
 type DCelsiusTemperature = DThermodynamicTemperature
@@ -220,7 +214,7 @@ type CelsiusTemperature  = Quantity DCelsiusTemperature
 type DActivity = DFrequency -- Activity of a radionuclide.
 type Activity  = Quantity DActivity
 
-type DAbsorbedDose   = 'Dim 'Pos2 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero
+type DAbsorbedDose   = 'Dim 'Pos2 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type DSpecificEnergy = DAbsorbedDose
 type DKerma          = DAbsorbedDose
 type AbsorbedDose    = Quantity DAbsorbedDose
@@ -238,7 +232,7 @@ type DirectionalDoseEquivalent  = DoseEquivalent
 type PersonalDoseEquivalent     = DoseEquivalent
 type EquivalentDose             = DoseEquivalent
 
-type DCatalyticActivity = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Pos1 'Zero
+type DCatalyticActivity = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero
 type CatalyticActivity  = Quantity DCatalyticActivity
 
 {- $table4
@@ -251,38 +245,38 @@ We use the same grouping as for table 2.
 
 -}
 
-type DAngularVelocity = DFrequency
+type DAngularVelocity = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
 type AngularVelocity  = Quantity DAngularVelocity
 
-type DAngularAcceleration = 'Dim 'Zero 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero
+type DAngularAcceleration = 'Dim 'Zero 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
 type AngularAcceleration  = Quantity DAngularAcceleration
 
-type DDynamicViscosity = 'Dim 'Neg1 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DDynamicViscosity = 'Dim 'Neg1 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type DynamicViscosity  = Quantity DDynamicViscosity
 
 type DMomentOfForce = DEnergy
 type MomentOfForce  = Quantity DMomentOfForce
 
-type DSurfaceTension = 'Dim 'Zero 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero
+type DSurfaceTension = 'Dim 'Zero 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type SurfaceTension  = Quantity DSurfaceTension
 
-type DHeatFluxDensity = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero
+type DHeatFluxDensity = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type DIrradiance      = DHeatFluxDensity
 type HeatFluxDensity  = Quantity DHeatFluxDensity
 type Irradiance       = Quantity DIrradiance
 
-type DRadiantIntensity = DPower
+type DRadiantIntensity = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Neg1
 type RadiantIntensity  = Quantity DRadiantIntensity
 
-type DRadiance = DIrradiance
+type DRadiance = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Neg1
 type Radiance  = Quantity DRadiance
 
-type DHeatCapacity = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Neg1 'Zero 'Zero
+type DHeatCapacity = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
 type DEntropy      = DHeatCapacity
 type HeatCapacity  = Quantity DHeatCapacity
 type Entropy       = Quantity DEntropy
 
-type DSpecificHeatCapacity = 'Dim 'Pos2 'Zero 'Neg2 'Zero 'Neg1 'Zero 'Zero
+type DSpecificHeatCapacity = 'Dim 'Pos2 'Zero 'Neg2 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
 type DSpecificEntropy      = DSpecificHeatCapacity
 type SpecificHeatCapacity  = Quantity DSpecificHeatCapacity
 type SpecificEntropy       = Quantity DSpecificEntropy
@@ -293,39 +287,39 @@ Specific energy was already defined in table 3.
 
 -}
 
-type DThermalConductivity = 'Dim 'Pos1 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero
+type DThermalConductivity = 'Dim 'Pos1 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
 type ThermalConductivity  = Quantity DThermalConductivity
 
 type DEnergyDensity = DPressure
 type EnergyDensity  = Quantity DEnergyDensity
 
-type DElectricFieldStrength = 'Dim 'Pos1 'Pos1 'Neg3 'Neg1 'Zero 'Zero 'Zero
+type DElectricFieldStrength = 'Dim 'Pos1 'Pos1 'Neg3 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type ElectricFieldStrength  = Quantity DElectricFieldStrength
 
-type DElectricChargeDensity = 'Dim 'Neg3 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero
+type DElectricChargeDensity = 'Dim 'Neg3 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
 type ElectricChargeDensity  = Quantity DElectricChargeDensity
 
-type DElectricFluxDensity = 'Dim 'Neg2 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero
+type DElectricFluxDensity = 'Dim 'Neg2 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
 type ElectricFluxDensity  = Quantity DElectricFluxDensity
 
-type DPermittivity = 'Dim 'Neg3 'Neg1 'Pos4 'Pos2 'Zero 'Zero 'Zero
+type DPermittivity = 'Dim 'Neg3 'Neg1 'Pos4 'Pos2 'Zero 'Zero 'Zero 'Zero 'Zero
 type Permittivity  = Quantity DPermittivity
 
-type DPermeability = 'Dim 'Pos1 'Pos1 'Neg2 'Neg2 'Zero 'Zero 'Zero
+type DPermeability = 'Dim 'Pos1 'Pos1 'Neg2 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
 type Permeability  = Quantity DPermeability
 
-type DMolarEnergy = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Neg1 'Zero
+type DMolarEnergy = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Neg1 'Zero 'Zero 'Zero
 type MolarEnergy  = Quantity DMolarEnergy
 
-type DMolarEntropy      = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Neg1 'Neg1 'Zero
+type DMolarEntropy      = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Neg1 'Neg1 'Zero 'Zero 'Zero
 type DMolarHeatCapacity = DMolarEntropy
 type MolarEntropy       = Quantity DMolarEntropy
 type MolarHeatCapacity  = Quantity DMolarHeatCapacity
 
-type DExposure = 'Dim 'Zero 'Neg1 'Pos1 'Pos1 'Zero 'Zero 'Zero
+type DExposure = 'Dim 'Zero 'Neg1 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
 type Exposure  = Quantity DExposure -- Exposure to x and gamma rays.
 
-type DAbsorbedDoseRate = 'Dim 'Pos2 'Zero 'Neg3 'Zero 'Zero 'Zero 'Zero
+type DAbsorbedDoseRate = 'Dim 'Pos2 'Zero 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type AbsorbedDoseRate  = Quantity DAbsorbedDoseRate
 
 {- $not-nist-guide
@@ -333,28 +327,28 @@ Here we define additional quantities on an as-needed basis. We also
 provide some synonyms that we anticipate will be useful.
 -}
 
-type DImpulse = 'Dim 'Pos1 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DImpulse = 'Dim 'Pos1 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type Impulse  = Quantity DImpulse
 
-type DMassFlow = 'Dim 'Zero 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DMassFlow = 'Dim 'Zero 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type MassFlow  = Quantity DMassFlow
 
-type DVolumeFlow = 'Dim 'Pos3 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DVolumeFlow = 'Dim 'Pos3 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type VolumeFlow = Quantity DVolumeFlow
 
-type DGravitationalParameter = 'Dim 'Pos3 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero
+type DGravitationalParameter = 'Dim 'Pos3 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type GravitationalParameter  = Quantity DGravitationalParameter
 
-type DKinematicViscosity = 'Dim 'Pos2 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DKinematicViscosity = 'Dim 'Pos2 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type KinematicViscosity  = Quantity DKinematicViscosity
 
-type DFirstMassMoment = 'Dim 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DFirstMassMoment = 'Dim 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type FirstMassMoment = Quantity DFirstMassMoment
 
-type DMomentOfInertia = 'Dim 'Pos2 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DMomentOfInertia = 'Dim 'Pos2 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type MomentOfInertia = Quantity DMomentOfInertia
 
-type DAngularMomentum = 'Dim 'Pos2 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DAngularMomentum = 'Dim 'Pos2 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
 type AngularMomentum = Quantity DAngularMomentum
 
 {-
@@ -363,7 +357,7 @@ The reciprocal of thermal conductivity.
 
 -}
 
-type DThermalResistivity = 'Dim 'Neg1 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero
+type DThermalResistivity = 'Dim 'Neg1 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
 type ThermalResistivity = Quantity DThermalResistivity
 
 {-
@@ -372,35 +366,35 @@ Thermal conductance and resistance quantities after http://en.wikipedia.org/wiki
 
 -}
 
-type DThermalConductance = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero
+type DThermalConductance = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
 type ThermalConductance = Quantity DThermalConductance
 
-type DThermalResistance = 'Dim 'Neg2 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero
+type DThermalResistance = 'Dim 'Neg2 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
 type ThermalResistance = Quantity DThermalResistance
 
-type DHeatTransferCoefficient = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero
+type DHeatTransferCoefficient = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
 type HeatTransferCoefficient = Quantity DHeatTransferCoefficient
 
 type DThermalAdmittance = DHeatTransferCoefficient
 type ThermalAdmittance = HeatTransferCoefficient
 
-type DThermalInsulance = 'Dim 'Zero 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero
+type DThermalInsulance = 'Dim 'Zero 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
 type ThermalInsulance = Quantity DThermalInsulance
 
-type DJerk = 'Dim 'Pos1 'Zero 'Neg3 'Zero 'Zero 'Zero 'Zero
+type DJerk = 'Dim 'Pos1 'Zero 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type Jerk = Quantity DJerk
 
-type Angle = PlaneAngle -- Abbreviation
 type DAngle = DPlaneAngle -- Abbreviation
+type Angle = Quantity DAngle -- Abbreviation
 
-type Thrust = Force
 type DThrust = DForce
+type Thrust = Force
 
+type DTorque = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Neg1 'Zero
 type Torque = MomentOfForce
-type DTorque = DMomentOfForce
 
-type EnergyPerUnitMass = SpecificEnergy
 type DEnergyPerUnitMass = DSpecificEnergy
+type EnergyPerUnitMass = SpecificEnergy
 
 {- $powers-of-length-units
 It is permissible to express powers of length units by prefixing
diff --git a/src/Numeric/Units/Dimensional/DK/SIUnits.hs b/src/Numeric/Units/Dimensional/DK/SIUnits.hs
index 02b3bb68..23c70cae 100644
--- a/src/Numeric/Units/Dimensional/DK/SIUnits.hs
+++ b/src/Numeric/Units/Dimensional/DK/SIUnits.hs
@@ -148,9 +148,9 @@ radian and steradian.
 -}
 
 radian :: Fractional a => Unit DPlaneAngle a
-radian = one -- meter * meter ^ neg1
+radian = siUnit
 steradian :: Fractional a => Unit DSolidAngle a
-steradian = one -- meter ^ pos2 * meter ^ neg2
+steradian = siUnit
 hertz :: Fractional a => Unit DFrequency a
 hertz = second ^ neg1
 newton :: Fractional a => Unit DForce a

From c15d0efb65318d928e43525c0db7a5dbaf777cf5 Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Thu, 16 Jul 2015 15:05:37 -0400
Subject: [PATCH 02/12] Added coerceAngles and removeAngles.

---
 src/Numeric/Units/Dimensional/DK.hs | 22 +++++++++++++++++++---
 1 file changed, 19 insertions(+), 3 deletions(-)

diff --git a/src/Numeric/Units/Dimensional/DK.hs b/src/Numeric/Units/Dimensional/DK.hs
index b558c0d5..464d8550 100644
--- a/src/Numeric/Units/Dimensional/DK.hs
+++ b/src/Numeric/Units/Dimensional/DK.hs
@@ -205,6 +205,8 @@ module Numeric.Units.Dimensional.DK
     negate, abs, nroot, sqrt, cbrt,
     -- ** Transcendental Functions
     exp, log, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh, atan2,
+    -- ** Dealing with Angles
+    removeAngles, coerceAngles,
     -- ** Operations on Collections
     -- $collections
     (*~~), (/~~), sum, mean, dimensionlessLength, nFromTo,
@@ -232,10 +234,11 @@ import Prelude
   )
 import qualified Prelude
 import Numeric.NumType.DK.Integers
-  ( TypeInt (Pos2, Pos3)
+  ( TypeInt (Zero, Pos2, Pos3)
   , pos2, pos3
   , KnownTypeInt, toNum
   )
+import Data.Coerce (coerce)
 import Data.Foldable (Foldable(foldr, foldl'))
 import Data.Monoid (Monoid(..))
 import Data.Ratio ((%))
@@ -591,15 +594,28 @@ asin  = (*~ siUnit) . Prelude.asin . (/~ one)
 acos  = (*~ siUnit) . Prelude.acos . (/~ one)
 atan  = (*~ siUnit) . Prelude.atan . (/~ one)
 
+-- | Removes angular dimensions from a dimensional value by equating radians
+-- and steradians with the dimensionless quantity one.
+removeAngles :: Dimensional v ('Dim l m t i th n j pa sa) a -> Dimensional v ('Dim l m t i th n j 'Zero 'Zero) a
+removeAngles = coerceAngles
+
+-- | Equates values whose dimensions differ in plane or solid angles by equating
+-- radians and steradians with the dimensionless quantity one.
+--
+-- See `removeAngles`, which offers a more specific result type, if you are only interested
+-- in ignoring plane and solid angle dimensions.
+coerceAngles :: Dimensional v ('Dim l m t i th n j pa sa) a -> Dimensional v ('Dim l m t i th n j pa' sa') a
+coerceAngles = coerce
+
 (**) :: Floating a => Dimensionless a -> Dimensionless a -> Dimensionless a
 Dimensional x ** Dimensional y = Dimensional (x Prelude.** y)
 
 {-
 For 'atan2' the operands need not be dimensionless but they must be
-of the same type. The result will of course always be dimensionless.
+of the same type. The result will of course always be a plane angle.
 -}
 
-atan2 :: RealFloat a => Quantity d a -> Quantity d a -> Dimensionless a
+atan2 :: RealFloat a => Quantity d a -> Quantity d a -> PlaneAngle a
 atan2 (Dimensional y) (Dimensional x) = Dimensional (Prelude.atan2 y x)
 
 -- | A polymorphic 'Unit' which can be used in place of the coherent

From d1531a3361454905d186b43cb72e456a79e1c284 Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Thu, 16 Jul 2015 15:16:59 -0400
Subject: [PATCH 03/12] Split baseUnit from siUnit.

---
 src/Numeric/Units/Dimensional/DK.hs         | 34 ++++++++++++++-------
 src/Numeric/Units/Dimensional/DK/SIUnits.hs | 18 +++++------
 2 files changed, 32 insertions(+), 20 deletions(-)

diff --git a/src/Numeric/Units/Dimensional/DK.hs b/src/Numeric/Units/Dimensional/DK.hs
index 464d8550..289dee52 100644
--- a/src/Numeric/Units/Dimensional/DK.hs
+++ b/src/Numeric/Units/Dimensional/DK.hs
@@ -220,7 +220,7 @@ module Numeric.Units.Dimensional.DK
     -- $constants
     _0, _1, _2, _3, _4, _5, _6, _7, _8, _9, pi, tau,
     -- * Constructing Units
-    prefix, siUnit, one,
+    prefix, baseUnit, siUnit, one,
     -- * On 'Functor', and Conversion Between Number Representations
     -- $functor
     dmap, changeRep
@@ -585,14 +585,14 @@ acosh = fmap Prelude.acosh
 atanh = fmap Prelude.atanh
 
 sin, cos, tan :: Floating a => PlaneAngle a -> Dimensionless a
-sin   = (*~ one) . Prelude.sin . (/~ siUnit)
-cos   = (*~ one) . Prelude.cos . (/~ siUnit)
-tan   = (*~ one) . Prelude.tan . (/~ siUnit)
+sin   = (*~ one) . Prelude.sin . (/~ baseUnit)
+cos   = (*~ one) . Prelude.cos . (/~ baseUnit)
+tan   = (*~ one) . Prelude.tan . (/~ baseUnit)
 
 asin, acos, atan :: Floating a => Dimensionless a -> PlaneAngle a
-asin  = (*~ siUnit) . Prelude.asin . (/~ one)
-acos  = (*~ siUnit) . Prelude.acos . (/~ one)
-atan  = (*~ siUnit) . Prelude.atan . (/~ one)
+asin  = (*~ baseUnit) . Prelude.asin . (/~ one)
+acos  = (*~ baseUnit) . Prelude.acos . (/~ one)
+atan  = (*~ baseUnit) . Prelude.atan . (/~ one)
 
 -- | Removes angular dimensions from a dimensional value by equating radians
 -- and steradians with the dimensionless quantity one.
@@ -618,11 +618,23 @@ of the same type. The result will of course always be a plane angle.
 atan2 :: RealFloat a => Quantity d a -> Quantity d a -> PlaneAngle a
 atan2 (Dimensional y) (Dimensional x) = Dimensional (Prelude.atan2 y x)
 
+-- | A polymorphic 'Unit' which can be used in place of the coherent
+-- base unit of any dimension. This allows polymorphic quantity
+-- creation and destruction without exposing the 'Dimensional' constructor.
+--
+-- `siUnit` is similar but does not include the radians or steradians associated
+-- with plane or solid angles.
+baseUnit :: Num a => Unit d a
+baseUnit = Dimensional 1
+
 -- | A polymorphic 'Unit' which can be used in place of the coherent
 -- SI base unit of any dimension. This allows polymorphic quantity
 -- creation and destruction without exposing the 'Dimensional' constructor.
-siUnit :: Num a => Unit d a
-siUnit = Dimensional 1
+--
+-- `baseUnit` is similar but includes the radians and steradians associated
+-- with plane or solid angles.
+siUnit :: Num a => Unit ('Dim l m t i th n j 'Zero 'Zero) a
+siUnit = removeAngles baseUnit
 
 {-
 The only unit we will define in this module is 'one'.
@@ -635,7 +647,7 @@ The only unit we will define in this module is 'one'.
 -- appear in expressions. However, for us it is necessary to use 'one'
 -- as we would any other unit to perform the "boxing" of dimensionless values.
 one :: Num a => Unit DOne a
-one = siUnit
+one = baseUnit
 
 {- $constants
 For convenience we define some constants for small integer values
@@ -648,7 +660,7 @@ good measure.
 -- it to express zero Length or Capacitance or Velocity etc, in addition
 -- to the dimensionless value zero.
 _0 :: Num a => Quantity d a
-_0 = 0 *~ siUnit
+_0 = 0 *~ baseUnit
 
 _1, _2, _3, _4, _5, _6, _7, _8, _9 :: (Num a) => Dimensionless a
 _1 = 1 *~ one
diff --git a/src/Numeric/Units/Dimensional/DK/SIUnits.hs b/src/Numeric/Units/Dimensional/DK/SIUnits.hs
index 23c70cae..71cb38fc 100644
--- a/src/Numeric/Units/Dimensional/DK/SIUnits.hs
+++ b/src/Numeric/Units/Dimensional/DK/SIUnits.hs
@@ -118,7 +118,7 @@ We define the SI base units in the order of table 1.
 -}
 
 metre, meter :: Num a => Unit DLength a
-metre = siUnit -- International English.
+metre = baseUnit -- International English.
 meter = metre         -- American English.
 
 {-
@@ -130,17 +130,17 @@ The drawback is that we are forced to use 'Fractional'.
 -}
 
 gram    :: Fractional a => Unit DMass a
-gram    = milli siUnit
+gram    = milli baseUnit
 second  :: Num a => Unit DTime a
-second  = siUnit
+second  = baseUnit
 ampere  :: Num a => Unit DElectricCurrent a
-ampere  = siUnit
+ampere  = baseUnit
 kelvin  :: Num a => Unit DThermodynamicTemperature a
-kelvin  = siUnit
+kelvin  = baseUnit
 mole    :: Num a => Unit DAmountOfSubstance a
-mole    = siUnit
+mole    = baseUnit
 candela :: Num a => Unit DLuminousIntensity a
-candela = siUnit
+candela = baseUnit
 
 {- $derived-units
 From Table 3, SI derived units with special names and symbols, including the
@@ -148,9 +148,9 @@ radian and steradian.
 -}
 
 radian :: Fractional a => Unit DPlaneAngle a
-radian = siUnit
+radian = baseUnit
 steradian :: Fractional a => Unit DSolidAngle a
-steradian = siUnit
+steradian = baseUnit
 hertz :: Fractional a => Unit DFrequency a
 hertz = second ^ neg1
 newton :: Fractional a => Unit DForce a

From 24a78392b9355b43d0e86728044b2a1f7e074473 Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Sat, 18 Jul 2015 14:36:22 -0400
Subject: [PATCH 04/12] Created .SI versions of modules which do not treat
 angles as dimensions.

---
 dimensional-dk.cabal                          |   6 +-
 src/Numeric/Units/Dimensional/DK/SI.hs        |  53 +++++++
 .../Units/Dimensional/DK/SI/Prelude.hs        |  44 ++++++
 .../Units/Dimensional/DK/SI/Quantities.hs     |  62 ++++++++
 .../Units/Dimensional/DK/SI/SIUnits.hs        |  45 ++++++
 src/Numeric/Units/Dimensional/DK/SIUnits.hs   |  86 ++++++------
 .../Units/Dimensional/DK/QuantitiesTest.hs    | 132 ++++++++++++------
 7 files changed, 340 insertions(+), 88 deletions(-)
 create mode 100644 src/Numeric/Units/Dimensional/DK/SI.hs
 create mode 100644 src/Numeric/Units/Dimensional/DK/SI/Prelude.hs
 create mode 100644 src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
 create mode 100644 src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs

diff --git a/dimensional-dk.cabal b/dimensional-dk.cabal
index 688924b8..1a699a8e 100644
--- a/dimensional-dk.cabal
+++ b/dimensional-dk.cabal
@@ -51,7 +51,11 @@ library
                        Numeric.Units.Dimensional.DK.Dimensions,
                        Numeric.Units.Dimensional.DK.Dimensions.TermLevel,
                        Numeric.Units.Dimensional.DK.Dimensions.TypeLevel,
-                       Numeric.Units.Dimensional.DK.Functor
+                       Numeric.Units.Dimensional.DK.Functor,
+                       Numeric.Units.Dimensional.DK.SI,
+                       Numeric.Units.Dimensional.DK.SI.Prelude,
+                       Numeric.Units.Dimensional.DK.SI.SIUnits,
+                       Numeric.Units.Dimensional.DK.SI.Quantities
 
 test-suite tests
   type:                exitcode-stdio-1.0
diff --git a/src/Numeric/Units/Dimensional/DK/SI.hs b/src/Numeric/Units/Dimensional/DK/SI.hs
new file mode 100644
index 00000000..ba62fd52
--- /dev/null
+++ b/src/Numeric/Units/Dimensional/DK/SI.hs
@@ -0,0 +1,53 @@
+{-# LANGUAGE DataKinds #-}
+
+{- |
+   Copyright  : Copyright (C) 2006-2015 Bjorn Buckwalter
+   License    : BSD3
+
+   Maintainer : bjorn@buckwalter.se
+   Stability  : Stable
+   Portability: GHC only
+
+= Summary
+
+This module defines a variant of "Numeric.Units.Dimensional.DK" where plane and solid angles are not treated as dimensions.
+
+Compare, e.g., 'sin' with 'Numeric.Units.Dimensional.DK.sin'
+
+-}
+module Numeric.Units.Dimensional.DK.SI
+(
+  module Numeric.Units.Dimensional.DK,
+  sin, cos, tan, asin, acos, atan, atan2,
+  DPlaneAngle, DSolidAngle,
+  PlaneAngle, SolidAngle,
+  baseUnit
+)
+where
+
+import qualified Numeric.Units.Dimensional.DK as A
+import Numeric.Units.Dimensional.DK hiding (sin, cos, tan, asin, acos, atan, atan2, DPlaneAngle, DSolidAngle, PlaneAngle, SolidAngle, baseUnit)
+import Numeric.NumType.DK.Integers
+  ( TypeInt (Zero)
+  )
+import Prelude hiding (sin, cos, tan, asin, acos, atan, atan2)
+
+sin, cos, tan, asin, acos, atan :: (Floating a) => Dimensionless a -> Dimensionless a
+sin = A.sin . coerceAngles
+cos = A.cos . coerceAngles
+tan = A.tan . coerceAngles
+asin = removeAngles . A.asin
+acos = removeAngles . A.acos
+atan = removeAngles . A.atan
+
+atan2 :: (RealFloat a) => Quantity d a -> Quantity d a -> Dimensionless a
+atan2 x y = removeAngles $ A.atan2 x y
+
+type DPlaneAngle = DOne
+type DSolidAngle = DOne
+
+type PlaneAngle = Quantity DPlaneAngle
+type SolidAngle = Quantity DSolidAngle
+
+baseUnit :: (Num a) => Unit ('Dim l m t i th n j 'Zero 'Zero) a
+baseUnit = siUnit
diff --git a/src/Numeric/Units/Dimensional/DK/SI/Prelude.hs b/src/Numeric/Units/Dimensional/DK/SI/Prelude.hs
new file mode 100644
index 00000000..849f077c
--- /dev/null
+++ b/src/Numeric/Units/Dimensional/DK/SI/Prelude.hs
@@ -0,0 +1,44 @@
+{- |
+   Copyright  : Copyright (C) 2006-2015 Bjorn Buckwalter
+   License    : BSD3
+
+   Maintainer : bjorn@buckwalter.se
+   Stability  : Stable
+   Portability: GHC only
+
+= Summary
+
+This module defines a variant of "Numeric.Units.Dimensional.DK.Prelude" where plane and solid angles are not treated as dimensions.
+
+-}
+module Numeric.Units.Dimensional.DK.SI.Prelude
+    ( module Numeric.Units.Dimensional.DK.SI
+    , module Numeric.Units.Dimensional.DK.SI.Quantities
+    , module Numeric.Units.Dimensional.DK.SI.SIUnits
+    , module Numeric.NumType.DK.Integers
+    , module Data.Foldable
+    , module Prelude
+    ) where
+
+import Numeric.Units.Dimensional.DK.SI hiding
+    ( dmap
+    )
+
+import Numeric.Units.Dimensional.DK.SI.Quantities
+
+import Numeric.Units.Dimensional.DK.SI.SIUnits
+
+import Numeric.NumType.DK.Integers
+    ( neg5, neg4, neg3, neg2, neg1, zero, pos1, pos2, pos3, pos4, pos5
+    )  -- Used in exponents.
+
+import Data.Foldable
+    ( product, minimum, maximum )
+
+import Prelude hiding
+    ( (+), (-), (*), (/), (^), (**)
+    , abs, negate, pi, exp, log, sqrt
+    , sin, cos, tan, asin, acos, atan, atan2
+    , sinh, cosh, tanh, asinh, acosh, atanh
+    , sum, product, minimum, maximum
+    )  -- Hide definitions overridden by 'Numeric.Dimensional'.
\ No newline at end of file
diff --git a/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs b/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
new file mode 100644
index 00000000..70d884a4
--- /dev/null
+++ b/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
@@ -0,0 +1,62 @@
+{-# LANGUAGE DataKinds #-}
+{-# LANGUAGE KindSignatures #-}
+{-# LANGUAGE TypeFamilies #-}
+
+{- |
+   Copyright  : Copyright (C) 2006-2015 Bjorn Buckwalter
+   License    : BSD3
+
+   Maintainer : bjorn@buckwalter.se
+   Stability  : Stable
+   Portability: GHC only
+
+= Summary
+
+This module defines a variant of "Numeric.Units.Dimensional.DK.Quantities" where plane and solid angles are not treated as dimensions.
+
+Compare, e.g., 'LuminousFlux' with 'Numeric.Units.Dimensional.DK.Quantities.LuminousFlux'
+
+-}
+module Numeric.Units.Dimensional.DK.SI.Quantities
+(
+  module Numeric.Units.Dimensional.DK.Quantities,
+
+  LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque,
+  DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque
+)
+where
+
+import Numeric.Units.Dimensional.DK.SI
+import qualified Numeric.Units.Dimensional.DK.Quantities as A
+import Numeric.Units.Dimensional.DK.Quantities hiding (LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque,
+                                                       DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque)
+import Numeric.NumType.DK.Integers
+  ( TypeInt (Zero)
+  )
+
+type family RemoveAngles (d :: Dimension) :: Dimension where
+  RemoveAngles ('Dim l m t i th n j pa sa) = 'Dim l m t i th n j 'Zero 'Zero
+
+type DLuminousFlux = RemoveAngles A.DLuminousFlux
+type LuminousFlux = Quantity DLuminousFlux
+
+type DIlluminance = RemoveAngles A.DIlluminance
+type Illuminance = Quantity DIlluminance
+
+type DAngularVelocity = RemoveAngles A.DAngularVelocity
+type AngularVelocity = Quantity DAngularVelocity
+
+type DAngularAcceleration = RemoveAngles A.DAngularAcceleration
+type AngularAcceleration = Quantity DAngularAcceleration
+
+type DRadiantIntensity = RemoveAngles A.DRadiantIntensity
+type RadiantIntensity = Quantity DRadiantIntensity
+
+type DRadiance = RemoveAngles A.DRadiance
+type Radiance = Quantity DRadiance
+
+type DAngularMomentum = RemoveAngles A.DAngularMomentum
+type AngularMomentum = Quantity DAngularMomentum
+
+type DTorque = RemoveAngles A.DTorque
+type Torque = Quantity DTorque
diff --git a/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs b/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs
new file mode 100644
index 00000000..4711a62d
--- /dev/null
+++ b/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs
@@ -0,0 +1,45 @@
+{- |
+   Copyright  : Copyright (C) 2006-2015 Bjorn Buckwalter
+   License    : BSD3
+
+   Maintainer : bjorn@buckwalter.se
+   Stability  : Stable
+   Portability: GHC only
+
+= Summary
+
+This module defines a variant of "Numeric.Units.Dimensional.DK.SIUnits" where plane and solid angles are not treated as dimensions.
+
+Compare, e.g., 'radian' with 'Numeric.Units.Dimensional.DK.SIUnits.radian'
+
+-}
+module Numeric.Units.Dimensional.DK.SI.SIUnits
+(
+  module Numeric.Units.Dimensional.DK.SIUnits,
+  radian, steradian,
+  degree, arcminute, arcsecond, degreeOfArc, minuteOfArc, secondOfArc,
+  lumen, lux
+)
+where
+
+import Numeric.Units.Dimensional.DK.SI
+import Numeric.Units.Dimensional.DK.SI.Quantities
+import qualified Numeric.Units.Dimensional.DK.SIUnits as A
+import Numeric.Units.Dimensional.DK.SIUnits hiding (radian, steradian, degree, arcminute, arcsecond, degreeOfArc, minuteOfArc, secondOfArc, lumen, lux)
+
+radian, steradian :: (Num a) => Unit DOne a
+radian = removeAngles A.radian
+steradian = removeAngles A.steradian
+
+degree, arcminute, arcsecond, degreeOfArc, minuteOfArc, secondOfArc :: (Floating a) => Unit DOne a
+degree = removeAngles A.degree
+arcminute = removeAngles A.arcminute
+arcsecond = removeAngles A.arcsecond
+degreeOfArc = removeAngles A.degreeOfArc
+minuteOfArc = removeAngles A.minuteOfArc
+secondOfArc = removeAngles A.secondOfArc
+
+lumen :: Num a => Unit DLuminousFlux a
+lumen = removeAngles A.lumen
+lux :: Num a => Unit DIlluminance a
+lux = removeAngles A.lux
diff --git a/src/Numeric/Units/Dimensional/DK/SIUnits.hs b/src/Numeric/Units/Dimensional/DK/SIUnits.hs
index 71cb38fc..351a69f8 100644
--- a/src/Numeric/Units/Dimensional/DK/SIUnits.hs
+++ b/src/Numeric/Units/Dimensional/DK/SIUnits.hs
@@ -60,7 +60,7 @@ where
 
 import Numeric.Units.Dimensional.DK
 import Numeric.Units.Dimensional.DK.Quantities
-import Numeric.NumType.DK.Integers ( neg1, neg2, pos2, pos3 )
+import Numeric.NumType.DK.Integers ( pos3 )
 import Prelude ( (.), Num, Real, realToFrac, Fractional, Floating, recip )
 import qualified Prelude
 
@@ -147,46 +147,46 @@ From Table 3, SI derived units with special names and symbols, including the
 radian and steradian.
 -}
 
-radian :: Fractional a => Unit DPlaneAngle a
-radian = baseUnit
-steradian :: Fractional a => Unit DSolidAngle a
-steradian = baseUnit
-hertz :: Fractional a => Unit DFrequency a
-hertz = second ^ neg1
-newton :: Fractional a => Unit DForce a
-newton = kilo gram * meter * second ^ neg2
-pascal :: Fractional a => Unit DPressure a
-pascal = newton / meter ^ pos2
-joule :: Fractional a => Unit DEnergy a
-joule = newton * meter
-watt :: Fractional a => Unit DPower a
-watt = joule / second
-coulomb :: Fractional a => Unit DElectricCharge a
-coulomb = second * ampere
-volt :: Fractional a => Unit DElectricPotential a
-volt = watt / ampere
-farad :: Fractional a => Unit DCapacitance a
-farad = coulomb / volt
-ohm :: Fractional a => Unit DElectricResistance a
-ohm = volt / ampere
-siemens :: Fractional a => Unit DElectricConductance a
-siemens = ampere / volt
-weber :: Fractional a => Unit DMagneticFlux a
-weber = volt * second
-tesla :: Fractional a => Unit DMagneticFluxDensity a
-tesla = weber / meter ^ pos2
-henry :: Fractional a => Unit DInductance a
-henry = weber / ampere
+radian :: Num a => Unit DPlaneAngle a
+radian = baseUnit -- meter * meter ^ neg1
+steradian :: Num a => Unit DSolidAngle a
+steradian = baseUnit -- meter ^ pos2 * meter ^ neg2
+hertz :: Num a => Unit DFrequency a
+hertz = siUnit -- second ^ neg1
+newton :: Num a => Unit DForce a
+newton = siUnit -- kilo gram * meter * second ^ neg2
+pascal :: Num a => Unit DPressure a
+pascal = siUnit -- newton / meter ^ pos2
+joule :: Num a => Unit DEnergy a
+joule = siUnit -- newton * meter
+watt :: Num a => Unit DPower a
+watt = siUnit -- joule / second
+coulomb :: Num a => Unit DElectricCharge a
+coulomb = siUnit -- second * ampere
+volt :: Num a => Unit DElectricPotential a
+volt = siUnit -- watt / ampere
+farad :: Num a => Unit DCapacitance a
+farad = siUnit -- coulomb / volt
+ohm :: Num a => Unit DElectricResistance a
+ohm = siUnit -- volt / ampere
+siemens :: Num a => Unit DElectricConductance a
+siemens = siUnit -- ampere / volt
+weber :: Num a => Unit DMagneticFlux a
+weber = siUnit -- volt * second
+tesla :: Num a => Unit DMagneticFluxDensity a
+tesla = siUnit -- weber / meter ^ pos2
+henry :: Num a => Unit DInductance a
+henry = siUnit -- weber / ampere
 
 {-
 We defer the definition of Celcius temperature to another section (would
 appear here if we stricly followed table 3).
 -}
 
-lumen :: Fractional a => Unit DLuminousFlux a
-lumen = candela * steradian
-lux :: Fractional a => Unit DIlluminance a
-lux = lumen / meter ^ pos2
+lumen :: Num a => Unit DLuminousFlux a
+lumen = baseUnit -- candela * steradian
+lux :: Num a => Unit DIlluminance a
+lux = baseUnit -- lumen / meter ^ pos2
 
 {- $celsius
 A problematic area is units which increase proportionally to the
@@ -216,14 +216,14 @@ The last units from Table 3 are SI derived units with special names and symbols
 of safeguarding human health.
 -}
 
-becquerel :: Fractional a => Unit DActivity a
-becquerel = second ^ neg1
-gray :: Fractional a => Unit DAbsorbedDose a
-gray = joule / kilo gram
-sievert :: Fractional a => Unit DDoseEquivalent a
-sievert = joule / kilo gram
-katal :: Fractional a => Unit DCatalyticActivity a
-katal = mole / second
+becquerel :: Num a => Unit DActivity a
+becquerel = siUnit -- second ^ neg1
+gray :: Num a => Unit DAbsorbedDose a
+gray = siUnit -- joule / kilo gram
+sievert :: Num a => Unit DDoseEquivalent a
+sievert = siUnit -- joule / kilo gram
+katal :: Num a => Unit DCatalyticActivity a
+katal = siUnit -- mole / second
 
 {- $accepted-units
 There are several units that are not strictly part of the SI but
diff --git a/tests/Numeric/Units/Dimensional/DK/QuantitiesTest.hs b/tests/Numeric/Units/Dimensional/DK/QuantitiesTest.hs
index 7217da19..52095160 100644
--- a/tests/Numeric/Units/Dimensional/DK/QuantitiesTest.hs
+++ b/tests/Numeric/Units/Dimensional/DK/QuantitiesTest.hs
@@ -30,54 +30,98 @@ x10 = 1 *~ (mole / meter ^ pos3)
 x11 :: Luminance Double
 x11 = 1 *~ (candela / meter ^ pos2)
 
--- Tables 3a and 3b are implicitely tested by the corresponding
--- unit definitions.
+-- These definitions simply verify that the type synonyms are
+-- consistent with the appropriate units from table 3. If the
+-- definitions compile the type synonyms are good.
+
+y1 :: PlaneAngle Double
+y1 = coerceAngles $ 1 *~ (meter / meter)
+y2 :: SolidAngle Double
+y2 = coerceAngles $ 1 *~ (meter ^ pos2 / meter ^ pos2)
+y3 :: Frequency Double
+y3 = 1 *~ (one / second)
+y4 :: Force Double
+y4 = 1 *~ (meter * kilo gram / second ^ pos2)
+y5 :: Pressure Double
+y5 = 1 *~ (newton / meter ^ pos2)
+y6 :: Energy Double
+y6 = 1 *~ (newton * meter)
+y7 :: Power Double
+y7 = 1 *~ (joule / second)
+y8 :: ElectricCharge Double
+y8 = 1 *~ (second * ampere)
+y9 :: ElectricPotential Double
+y9 = 1 *~ (watt / ampere)
+y10 :: Capacitance Double
+y10 = 1 *~ (coulomb / volt)
+y11 :: ElectricResistance Double
+y11 = 1 *~ (volt / ampere)
+y12 :: ElectricConductance Double
+y12 = 1 *~ (ampere / volt)
+y13 :: MagneticFlux Double
+y13 = 1 *~ (volt * second)
+y14 :: MagneticFluxDensity Double
+y14 = 1 *~ (weber / meter ^ pos2)
+y15 :: Inductance Double
+y15 = 1 *~ (weber / ampere)
+y16 :: LuminousFlux Double
+y16 = 1 *~ (candela * steradian)
+y17 :: Illuminance Double
+y17 = 1 *~ (lumen / meter ^ pos2)
+y18 :: Activity Double
+y18 = 1 *~ (one / second)
+y19 :: AbsorbedDose Double
+y19 = 1 *~ (joule / kilo gram)
+y20 :: DoseEquivalent Double
+y20 = 1 *~ (joule / kilo gram)
+y21 :: CatalyticActivity Double
+y21 = 1 *~ (mole / second)
 
 -- Verification of table 4. If the definitions compile the type
 -- synonyms are good.
 
-y1 :: AngularVelocity Double
-y1 = 1 *~ (radian / second)
-y2 :: AngularAcceleration Double
-y2 = 1 *~ (radian / second ^ pos2)
-y3 :: DynamicViscosity Double
-y3 = 1 *~ (pascal * second)
-y4 :: MomentOfForce Double
-y4 = 1 *~ (newton * meter)
-y5 :: SurfaceTension Double
-y5 = 1 *~ (newton / meter)
-y6 :: HeatFluxDensity Double
-y6 = 1 *~ (watt / meter ^ pos2)
-y7 :: RadiantIntensity Double
-y7 = 1 *~ (watt / steradian)
-y8 :: Radiance Double
-y8 = 1 *~ (watt / (meter ^ pos2 * steradian))
-y9 :: HeatCapacity Double
-y9 = 1 *~ (joule / kelvin)
-y10 :: SpecificHeatCapacity Double
-y10 = 1 *~ (joule / (kilo gram * kelvin))
-y11 :: ThermalConductivity Double
-y11 = 1 *~ (watt / (meter * kelvin))
-y12 :: EnergyDensity Double
-y12 = 1 *~ (joule / meter ^ pos3)
-y13 :: ElectricFieldStrength Double
-y13 = 1 *~ (volt / meter)
-y14 :: ElectricChargeDensity Double
-y14 = 1 *~ (coulomb / meter ^ pos3)
-y15 :: ElectricFluxDensity Double
-y15 = 1 *~ (coulomb / meter ^ pos2)
-y16 :: Permittivity Double
-y16 = 1 *~ (farad / meter)
-y17 :: Permeability Double
-y17 = 1 *~ (henry / meter)
-y18 :: MolarEnergy Double
-y18 = 1 *~ (joule / mole)
-y19 :: MolarEntropy Double
-y19 = 1 *~ (joule / (mole * kelvin))
-y20 :: Exposure Double
-y20 = 1 *~ (coulomb / kilo gram)
-y21 :: AbsorbedDoseRate Double
-y21 = 1 *~ (gray / second)
+z1 :: AngularVelocity Double
+z1 = 1 *~ (radian / second)
+z2 :: AngularAcceleration Double
+z2 = 1 *~ (radian / second ^ pos2)
+z3 :: DynamicViscosity Double
+z3 = 1 *~ (pascal * second)
+z4 :: MomentOfForce Double
+z4 = 1 *~ (newton * meter)
+z5 :: SurfaceTension Double
+z5 = 1 *~ (newton / meter)
+z6 :: HeatFluxDensity Double
+z6 = 1 *~ (watt / meter ^ pos2)
+z7 :: RadiantIntensity Double
+z7 = 1 *~ (watt / steradian)
+z8 :: Radiance Double
+z8 = 1 *~ (watt / (meter ^ pos2 * steradian))
+z9 :: HeatCapacity Double
+z9 = 1 *~ (joule / kelvin)
+z10 :: SpecificHeatCapacity Double
+z10 = 1 *~ (joule / (kilo gram * kelvin))
+z11 :: ThermalConductivity Double
+z11 = 1 *~ (watt / (meter * kelvin))
+z12 :: EnergyDensity Double
+z12 = 1 *~ (joule / meter ^ pos3)
+z13 :: ElectricFieldStrength Double
+z13 = 1 *~ (volt / meter)
+z14 :: ElectricChargeDensity Double
+z14 = 1 *~ (coulomb / meter ^ pos3)
+z15 :: ElectricFluxDensity Double
+z15 = 1 *~ (coulomb / meter ^ pos2)
+z16 :: Permittivity Double
+z16 = 1 *~ (farad / meter)
+z17 :: Permeability Double
+z17 = 1 *~ (henry / meter)
+z18 :: MolarEnergy Double
+z18 = 1 *~ (joule / mole)
+z19 :: MolarEntropy Double
+z19 = 1 *~ (joule / (mole * kelvin))
+z20 :: Exposure Double
+z20 = 1 *~ (coulomb / kilo gram)
+z21 :: AbsorbedDoseRate Double
+z21 = 1 *~ (gray / second)
 
 -- Other quantitites.
 mu :: GravitationalParameter Double

From 475a932aeed6d5842b6f376f8b40fec14edc506e Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Sat, 18 Jul 2015 17:24:45 -0400
Subject: [PATCH 05/12] Hid coerceAngles and removeAngles from the Prelude.

---
 src/Numeric/Units/Dimensional/DK/Prelude.hs          | 1 +
 tests/Numeric/Units/Dimensional/DK/QuantitiesTest.hs | 1 +
 2 files changed, 2 insertions(+)

diff --git a/src/Numeric/Units/Dimensional/DK/Prelude.hs b/src/Numeric/Units/Dimensional/DK/Prelude.hs
index d3e0b85f..390f5846 100644
--- a/src/Numeric/Units/Dimensional/DK/Prelude.hs
+++ b/src/Numeric/Units/Dimensional/DK/Prelude.hs
@@ -25,6 +25,7 @@ module Numeric.Units.Dimensional.DK.Prelude
 
 import Numeric.Units.Dimensional.DK hiding
     ( dmap
+    , removeAngles, coerceAngles
     )
 
 import Numeric.Units.Dimensional.DK.Quantities
diff --git a/tests/Numeric/Units/Dimensional/DK/QuantitiesTest.hs b/tests/Numeric/Units/Dimensional/DK/QuantitiesTest.hs
index 52095160..c6313f63 100644
--- a/tests/Numeric/Units/Dimensional/DK/QuantitiesTest.hs
+++ b/tests/Numeric/Units/Dimensional/DK/QuantitiesTest.hs
@@ -1,6 +1,7 @@
 module Numeric.Units.Dimensional.DK.QuantitiesTest where
 
 import Numeric.Units.Dimensional.DK.Prelude
+import Numeric.Units.Dimensional.DK
 import qualified Prelude
 
 -- These definitions simply verify that the type synonyms are

From 4510cd6f8937b7164e72a33b58742f1394d78da1 Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Sat, 18 Jul 2015 17:30:11 -0400
Subject: [PATCH 06/12] Added angular units to displayed form of quantities.

---
 src/Numeric/Units/Dimensional/DK.hs | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Numeric/Units/Dimensional/DK.hs b/src/Numeric/Units/Dimensional/DK.hs
index 289dee52..6856859f 100644
--- a/src/Numeric/Units/Dimensional/DK.hs
+++ b/src/Numeric/Units/Dimensional/DK.hs
@@ -725,7 +725,7 @@ even a requirement.
 -}
 instance (KnownDimension d, Show a) => Show (Quantity d a) where
       show q@(Dimensional x) = let powers = asList $ dimension q
-                                   units = ["m", "kg", "s", "A", "K", "mol", "cd"]
+                                   units = ["m", "kg", "s", "A", "K", "mol", "cd", "rad", "sr"]
                                    dims = concat $ zipWith dimUnit units powers
                                in show x ++ dims
 

From 4db8b5a1c1b272e383c3c8eda2d3e8bbafa878f0 Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Sat, 18 Jul 2015 17:30:27 -0400
Subject: [PATCH 07/12] Corrected definition of Torque to use angular basis.

---
 src/Numeric/Units/Dimensional/DK/Quantities.hs | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Numeric/Units/Dimensional/DK/Quantities.hs b/src/Numeric/Units/Dimensional/DK/Quantities.hs
index 95d6dce4..eab92406 100644
--- a/src/Numeric/Units/Dimensional/DK/Quantities.hs
+++ b/src/Numeric/Units/Dimensional/DK/Quantities.hs
@@ -391,7 +391,7 @@ type DThrust = DForce
 type Thrust = Force
 
 type DTorque = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Neg1 'Zero
-type Torque = MomentOfForce
+type Torque = Quantity DTorque
 
 type DEnergyPerUnitMass = DSpecificEnergy
 type EnergyPerUnitMass = SpecificEnergy

From 0f79c8569e1ba90be0d68f0c385bb33003301acc Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Sun, 19 Jul 2015 09:42:57 -0400
Subject: [PATCH 08/12] Refactoring and documentation of ToSIDim type family.

---
 src/Numeric/Units/Dimensional/DK.hs           |  8 +++----
 .../Dimensional/DK/Dimensions/TypeLevel.hs    | 20 +++++++++++++++++
 src/Numeric/Units/Dimensional/DK/SI.hs        |  5 +----
 .../Units/Dimensional/DK/SI/Quantities.hs     | 22 +++++++------------
 .../Units/Dimensional/DK/SI/SIUnits.hs        |  4 ++--
 5 files changed, 35 insertions(+), 24 deletions(-)

diff --git a/src/Numeric/Units/Dimensional/DK.hs b/src/Numeric/Units/Dimensional/DK.hs
index 6856859f..7009a10f 100644
--- a/src/Numeric/Units/Dimensional/DK.hs
+++ b/src/Numeric/Units/Dimensional/DK.hs
@@ -195,7 +195,7 @@ module Numeric.Units.Dimensional.DK
     Dimension (Dim),
     -- ** Dimension Arithmetic
     -- $dimension-arithmetic
-    type (*), type (/), type (^), Root, Recip,
+    type (*), type (/), type (^), Root, Recip, type SIDim, type ToSIDim,
     -- ** Term Level Representation of Dimensions
     -- $dimension-terms
     Dimension' (Dim'), HasDimension(..), KnownDimension,
@@ -234,7 +234,7 @@ import Prelude
   )
 import qualified Prelude
 import Numeric.NumType.DK.Integers
-  ( TypeInt (Zero, Pos2, Pos3)
+  ( TypeInt (Pos2, Pos3)
   , pos2, pos3
   , KnownTypeInt, toNum
   )
@@ -596,7 +596,7 @@ atan  = (*~ baseUnit) . Prelude.atan . (/~ one)
 
 -- | Removes angular dimensions from a dimensional value by equating radians
 -- and steradians with the dimensionless quantity one.
-removeAngles :: Dimensional v ('Dim l m t i th n j pa sa) a -> Dimensional v ('Dim l m t i th n j 'Zero 'Zero) a
+removeAngles :: Dimensional v ('Dim l m t i th n j pa sa) a -> Dimensional v (SIDim l m t i th n j) a
 removeAngles = coerceAngles
 
 -- | Equates values whose dimensions differ in plane or solid angles by equating
@@ -633,7 +633,7 @@ baseUnit = Dimensional 1
 --
 -- `baseUnit` is similar but includes the radians and steradians associated
 -- with plane or solid angles.
-siUnit :: Num a => Unit ('Dim l m t i th n j 'Zero 'Zero) a
+siUnit :: Num a => Unit (SIDim l m t i th n j) a
 siUnit = removeAngles baseUnit
 
 {-
diff --git a/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs b/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs
index 25ab7038..e7441e98 100644
--- a/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs
+++ b/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs
@@ -31,6 +31,8 @@ module Numeric.Units.Dimensional.DK.Dimensions.TypeLevel
   -- * Synonyms for Base Dimensions
   DOne,
   DLength, DMass, DTime, DElectricCurrent, DThermodynamicTemperature, DAmountOfSubstance, DLuminousIntensity, DPlaneAngle, DSolidAngle,
+  -- * Conversion to SI Basis
+  type SIDim, type ToSIDim,
   -- * Conversion to Term Level
   type KnownDimension
 )
@@ -125,6 +127,24 @@ type family Root (d::Dimension) (x::TypeInt) where
   Root ('Dim l  m  t  i  th  n  j pa sa) x
     = 'Dim (l N./ x) (m N./ x) (t N./ x) (i N./ x) (th N./ x) (n N./ x) (j N./ x) (pa N./ x) (sa N./ x)
 
+-- | Represents a physical dimension in the basis of the 7 SI base dimensions, 
+-- where the respective dimensions are represented by type variables
+-- using the following convention.
+--
+--  * l: Length
+--  * m: Mass
+--  * t: Time
+--  * i: Electric current
+--  * th: Thermodynamic temperature
+--  * n: Amount of substance
+--  * j: Luminous intensity
+type SIDim l m t i th n j = 'Dim l m t i th n j 'Zero 'Zero
+
+-- | Projects a 'Dimension' in the extended basis including plane and solid angles to
+-- the SI basis.
+type family ToSIDim (d :: Dimension) :: Dimension where
+  ToSIDim ('Dim l m t i th n j pa sa) = SIDim l m t i th n j
+
 -- | A KnownDimension is one for which we can construct a term-level representation.
 -- Each validly constructed type of kind 'Dimension' has a 'KnownDimension' instance.
 --
diff --git a/src/Numeric/Units/Dimensional/DK/SI.hs b/src/Numeric/Units/Dimensional/DK/SI.hs
index ba62fd52..5de24901 100644
--- a/src/Numeric/Units/Dimensional/DK/SI.hs
+++ b/src/Numeric/Units/Dimensional/DK/SI.hs
@@ -27,9 +27,6 @@ where
 
 import qualified Numeric.Units.Dimensional.DK as A
 import Numeric.Units.Dimensional.DK hiding (sin, cos, tan, asin, acos, atan, atan2, DPlaneAngle, DSolidAngle, PlaneAngle, SolidAngle, baseUnit)
-import Numeric.NumType.DK.Integers
-  ( TypeInt (Zero)
-  )
 import Prelude hiding (sin, cos, tan, asin, acos, atan, atan2)
 
 sin, cos, tan, asin, acos, atan :: (Floating a) => Dimensionless a -> Dimensionless a
@@ -49,5 +46,5 @@ type DSolidAngle = DOne
 type PlaneAngle = Quantity DPlaneAngle
 type SolidAngle = Quantity DSolidAngle
 
-baseUnit :: (Num a) => Unit ('Dim l m t i th n j 'Zero 'Zero) a
+baseUnit :: (Num a) => Unit (SIDim l m t i th n j) a
 baseUnit = siUnit
diff --git a/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs b/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
index 70d884a4..68aebe30 100644
--- a/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
+++ b/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
@@ -30,33 +30,27 @@ import Numeric.Units.Dimensional.DK.SI
 import qualified Numeric.Units.Dimensional.DK.Quantities as A
 import Numeric.Units.Dimensional.DK.Quantities hiding (LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque,
                                                        DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque)
-import Numeric.NumType.DK.Integers
-  ( TypeInt (Zero)
-  )
 
-type family RemoveAngles (d :: Dimension) :: Dimension where
-  RemoveAngles ('Dim l m t i th n j pa sa) = 'Dim l m t i th n j 'Zero 'Zero
-
-type DLuminousFlux = RemoveAngles A.DLuminousFlux
+type DLuminousFlux = ToSIDim A.DLuminousFlux
 type LuminousFlux = Quantity DLuminousFlux
 
-type DIlluminance = RemoveAngles A.DIlluminance
+type DIlluminance = ToSIDim A.DIlluminance
 type Illuminance = Quantity DIlluminance
 
-type DAngularVelocity = RemoveAngles A.DAngularVelocity
+type DAngularVelocity = ToSIDim A.DAngularVelocity
 type AngularVelocity = Quantity DAngularVelocity
 
-type DAngularAcceleration = RemoveAngles A.DAngularAcceleration
+type DAngularAcceleration = ToSIDim A.DAngularAcceleration
 type AngularAcceleration = Quantity DAngularAcceleration
 
-type DRadiantIntensity = RemoveAngles A.DRadiantIntensity
+type DRadiantIntensity = ToSIDim A.DRadiantIntensity
 type RadiantIntensity = Quantity DRadiantIntensity
 
-type DRadiance = RemoveAngles A.DRadiance
+type DRadiance = ToSIDim A.DRadiance
 type Radiance = Quantity DRadiance
 
-type DAngularMomentum = RemoveAngles A.DAngularMomentum
+type DAngularMomentum = ToSIDim A.DAngularMomentum
 type AngularMomentum = Quantity DAngularMomentum
 
-type DTorque = RemoveAngles A.DTorque
+type DTorque = ToSIDim A.DTorque
 type Torque = Quantity DTorque
diff --git a/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs b/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs
index 4711a62d..3bab7ba6 100644
--- a/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs
+++ b/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs
@@ -27,11 +27,11 @@ import Numeric.Units.Dimensional.DK.SI.Quantities
 import qualified Numeric.Units.Dimensional.DK.SIUnits as A
 import Numeric.Units.Dimensional.DK.SIUnits hiding (radian, steradian, degree, arcminute, arcsecond, degreeOfArc, minuteOfArc, secondOfArc, lumen, lux)
 
-radian, steradian :: (Num a) => Unit DOne a
+radian, steradian :: (Num a) => Unit DPlaneAngle a
 radian = removeAngles A.radian
 steradian = removeAngles A.steradian
 
-degree, arcminute, arcsecond, degreeOfArc, minuteOfArc, secondOfArc :: (Floating a) => Unit DOne a
+degree, arcminute, arcsecond, degreeOfArc, minuteOfArc, secondOfArc :: (Floating a) => Unit DPlaneAngle a
 degree = removeAngles A.degree
 arcminute = removeAngles A.arcminute
 arcsecond = removeAngles A.arcsecond

From 42b5259fa60df037c7e2c02b451263f29c195cec Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Fri, 18 Sep 2015 22:29:50 -0400
Subject: [PATCH 09/12] Corrected dimensions of angular momentum and moment of
 inertia.

---
 src/Numeric/Units/Dimensional/DK/Quantities.hs | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/Numeric/Units/Dimensional/DK/Quantities.hs b/src/Numeric/Units/Dimensional/DK/Quantities.hs
index eab92406..b2e0ca49 100644
--- a/src/Numeric/Units/Dimensional/DK/Quantities.hs
+++ b/src/Numeric/Units/Dimensional/DK/Quantities.hs
@@ -345,10 +345,10 @@ type KinematicViscosity  = Quantity DKinematicViscosity
 type DFirstMassMoment = 'Dim 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type FirstMassMoment = Quantity DFirstMassMoment
 
-type DMomentOfInertia = 'Dim 'Pos2 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DMomentOfInertia = 'Dim 'Pos2 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Neg2 'Zero
 type MomentOfInertia = Quantity DMomentOfInertia
 
-type DAngularMomentum = 'Dim 'Pos2 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
+type DAngularMomentum = 'Dim 'Pos2 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Neg1 'Zero
 type AngularMomentum = Quantity DAngularMomentum
 
 {-

From 650d8b5f15a5b40d798847e7f4cdff6b6f6e9d7e Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Sat, 19 Sep 2015 12:10:20 -0400
Subject: [PATCH 10/12] Added SI definition of DMomentOfInertia, mistakenly
 omitted from the previous commit.

---
 src/Numeric/Units/Dimensional/DK/SI/Quantities.hs | 11 +++++++----
 1 file changed, 7 insertions(+), 4 deletions(-)

diff --git a/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs b/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
index 68aebe30..f61f6e3e 100644
--- a/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
+++ b/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
@@ -21,15 +21,15 @@ module Numeric.Units.Dimensional.DK.SI.Quantities
 (
   module Numeric.Units.Dimensional.DK.Quantities,
 
-  LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque,
-  DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque
+  LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque, MomentOfInertia,
+  DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque, DMomentOfInertia
 )
 where
 
 import Numeric.Units.Dimensional.DK.SI
 import qualified Numeric.Units.Dimensional.DK.Quantities as A
-import Numeric.Units.Dimensional.DK.Quantities hiding (LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque,
-                                                       DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque)
+import Numeric.Units.Dimensional.DK.Quantities hiding (LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque, MomentOfInertia,
+                                                       DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque, DMomentOfInertia)
 
 type DLuminousFlux = ToSIDim A.DLuminousFlux
 type LuminousFlux = Quantity DLuminousFlux
@@ -54,3 +54,6 @@ type AngularMomentum = Quantity DAngularMomentum
 
 type DTorque = ToSIDim A.DTorque
 type Torque = Quantity DTorque
+
+type DMomentOfInertia = ToSIDim A.DMomentOfInertia
+type MomentOfInertia = Quantity DMomentOfInertia

From 48fd3746e752f7e94d3f670f320ee8cf7763c78d Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Wed, 21 Oct 2015 10:39:25 -0400
Subject: [PATCH 11/12] Treat solid angle as the square of plane angle.

---
 src/Numeric/Units/Dimensional/DK.hs           |  25 ++--
 .../Dimensional/DK/Dimensions/TermLevel.hs    |  41 +++---
 .../Dimensional/DK/Dimensions/TypeLevel.hs    |  55 ++++----
 .../Units/Dimensional/DK/Quantities.hs        | 131 +++++++++---------
 src/Numeric/Units/Dimensional/DK/SI.hs        |  10 +-
 .../Units/Dimensional/DK/SI/Prelude.hs        |   5 +-
 .../Units/Dimensional/DK/SI/Quantities.hs     |  13 +-
 .../Units/Dimensional/DK/SI/SIUnits.hs        |   2 +-
 8 files changed, 140 insertions(+), 142 deletions(-)

diff --git a/src/Numeric/Units/Dimensional/DK.hs b/src/Numeric/Units/Dimensional/DK.hs
index 7ceff6de..108fa88c 100644
--- a/src/Numeric/Units/Dimensional/DK.hs
+++ b/src/Numeric/Units/Dimensional/DK.hs
@@ -216,10 +216,10 @@ module Numeric.Units.Dimensional.DK
     (*~~), (/~~), sum, mean, dimensionlessLength, nFromTo,
     -- * Dimension Synonyms
     -- $dimension-synonyms
-    DOne, DLength, DMass, DTime, DElectricCurrent, DThermodynamicTemperature, DAmountOfSubstance, DLuminousIntensity, DPlaneAngle, DSolidAngle,
+    DOne, DLength, DMass, DTime, DElectricCurrent, DThermodynamicTemperature, DAmountOfSubstance, DLuminousIntensity, DPlaneAngle,
     -- * Quantity Synonyms
     -- $quantity-synonyms
-    Dimensionless, Length, Mass, Time, ElectricCurrent, ThermodynamicTemperature, AmountOfSubstance, LuminousIntensity, PlaneAngle, SolidAngle,
+    Dimensionless, Length, Mass, Time, ElectricCurrent, ThermodynamicTemperature, AmountOfSubstance, LuminousIntensity, PlaneAngle,
     -- * Constants
     -- $constants
     _0, _1, _2, _3, _4, _5, _6, _7, _8, _9, pi, tau,
@@ -388,7 +388,6 @@ type ThermodynamicTemperature = Quantity DThermodynamicTemperature
 type AmountOfSubstance        = Quantity DAmountOfSubstance
 type LuminousIntensity        = Quantity DLuminousIntensity
 type PlaneAngle               = Quantity DPlaneAngle
-type SolidAngle               = Quantity DSolidAngle
 
 {- $dimension-arithmetic
 When performing arithmetic on units and quantities the arithmetics
@@ -607,16 +606,16 @@ acos  = (*~ baseUnit) . Prelude.acos . (/~ one)
 atan  = (*~ baseUnit) . Prelude.atan . (/~ one)
 
 -- | Removes angular dimensions from a dimensional value by equating radians
--- and steradians with the dimensionless quantity one.
-removeAngles :: Dimensional v ('Dim l m t i th n j pa sa) a -> Dimensional v (SIDim l m t i th n j) a
+-- with the dimensionless quantity one.
+removeAngles :: Dimensional v ('Dim l m t i th n j pa) a -> Dimensional v (SIDim l m t i th n j) a
 removeAngles = coerceAngles
 
--- | Equates values whose dimensions differ in plane or solid angles by equating
--- radians and steradians with the dimensionless quantity one.
+-- | Equates values whose dimensions differ in plane angles by equating
+-- radians with the dimensionless quantity one.
 --
 -- See `removeAngles`, which offers a more specific result type, if you are only interested
--- in ignoring plane and solid angle dimensions.
-coerceAngles :: Dimensional v ('Dim l m t i th n j pa sa) a -> Dimensional v ('Dim l m t i th n j pa' sa') a
+-- in ignoring angle dimensions.
+coerceAngles :: Dimensional v ('Dim l m t i th n j pa) a -> Dimensional v ('Dim l m t i th n j pa') a
 coerceAngles = coerce
 
 (**) :: Floating a => Dimensionless a -> Dimensionless a -> Dimensionless a
@@ -634,8 +633,8 @@ atan2 (Dimensional y) (Dimensional x) = Dimensional (Prelude.atan2 y x)
 -- base unit of any dimension. This allows polymorphic quantity
 -- creation and destruction without exposing the 'Dimensional' constructor.
 --
--- `siUnit` is similar but does not include the radians or steradians associated
--- with plane or solid angles.
+-- `siUnit` is similar but does not include the radians associated
+-- with plane angles.
 baseUnit :: Num a => Unit d a
 baseUnit = Dimensional 1
 
@@ -643,8 +642,8 @@ baseUnit = Dimensional 1
 -- SI base unit of any dimension. This allows polymorphic quantity
 -- creation and destruction without exposing the 'Dimensional' constructor.
 --
--- `baseUnit` is similar but includes the radians and steradians associated
--- with plane or solid angles.
+-- `baseUnit` is similar but includes the radians associated
+-- with plane angles.
 siUnit :: Num a => Unit (SIDim l m t i th n j) a
 siUnit = removeAngles baseUnit
 
diff --git a/src/Numeric/Units/Dimensional/DK/Dimensions/TermLevel.hs b/src/Numeric/Units/Dimensional/DK/Dimensions/TermLevel.hs
index 9608ebea..0bfa1298 100644
--- a/src/Numeric/Units/Dimensional/DK/Dimensions/TermLevel.hs
+++ b/src/Numeric/Units/Dimensional/DK/Dimensions/TermLevel.hs
@@ -9,7 +9,7 @@
    Portability: GHC only
 
 This module defines physical dimensions expressed in terms of
-the SI base dimensions along with plane and solid angles, including arithmetic.
+the SI base dimensions along with plane angles, including arithmetic.
 
 -}
 module Numeric.Units.Dimensional.DK.Dimensions.TermLevel
@@ -22,7 +22,7 @@ module Numeric.Units.Dimensional.DK.Dimensions.TermLevel
   (*), (/), (^), recip,
   -- * Synonyms for Base Dimensions
   dOne,
-  dLength, dMass, dTime, dElectricCurrent, dThermodynamicTemperature, dAmountOfSubstance, dLuminousIntensity, dPlaneAngle, dSolidAngle,
+  dLength, dMass, dTime, dElectricCurrent, dThermodynamicTemperature, dAmountOfSubstance, dLuminousIntensity, dPlaneAngle,
   -- * Deconstruction
   asList
 )
@@ -32,11 +32,11 @@ import Data.Monoid (Monoid(..))
 import Prelude (id, (+), (-), Int, Show, Eq, Ord)
 import qualified Prelude as P
 
--- | A physical dimension, encoded as 9 integers, representing a factorization of the dimension into the
--- 7 SI base dimensions, along with the dimensions of plane and solid angles.
+-- | A physical dimension, encoded as 8 integers, representing a factorization of the dimension into the
+-- 7 SI base dimensions, along with the dimensions of plane angle.
 -- By convention they are stored in the same order as 
 -- in the 'Numeric.Units.Dimensional.DK.Dimensions.TypeLevel.Dimension' data kind.
-data Dimension' = Dim' !Int !Int !Int !Int !Int !Int !Int !Int !Int
+data Dimension' = Dim' !Int !Int !Int !Int !Int !Int !Int !Int
   deriving (Show, Eq, Ord)
 
 -- | The monoid of dimensions under multiplication.
@@ -54,18 +54,17 @@ instance HasDimension Dimension' where
 
 -- | The dimension of dimensionless values.
 dOne :: Dimension'
-dOne = Dim' 0 0 0 0 0 0 0 0 0
-
-dLength, dMass, dTime, dElectricCurrent, dThermodynamicTemperature, dAmountOfSubstance, dLuminousIntensity, dPlaneAngle, dSolidAngle :: Dimension'
-dLength                   = Dim' 1 0 0 0 0 0 0 0 0
-dMass                     = Dim' 0 1 0 0 0 0 0 0 0
-dTime                     = Dim' 0 0 1 0 0 0 0 0 0
-dElectricCurrent          = Dim' 0 0 0 1 0 0 0 0 0
-dThermodynamicTemperature = Dim' 0 0 0 0 1 0 0 0 0
-dAmountOfSubstance        = Dim' 0 0 0 0 0 1 0 0 0
-dLuminousIntensity        = Dim' 0 0 0 0 0 0 1 0 0
-dPlaneAngle               = Dim' 0 0 0 0 0 0 0 1 0
-dSolidAngle               = Dim' 0 0 0 0 0 0 0 0 1
+dOne = Dim' 0 0 0 0 0 0 0 0
+
+dLength, dMass, dTime, dElectricCurrent, dThermodynamicTemperature, dAmountOfSubstance, dLuminousIntensity, dPlaneAngle :: Dimension'
+dLength                   = Dim' 1 0 0 0 0 0 0 0
+dMass                     = Dim' 0 1 0 0 0 0 0 0
+dTime                     = Dim' 0 0 1 0 0 0 0 0
+dElectricCurrent          = Dim' 0 0 0 1 0 0 0 0
+dThermodynamicTemperature = Dim' 0 0 0 0 1 0 0 0
+dAmountOfSubstance        = Dim' 0 0 0 0 0 1 0 0
+dLuminousIntensity        = Dim' 0 0 0 0 0 0 1 0
+dPlaneAngle               = Dim' 0 0 0 0 0 0 0 1
 
 {-
 We will reuse the operators and function names from the Prelude.
@@ -78,15 +77,15 @@ infixl 7  *, /
 
 -- | Forms the product of two dimensions.
 (*) :: Dimension' -> Dimension' -> Dimension'
-(Dim' l m t i th n j pa sa) * (Dim' l' m' t' i' th' n' j' pa' sa') = Dim' (l + l') (m + m') (t + t') (i + i') (th + th') (n + n') (j + j') (pa + pa') (sa + sa')
+(Dim' l m t i th n j pa) * (Dim' l' m' t' i' th' n' j' pa') = Dim' (l + l') (m + m') (t + t') (i + i') (th + th') (n + n') (j + j') (pa + pa')
 
 -- | Forms the quotient of two dimensions.
 (/) :: Dimension' -> Dimension' -> Dimension'
-(Dim' l m t i th n j pa sa) / (Dim' l' m' t' i' th' n' j' pa' sa') = Dim' (l - l') (m - m') (t - t') (i - i') (th - th') (n - n') (j - j') (pa - pa') (sa - sa')
+(Dim' l m t i th n j pa) / (Dim' l' m' t' i' th' n' j' pa') = Dim' (l - l') (m - m') (t - t') (i - i') (th - th') (n - n') (j - j') (pa - pa')
 
 -- | Raises a dimension to an integer power.
 (^) :: Dimension' -> Int -> Dimension'
-(Dim' l m t i th n j pa sa) ^ x = Dim' (x P.* l) (x P.* m) (x P.* t) (x P.* i) (x P.* th) (x P.* n) (x P.* j) (x P.* pa) (x P.* sa)
+(Dim' l m t i th n j pa) ^ x = Dim' (x P.* l) (x P.* m) (x P.* t) (x P.* i) (x P.* th) (x P.* n) (x P.* j) (x P.* pa)
 
 -- | Forms the reciprocal of a dimension.
 recip :: Dimension' -> Dimension'
@@ -95,4 +94,4 @@ recip = (dOne /)
 -- | Converts a dimension to a list of 9 integers, representing the exponent associated with each
 -- of the 7 SI base dimensions in the standard order, followed by the dimensions of plane and spherical angles.
 asList :: Dimension' -> [Int]
-asList (Dim' l m t i th n j pa sa) = [l, m, t, i, th, n, j, pa, sa]
+asList (Dim' l m t i th n j pa) = [l, m, t, i, th, n, j, pa]
diff --git a/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs b/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs
index e7441e98..0a6684b4 100644
--- a/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs
+++ b/src/Numeric/Units/Dimensional/DK/Dimensions/TypeLevel.hs
@@ -18,7 +18,7 @@
    Portability: GHC only
 
 This module defines type-level physical dimensions expressed in terms of
-the SI base dimensions along with plane and solid angles using 'Numeric.NumType.DK.NumType' for type-level integers.
+the SI base dimensions along with plane angles using 'Numeric.NumType.DK.NumType' for type-level integers.
 
 Type-level arithmetic, synonyms for the base dimensions, and conversion to the term-level are included.
 -}
@@ -30,7 +30,7 @@ module Numeric.Units.Dimensional.DK.Dimensions.TypeLevel
   type (*), type (/), type (^), type Recip, type Root,
   -- * Synonyms for Base Dimensions
   DOne,
-  DLength, DMass, DTime, DElectricCurrent, DThermodynamicTemperature, DAmountOfSubstance, DLuminousIntensity, DPlaneAngle, DSolidAngle,
+  DLength, DMass, DTime, DElectricCurrent, DThermodynamicTemperature, DAmountOfSubstance, DLuminousIntensity, DPlaneAngle,
   -- * Conversion to SI Basis
   type SIDim, type ToSIDim,
   -- * Conversion to Term Level
@@ -47,7 +47,7 @@ import qualified Numeric.NumType.DK.Integers as N
 import Numeric.Units.Dimensional.DK.Dimensions.TermLevel
 
 -- | Represents a physical dimension in the basis of the 7 SI base dimensions, 
--- along with the dimensions of plane and solid angles,
+-- along with the dimensions of plane angle,
 -- where the respective dimensions are represented by type variables
 -- using the following convention.
 --
@@ -59,22 +59,20 @@ import Numeric.Units.Dimensional.DK.Dimensions.TermLevel
 --  * n: Amount of substance
 --  * j: Luminous intensity
 --  * pa: Plane angle
---  * sa: Solid angle
 --
 -- For the equivalent term-level representation, see 'Dimension''
-data Dimension = Dim TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt
+data Dimension = Dim TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt TypeInt
 
 -- | The type-level dimensions of dimensionless values.
-type DOne                      = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
-type DLength                   = 'Dim 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
-type DMass                     = 'Dim 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
-type DTime                     = 'Dim 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
-type DElectricCurrent          = 'Dim 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
-type DThermodynamicTemperature = 'Dim 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
-type DAmountOfSubstance        = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero
-type DLuminousIntensity        = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero
-type DPlaneAngle               = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
-type DSolidAngle               = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1
+type DOne                      = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DLength                   = 'Dim 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DMass                     = 'Dim 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DTime                     = 'Dim 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DElectricCurrent          = 'Dim 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
+type DThermodynamicTemperature = 'Dim 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero
+type DAmountOfSubstance        = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero
+type DLuminousIntensity        = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
+type DPlaneAngle               = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1
 
 {-
 We will reuse the operators and function names from the Prelude.
@@ -90,16 +88,16 @@ infixl 7  *, /
 type family (a::Dimension) * (b::Dimension) where
   DOne * d = d
   d * DOne = d
-  ('Dim l  m  t  i  th  n  j pa sa) * ('Dim l' m' t' i' th' n' j' pa' sa')
-    = 'Dim (l + l') (m + m') (t + t') (i + i') (th + th') (n + n') (j + j') (pa + pa') (sa + sa')
+  ('Dim l  m  t  i  th  n  j pa) * ('Dim l' m' t' i' th' n' j' pa')
+    = 'Dim (l + l') (m + m') (t + t') (i + i') (th + th') (n + n') (j + j') (pa + pa')
 
 -- | Division of dimensions corresponds to subtraction of the base
 -- dimensions' exponents.
 type family (a::Dimension) / (d::Dimension) where
   d / DOne = d
   d / d = DOne
-  ('Dim l  m  t  i  th  n  j pa sa) / ('Dim l' m' t' i' th' n' j' pa' sa')
-    = 'Dim (l - l') (m - m') (t - t') (i - i') (th - th') (n - n') (j - j') (pa - pa') (sa - sa')
+  ('Dim l  m  t  i  th  n  j pa) / ('Dim l' m' t' i' th' n' j' pa')
+    = 'Dim (l - l') (m - m') (t - t') (i - i') (th - th') (n - n') (j - j') (pa - pa')
 
 -- | The reciprocal of a dimension is defined as the result of dividing 'DOne' by it,
 -- or of negating each of the base dimensions' exponents.
@@ -114,8 +112,8 @@ type family (d::Dimension) ^ (x::TypeInt) where
   DOne ^ x = DOne
   d ^ 'Zero = DOne
   d ^ 'Pos1 = d
-  ('Dim l  m  t  i  th  n  j pa sa) ^ x
-    = 'Dim (l N.* x) (m N.* x) (t N.* x) (i N.* x) (th N.* x) (n N.* x) (j N.* x) (pa N.* x) (sa N.* x)
+  ('Dim l  m  t  i  th  n  j pa) ^ x
+    = 'Dim (l N.* x) (m N.* x) (t N.* x) (i N.* x) (th N.* x) (n N.* x) (j N.* x) (pa N.* x)
 
 -- | Roots of dimensions corresponds to division of the base dimensions'
 -- exponents by the order(?) of the root.
@@ -124,8 +122,8 @@ type family (d::Dimension) ^ (x::TypeInt) where
 type family Root (d::Dimension) (x::TypeInt) where
   Root DOne x = DOne
   Root d 'Pos1 = d
-  Root ('Dim l  m  t  i  th  n  j pa sa) x
-    = 'Dim (l N./ x) (m N./ x) (t N./ x) (i N./ x) (th N./ x) (n N./ x) (j N./ x) (pa N./ x) (sa N./ x)
+  Root ('Dim l  m  t  i  th  n  j pa) x
+    = 'Dim (l N./ x) (m N./ x) (t N./ x) (i N./ x) (th N./ x) (n N./ x) (j N./ x) (pa N./ x)
 
 -- | Represents a physical dimension in the basis of the 7 SI base dimensions, 
 -- where the respective dimensions are represented by type variables
@@ -138,12 +136,11 @@ type family Root (d::Dimension) (x::TypeInt) where
 --  * th: Thermodynamic temperature
 --  * n: Amount of substance
 --  * j: Luminous intensity
-type SIDim l m t i th n j = 'Dim l m t i th n j 'Zero 'Zero
+type SIDim l m t i th n j = 'Dim l m t i th n j 'Zero
 
--- | Projects a 'Dimension' in the extended basis including plane and solid angles to
--- the SI basis.
+-- | Projects a 'Dimension' in the extended basis including plane angles to the SI basis.
 type family ToSIDim (d :: Dimension) :: Dimension where
-  ToSIDim ('Dim l m t i th n j pa sa) = SIDim l m t i th n j
+  ToSIDim ('Dim l m t i th n j pa) = SIDim l m t i th n j
 
 -- | A KnownDimension is one for which we can construct a term-level representation.
 -- Each validly constructed type of kind 'Dimension' has a 'KnownDimension' instance.
@@ -160,8 +157,7 @@ instance ( KnownTypeInt l
          , KnownTypeInt n
          , KnownTypeInt j
          , KnownTypeInt pa
-         , KnownTypeInt sa
-         ) => HasDimension (Proxy ('Dim l m t i th n j pa sa))
+         ) => HasDimension (Proxy ('Dim l m t i th n j pa))
   where 
     dimension _ = Dim'
                 (toNum (Proxy :: Proxy l))
@@ -172,4 +168,3 @@ instance ( KnownTypeInt l
                 (toNum (Proxy :: Proxy n))
                 (toNum (Proxy :: Proxy j))
                 (toNum (Proxy :: Proxy pa))
-                (toNum (Proxy :: Proxy sa))
diff --git a/src/Numeric/Units/Dimensional/DK/Quantities.hs b/src/Numeric/Units/Dimensional/DK/Quantities.hs
index b2e0ca49..37751cbe 100644
--- a/src/Numeric/Units/Dimensional/DK/Quantities.hs
+++ b/src/Numeric/Units/Dimensional/DK/Quantities.hs
@@ -49,7 +49,7 @@ module Numeric.Units.Dimensional.DK.Quantities
   -- $not-nist-guide
   Impulse, MassFlow, VolumeFlow, GravitationalParameter, KinematicViscosity, FirstMassMoment, MomentOfInertia, AngularMomentum,
   ThermalResistivity, ThermalConductance, ThermalResistance, HeatTransferCoefficient, ThermalAdmittance, ThermalInsulance,
-  Jerk, Angle, Thrust, Torque, EnergyPerUnitMass,
+  Jerk, Angle, Thrust, Torque, EnergyPerUnitMass, SolidAngle,
   -- * Powers of Unit Lengths
   -- $powers-of-length-units
   square, cubic,
@@ -68,7 +68,7 @@ module Numeric.Units.Dimensional.DK.Quantities
   DMolarEnergy, DMolarEntropy, DMolarHeatCapacity, DExposure, DAbsorbedDoseRate,
   DImpulse, DMassFlow, DVolumeFlow, DGravitationalParameter, DKinematicViscosity, DFirstMassMoment, DMomentOfInertia, DAngularMomentum,
   DThermalResistivity, DThermalConductance, DThermalResistance, DHeatTransferCoefficient, DThermalAdmittance, DThermalInsulance,
-  DJerk, DAngle, DThrust, DTorque, DEnergyPerUnitMass
+  DJerk, DAngle, DThrust, DTorque, DEnergyPerUnitMass, DSolidAngle
 )
 where
 
@@ -104,41 +104,41 @@ For each 'Quantity' alias supplied above, we also supply a corresponding 'Dimens
 These dimension aliases may be convenient for supplying type signatures for 'Unit's or for other type-level dimensional programming.
 -}
 
-type DArea = 'Dim 'Pos2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DArea = 'Dim 'Pos2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type Area  = Quantity DArea
 
-type DVolume = 'Dim 'Pos3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DVolume = 'Dim 'Pos3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type Volume  = Quantity DVolume
 
-type DVelocity = 'Dim 'Pos1 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DVelocity = 'Dim 'Pos1 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type Velocity  = Quantity DVelocity
 
-type DAcceleration = 'Dim 'Pos1 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DAcceleration = 'Dim 'Pos1 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
 type Acceleration  = Quantity DAcceleration
 
-type DWaveNumber = 'Dim 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DWaveNumber = 'Dim 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type WaveNumber  = Quantity DWaveNumber
 
-type DMassDensity = 'Dim 'Neg3 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DMassDensity = 'Dim 'Neg3 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type DDensity     = DMassDensity
 type MassDensity  = Quantity DMassDensity
 type Density      = MassDensity -- Short name.
 
-type DSpecificVolume = 'Dim 'Pos3 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DSpecificVolume = 'Dim 'Pos3 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type SpecificVolume  = Quantity DSpecificVolume
 
-type DCurrentDensity = 'Dim 'Neg2 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DCurrentDensity = 'Dim 'Neg2 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
 type CurrentDensity  = Quantity DCurrentDensity
 
-type DMagneticFieldStrength = 'Dim 'Neg1 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DMagneticFieldStrength = 'Dim 'Neg1 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
 type MagneticFieldStrength  = Quantity DMagneticFieldStrength
 
-type DAmountOfSubstanceConcentration = 'Dim 'Neg3 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero
+type DAmountOfSubstanceConcentration = 'Dim 'Neg3 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero
 type DConcentration                  = DAmountOfSubstanceConcentration
 type AmountOfSubstanceConcentration  = Quantity DAmountOfSubstanceConcentration
 type Concentration                   = AmountOfSubstanceConcentration -- Short name.
 
-type DLuminance = 'Dim 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Zero
+type DLuminance = 'Dim 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
 type Luminance  = Quantity DLuminance
 
 
@@ -149,63 +149,63 @@ SI coherent derived units with special names and symbols.
 
 -}
 
-type DFrequency = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DFrequency = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type Frequency  = Quantity DFrequency
 
-type DForce = 'Dim 'Pos1 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DForce = 'Dim 'Pos1 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
 type Force  = Quantity DForce
 
-type DPressure = 'Dim 'Neg1 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DPressure = 'Dim 'Neg1 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
 type DStress   = DPressure
 type Pressure  = Quantity DPressure
 type Stress    = Quantity DStress
 
-type DEnergy         = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DEnergy         = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
 type DWork           = DEnergy
 type DQuantityOfHeat = DEnergy
 type Energy          = Quantity DEnergy
 type Work            = Quantity DWork
 type QuantityOfHeat  = Quantity DQuantityOfHeat
 
-type DPower       = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DPower       = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero
 type DRadiantFlux = DPower
 type Power        = Quantity DPower
 type RadiantFlux  = Quantity DRadiantFlux
 
-type DElectricCharge        = 'Dim 'Zero 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DElectricCharge        = 'Dim 'Zero 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero
 type DQuantityOfElectricity = DElectricCharge
 type ElectricCharge         = Quantity DElectricCharge
 type QuantityOfElectricity  = Quantity DQuantityOfElectricity
 
-type DElectricPotential   = 'Dim 'Pos2 'Pos1 'Neg3 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DElectricPotential   = 'Dim 'Pos2 'Pos1 'Neg3 'Neg1 'Zero 'Zero 'Zero 'Zero
 type DPotentialDifference = DElectricPotential
 type DElectromotiveForce  = DElectricPotential
 type ElectricPotential    = Quantity DElectricPotential
 type PotentialDifference  = Quantity DPotentialDifference
 type ElectromotiveForce   = Quantity DElectromotiveForce
 
-type DCapacitance = 'Dim 'Neg2 'Neg1 'Pos4 'Pos2 'Zero 'Zero 'Zero 'Zero 'Zero
+type DCapacitance = 'Dim 'Neg2 'Neg1 'Pos4 'Pos2 'Zero 'Zero 'Zero 'Zero
 type Capacitance  = Quantity DCapacitance
 
-type DElectricResistance = 'Dim 'Pos2 'Pos1 'Neg3 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
+type DElectricResistance = 'Dim 'Pos2 'Pos1 'Neg3 'Neg2 'Zero 'Zero 'Zero 'Zero
 type ElectricResistance  = Quantity DElectricResistance
 
-type DElectricConductance = 'Dim 'Neg2 'Neg1 'Pos3 'Pos2 'Zero 'Zero 'Zero 'Zero 'Zero
+type DElectricConductance = 'Dim 'Neg2 'Neg1 'Pos3 'Pos2 'Zero 'Zero 'Zero 'Zero
 type ElectricConductance  = Quantity DElectricConductance
 
-type DMagneticFlux = 'Dim 'Pos2 'Pos1 'Neg2 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DMagneticFlux = 'Dim 'Pos2 'Pos1 'Neg2 'Neg1 'Zero 'Zero 'Zero 'Zero
 type MagneticFlux  = Quantity DMagneticFlux
 
-type DMagneticFluxDensity = 'Dim 'Zero 'Pos1 'Neg2 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DMagneticFluxDensity = 'Dim 'Zero 'Pos1 'Neg2 'Neg1 'Zero 'Zero 'Zero 'Zero
 type MagneticFluxDensity  = Quantity DMagneticFluxDensity
 
-type DInductance = 'Dim 'Pos2 'Pos1 'Neg2 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
+type DInductance = 'Dim 'Pos2 'Pos1 'Neg2 'Neg2 'Zero 'Zero 'Zero 'Zero
 type Inductance  = Quantity DInductance
 
-type DLuminousFlux = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Pos1
+type DLuminousFlux = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Pos2
 type LuminousFlux  = Quantity DLuminousFlux
 
-type DIlluminance = 'Dim 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero 'Pos1
+type DIlluminance = 'Dim 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos1 'Pos2
 type Illuminance  = Quantity DIlluminance
 
 type DCelsiusTemperature = DThermodynamicTemperature
@@ -214,7 +214,7 @@ type CelsiusTemperature  = Quantity DCelsiusTemperature
 type DActivity = DFrequency -- Activity of a radionuclide.
 type Activity  = Quantity DActivity
 
-type DAbsorbedDose   = 'Dim 'Pos2 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DAbsorbedDose   = 'Dim 'Pos2 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
 type DSpecificEnergy = DAbsorbedDose
 type DKerma          = DAbsorbedDose
 type AbsorbedDose    = Quantity DAbsorbedDose
@@ -232,7 +232,7 @@ type DirectionalDoseEquivalent  = DoseEquivalent
 type PersonalDoseEquivalent     = DoseEquivalent
 type EquivalentDose             = DoseEquivalent
 
-type DCatalyticActivity = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Pos1 'Zero 'Zero 'Zero
+type DCatalyticActivity = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Pos1 'Zero 'Zero
 type CatalyticActivity  = Quantity DCatalyticActivity
 
 {- $table4
@@ -245,38 +245,38 @@ We use the same grouping as for table 2.
 
 -}
 
-type DAngularVelocity = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
+type DAngularVelocity = 'Dim 'Zero 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Pos1
 type AngularVelocity  = Quantity DAngularVelocity
 
-type DAngularAcceleration = 'Dim 'Zero 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Pos1 'Zero
+type DAngularAcceleration = 'Dim 'Zero 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Pos1
 type AngularAcceleration  = Quantity DAngularAcceleration
 
-type DDynamicViscosity = 'Dim 'Neg1 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DDynamicViscosity = 'Dim 'Neg1 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type DynamicViscosity  = Quantity DDynamicViscosity
 
 type DMomentOfForce = DEnergy
 type MomentOfForce  = Quantity DMomentOfForce
 
-type DSurfaceTension = 'Dim 'Zero 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DSurfaceTension = 'Dim 'Zero 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
 type SurfaceTension  = Quantity DSurfaceTension
 
-type DHeatFluxDensity = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DHeatFluxDensity = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero
 type DIrradiance      = DHeatFluxDensity
 type HeatFluxDensity  = Quantity DHeatFluxDensity
 type Irradiance       = Quantity DIrradiance
 
-type DRadiantIntensity = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Neg1
+type DRadiantIntensity = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Neg2
 type RadiantIntensity  = Quantity DRadiantIntensity
 
-type DRadiance = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Neg1
+type DRadiance = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Zero 'Zero 'Zero 'Neg2
 type Radiance  = Quantity DRadiance
 
-type DHeatCapacity = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DHeatCapacity = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Neg1 'Zero 'Zero 'Zero
 type DEntropy      = DHeatCapacity
 type HeatCapacity  = Quantity DHeatCapacity
 type Entropy       = Quantity DEntropy
 
-type DSpecificHeatCapacity = 'Dim 'Pos2 'Zero 'Neg2 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DSpecificHeatCapacity = 'Dim 'Pos2 'Zero 'Neg2 'Zero 'Neg1 'Zero 'Zero 'Zero
 type DSpecificEntropy      = DSpecificHeatCapacity
 type SpecificHeatCapacity  = Quantity DSpecificHeatCapacity
 type SpecificEntropy       = Quantity DSpecificEntropy
@@ -287,39 +287,39 @@ Specific energy was already defined in table 3.
 
 -}
 
-type DThermalConductivity = 'Dim 'Pos1 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DThermalConductivity = 'Dim 'Pos1 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero 'Zero
 type ThermalConductivity  = Quantity DThermalConductivity
 
 type DEnergyDensity = DPressure
 type EnergyDensity  = Quantity DEnergyDensity
 
-type DElectricFieldStrength = 'Dim 'Pos1 'Pos1 'Neg3 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DElectricFieldStrength = 'Dim 'Pos1 'Pos1 'Neg3 'Neg1 'Zero 'Zero 'Zero 'Zero
 type ElectricFieldStrength  = Quantity DElectricFieldStrength
 
-type DElectricChargeDensity = 'Dim 'Neg3 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DElectricChargeDensity = 'Dim 'Neg3 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero
 type ElectricChargeDensity  = Quantity DElectricChargeDensity
 
-type DElectricFluxDensity = 'Dim 'Neg2 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DElectricFluxDensity = 'Dim 'Neg2 'Zero 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero
 type ElectricFluxDensity  = Quantity DElectricFluxDensity
 
-type DPermittivity = 'Dim 'Neg3 'Neg1 'Pos4 'Pos2 'Zero 'Zero 'Zero 'Zero 'Zero
+type DPermittivity = 'Dim 'Neg3 'Neg1 'Pos4 'Pos2 'Zero 'Zero 'Zero 'Zero
 type Permittivity  = Quantity DPermittivity
 
-type DPermeability = 'Dim 'Pos1 'Pos1 'Neg2 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
+type DPermeability = 'Dim 'Pos1 'Pos1 'Neg2 'Neg2 'Zero 'Zero 'Zero 'Zero
 type Permeability  = Quantity DPermeability
 
-type DMolarEnergy = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Neg1 'Zero 'Zero 'Zero
+type DMolarEnergy = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Neg1 'Zero 'Zero
 type MolarEnergy  = Quantity DMolarEnergy
 
-type DMolarEntropy      = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Neg1 'Neg1 'Zero 'Zero 'Zero
+type DMolarEntropy      = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Neg1 'Neg1 'Zero 'Zero
 type DMolarHeatCapacity = DMolarEntropy
 type MolarEntropy       = Quantity DMolarEntropy
 type MolarHeatCapacity  = Quantity DMolarHeatCapacity
 
-type DExposure = 'Dim 'Zero 'Neg1 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero
+type DExposure = 'Dim 'Zero 'Neg1 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero
 type Exposure  = Quantity DExposure -- Exposure to x and gamma rays.
 
-type DAbsorbedDoseRate = 'Dim 'Pos2 'Zero 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DAbsorbedDoseRate = 'Dim 'Pos2 'Zero 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero
 type AbsorbedDoseRate  = Quantity DAbsorbedDoseRate
 
 {- $not-nist-guide
@@ -327,28 +327,28 @@ Here we define additional quantities on an as-needed basis. We also
 provide some synonyms that we anticipate will be useful.
 -}
 
-type DImpulse = 'Dim 'Pos1 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DImpulse = 'Dim 'Pos1 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type Impulse  = Quantity DImpulse
 
-type DMassFlow = 'Dim 'Zero 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DMassFlow = 'Dim 'Zero 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type MassFlow  = Quantity DMassFlow
 
-type DVolumeFlow = 'Dim 'Pos3 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DVolumeFlow = 'Dim 'Pos3 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type VolumeFlow = Quantity DVolumeFlow
 
-type DGravitationalParameter = 'Dim 'Pos3 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DGravitationalParameter = 'Dim 'Pos3 'Zero 'Neg2 'Zero 'Zero 'Zero 'Zero 'Zero
 type GravitationalParameter  = Quantity DGravitationalParameter
 
-type DKinematicViscosity = 'Dim 'Pos2 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DKinematicViscosity = 'Dim 'Pos2 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero 'Zero
 type KinematicViscosity  = Quantity DKinematicViscosity
 
-type DFirstMassMoment = 'Dim 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DFirstMassMoment = 'Dim 'Pos1 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
 type FirstMassMoment = Quantity DFirstMassMoment
 
-type DMomentOfInertia = 'Dim 'Pos2 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Neg2 'Zero
+type DMomentOfInertia = 'Dim 'Pos2 'Pos1 'Zero 'Zero 'Zero 'Zero 'Zero 'Neg2
 type MomentOfInertia = Quantity DMomentOfInertia
 
-type DAngularMomentum = 'Dim 'Pos2 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Neg1 'Zero
+type DAngularMomentum = 'Dim 'Pos2 'Pos1 'Neg1 'Zero 'Zero 'Zero 'Zero 'Neg1
 type AngularMomentum = Quantity DAngularMomentum
 
 {-
@@ -357,7 +357,7 @@ The reciprocal of thermal conductivity.
 
 -}
 
-type DThermalResistivity = 'Dim 'Neg1 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
+type DThermalResistivity = 'Dim 'Neg1 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero 'Zero
 type ThermalResistivity = Quantity DThermalResistivity
 
 {-
@@ -366,22 +366,22 @@ Thermal conductance and resistance quantities after http://en.wikipedia.org/wiki
 
 -}
 
-type DThermalConductance = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DThermalConductance = 'Dim 'Pos2 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero 'Zero
 type ThermalConductance = Quantity DThermalConductance
 
-type DThermalResistance = 'Dim 'Neg2 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
+type DThermalResistance = 'Dim 'Neg2 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero 'Zero
 type ThermalResistance = Quantity DThermalResistance
 
-type DHeatTransferCoefficient = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero 'Zero 'Zero
+type DHeatTransferCoefficient = 'Dim 'Zero 'Pos1 'Neg3 'Zero 'Neg1 'Zero 'Zero 'Zero
 type HeatTransferCoefficient = Quantity DHeatTransferCoefficient
 
 type DThermalAdmittance = DHeatTransferCoefficient
 type ThermalAdmittance = HeatTransferCoefficient
 
-type DThermalInsulance = 'Dim 'Zero 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero 'Zero 'Zero
+type DThermalInsulance = 'Dim 'Zero 'Neg1 'Pos3 'Zero 'Pos1 'Zero 'Zero 'Zero
 type ThermalInsulance = Quantity DThermalInsulance
 
-type DJerk = 'Dim 'Pos1 'Zero 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero
+type DJerk = 'Dim 'Pos1 'Zero 'Neg3 'Zero 'Zero 'Zero 'Zero 'Zero
 type Jerk = Quantity DJerk
 
 type DAngle = DPlaneAngle -- Abbreviation
@@ -390,12 +390,15 @@ type Angle = Quantity DAngle -- Abbreviation
 type DThrust = DForce
 type Thrust = Force
 
-type DTorque = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Neg1 'Zero
+type DTorque = 'Dim 'Pos2 'Pos1 'Neg2 'Zero 'Zero 'Zero 'Zero 'Neg1
 type Torque = Quantity DTorque
 
 type DEnergyPerUnitMass = DSpecificEnergy
 type EnergyPerUnitMass = SpecificEnergy
 
+type DSolidAngle = 'Dim 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Zero 'Pos2
+type SolidAngle = Quantity DSolidAngle
+
 {- $powers-of-length-units
 It is permissible to express powers of length units by prefixing
 'square' and 'cubic' (see section 9.6 "Spelling unit names raised
diff --git a/src/Numeric/Units/Dimensional/DK/SI.hs b/src/Numeric/Units/Dimensional/DK/SI.hs
index 5de24901..81ee4af6 100644
--- a/src/Numeric/Units/Dimensional/DK/SI.hs
+++ b/src/Numeric/Units/Dimensional/DK/SI.hs
@@ -10,7 +10,7 @@
 
 = Summary
 
-This module defines a variant of "Numeric.Units.Dimensional.DK" where plane and solid angles are not treated as dimensions.
+This module defines a variant of "Numeric.Units.Dimensional.DK" where plane angles are treated as dimensionless.
 
 Compare, e.g., 'sin' with 'Numeric.Units.Dimensional.DK.sin'
 
@@ -19,14 +19,14 @@ module Numeric.Units.Dimensional.DK.SI
 (
   module Numeric.Units.Dimensional.DK,
   sin, cos, tan, asin, acos, atan, atan2,
-  DPlaneAngle, DSolidAngle,
-  PlaneAngle, SolidAngle,
+  DPlaneAngle,
+  PlaneAngle,
   baseUnit
 )
 where
 
 import qualified Numeric.Units.Dimensional.DK as A
-import Numeric.Units.Dimensional.DK hiding (sin, cos, tan, asin, acos, atan, atan2, DPlaneAngle, DSolidAngle, PlaneAngle, SolidAngle, baseUnit)
+import Numeric.Units.Dimensional.DK hiding (sin, cos, tan, asin, acos, atan, atan2, DPlaneAngle, PlaneAngle, baseUnit)
 import Prelude hiding (sin, cos, tan, asin, acos, atan, atan2)
 
 sin, cos, tan, asin, acos, atan :: (Floating a) => Dimensionless a -> Dimensionless a
@@ -41,10 +41,8 @@ atan2 :: (RealFloat a) => Quantity d a -> Quantity d a -> Dimensionless a
 atan2 x y = removeAngles $ A.atan2 x y
 
 type DPlaneAngle = DOne
-type DSolidAngle = DOne
 
 type PlaneAngle = Quantity DPlaneAngle
-type SolidAngle = Quantity DSolidAngle
 
 baseUnit :: (Num a) => Unit (SIDim l m t i th n j) a
 baseUnit = siUnit
diff --git a/src/Numeric/Units/Dimensional/DK/SI/Prelude.hs b/src/Numeric/Units/Dimensional/DK/SI/Prelude.hs
index 849f077c..17dbc7ab 100644
--- a/src/Numeric/Units/Dimensional/DK/SI/Prelude.hs
+++ b/src/Numeric/Units/Dimensional/DK/SI/Prelude.hs
@@ -8,7 +8,7 @@
 
 = Summary
 
-This module defines a variant of "Numeric.Units.Dimensional.DK.Prelude" where plane and solid angles are not treated as dimensions.
+This module defines a variant of "Numeric.Units.Dimensional.DK.Prelude" where plane angles are treated as dimensionless.
 
 -}
 module Numeric.Units.Dimensional.DK.SI.Prelude
@@ -41,4 +41,5 @@ import Prelude hiding
     , sin, cos, tan, asin, acos, atan, atan2
     , sinh, cosh, tanh, asinh, acosh, atanh
     , sum, product, minimum, maximum
-    )  -- Hide definitions overridden by 'Numeric.Dimensional'.
\ No newline at end of file
+    )  -- Hide definitions overridden by 'Numeric.Dimensional'.
+    
\ No newline at end of file
diff --git a/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs b/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
index f61f6e3e..317d11f4 100644
--- a/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
+++ b/src/Numeric/Units/Dimensional/DK/SI/Quantities.hs
@@ -12,7 +12,7 @@
 
 = Summary
 
-This module defines a variant of "Numeric.Units.Dimensional.DK.Quantities" where plane and solid angles are not treated as dimensions.
+This module defines a variant of "Numeric.Units.Dimensional.DK.Quantities" where plane angles are treated as dimensionless.
 
 Compare, e.g., 'LuminousFlux' with 'Numeric.Units.Dimensional.DK.Quantities.LuminousFlux'
 
@@ -21,15 +21,15 @@ module Numeric.Units.Dimensional.DK.SI.Quantities
 (
   module Numeric.Units.Dimensional.DK.Quantities,
 
-  LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque, MomentOfInertia,
-  DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque, DMomentOfInertia
+  LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque, MomentOfInertia, SolidAngle,
+  DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque, DMomentOfInertia, DSolidAngle
 )
 where
 
 import Numeric.Units.Dimensional.DK.SI
 import qualified Numeric.Units.Dimensional.DK.Quantities as A
-import Numeric.Units.Dimensional.DK.Quantities hiding (LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque, MomentOfInertia,
-                                                       DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque, DMomentOfInertia)
+import Numeric.Units.Dimensional.DK.Quantities hiding (LuminousFlux,  Illuminance,  AngularVelocity,  AngularAcceleration,  RadiantIntensity,  Radiance,  AngularMomentum,  Torque,  MomentOfInertia,  SolidAngle,
+                                                       DLuminousFlux, DIlluminance, DAngularVelocity, DAngularAcceleration, DRadiantIntensity, DRadiance, DAngularMomentum, DTorque, DMomentOfInertia, DSolidAngle)
 
 type DLuminousFlux = ToSIDim A.DLuminousFlux
 type LuminousFlux = Quantity DLuminousFlux
@@ -57,3 +57,6 @@ type Torque = Quantity DTorque
 
 type DMomentOfInertia = ToSIDim A.DMomentOfInertia
 type MomentOfInertia = Quantity DMomentOfInertia
+
+type DSolidAngle = ToSIDim A.DSolidAngle
+type SolidAngle = Quantity DSolidAngle
diff --git a/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs b/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs
index 3bab7ba6..7e5ac22f 100644
--- a/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs
+++ b/src/Numeric/Units/Dimensional/DK/SI/SIUnits.hs
@@ -8,7 +8,7 @@
 
 = Summary
 
-This module defines a variant of "Numeric.Units.Dimensional.DK.SIUnits" where plane and solid angles are not treated as dimensions.
+This module defines a variant of "Numeric.Units.Dimensional.DK.SIUnits" where plane angles are treated as dimensionless.
 
 Compare, e.g., 'radian' with 'Numeric.Units.Dimensional.DK.SIUnits.radian'
 

From 5f2003409c0ba33f640f818902ed1c241f6c010a Mon Sep 17 00:00:00 2001
From: Douglas McClean <douglas.mcclean@gmail.com>
Date: Thu, 22 Oct 2015 15:06:30 -0400
Subject: [PATCH 12/12] Converted newtonian mechanics example to work with
 angles branch.

---
 examples/NewtonianMechanics.hs | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/examples/NewtonianMechanics.hs b/examples/NewtonianMechanics.hs
index 6848ab07..a6312ee6 100644
--- a/examples/NewtonianMechanics.hs
+++ b/examples/NewtonianMechanics.hs
@@ -33,8 +33,8 @@ rotationalMomentum i w = i * w
 rotationalPower :: (Num a) => Torque a -> AngularVelocity a -> Power a
 rotationalPower t w = t * w
 
-torque :: (Num a) => Force a -> Length a -> Torque a
-torque r f = r * f
+torque :: (Fractional a) => Force a -> Length a -> Torque a
+torque r f = r * f / (1 *~ radian)
 
 torqueFromChangeInMomentum :: (Fractional a) => AngularMomentum a -> Time a -> Torque a
 torqueFromChangeInMomentum dL dt = dL / dt