diff --git a/Numeric/Units/Dimensional/CGS.lhs b/Numeric/Units/Dimensional/CGS.lhs new file mode 100644 --- /dev/null +++ b/Numeric/Units/Dimensional/CGS.lhs @@ -0,0 +1,326 @@ +Numeric.Dimensional.CGS -- CGS system of units +Bjorn Buckwalter, bjorn.buckwalter@gmail.com +License: BSD3 + +*** EXPERIMENTAL *** + + += Introduction = + +This module was prompted by an email from Chuck Blake[1]. He asked if +the Dimensional library could support other systems of units than +SI, in particular systems such as the centimeter-gram-second (CGS) +system where fractional exponents of dimensions occur. He also +wondered whether it was possible to convert quantities between +different systems while statically ensuring that a given conversion +was valid. + +In this module we show that we can in a straight forward manner +support systems with rational exponents, provided that the rationals +that may be encountered are known a priori. As an example we provide +a rudimentary implementation of the CGS system. + +We also show that we can indeed statically prohibit invalid conversions +between different systems. + + += Caveats = + +I'm ignorantly assuming that when working with the CGS (or MKS) +system you will only (meaningfully?) encounter half-exponents and +only of the length and mass dimensions. Of course, in other systems +other rational exponents may be encountered. + +I am also assuming that the CGS system would not be employed when +working with temperature, amount or luminosity. This is evident in +the below type signatures where I have assumed zero extent in the +temperature, amount and luminosity dimensions. If this is incorrect +I would appreciate pointers to the CGS representation of these +dimensions. + +Please correct and inform me if my assumptions are wrong! + + += Preliminaries = + +> {-# LANGUAGE UndecidableInstances +> , ScopedTypeVariables +> , EmptyDataDecls +> , MultiParamTypeClasses +> , FlexibleInstances +> , FlexibleContexts +> #-} + +> module Numeric.Units.Dimensional.CGS where + +> import Prelude ( undefined, Num, Fractional, Floating, Show, recip, Double ) +> import qualified Prelude +> import Numeric.Units.Dimensional hiding ( DLength, DMass, DTime, DElectricCurrent ) +> import Numeric.Units.Dimensional.Quantities as SIQ +> import qualified Numeric.Units.Dimensional.SIUnits as SI +> import qualified Numeric.NumType as N +> import Numeric.NumType ( Neg2, Neg1, Zero, Pos, Pos1, Pos2, Pos3, NumType ) +> import Numeric.NumType ( neg2, neg1, zero, pos1, pos2, pos3 ) +> import Data.Maybe (catMaybes) + + += Dimensions = + +Analogously with the SI we collect the base dimensions of the CGS +system in the data type 'CGSDim'. + +> data CGSDim lh mh t + +In the above 'lh' and 'mh' represent the number of half-exponents +of length and mass respectively while 't' represents the number of +whole-exponents. The base dimensions illustrate this. + +> type DLength = CGSDim Pos2 Zero Zero +> type DMass = CGSDim Zero Pos2 Zero +> type DTime = CGSDim Zero Zero Pos1 + +We add a few non-base dimensions for the sake of example. Charge +is particularly interesting as it illustrates the need for +half-exponents as described in [2]. + +> type DElectricCurrent = CGSDim Pos3 Pos1 Neg2 +> type DCharge = CGSDim Pos3 Pos1 Neg1 + + += 'Mul', 'Div', 'Pow' and 'Root' instances = + +The 'Mul', 'Div', 'Pow' and 'Root' instances are strictly analogous +with the SI. + +> instance ( N.Sum lh lh' lh'' +> , N.Sum mh mh' mh'' +> , N.Sum t t' t'' ) => Mul (CGSDim lh mh t) +> (CGSDim lh' mh' t') +> (CGSDim lh'' mh'' t'') + +> instance ( N.Sum lh lh' lh'' +> , N.Sum mh mh' mh'' +> , N.Sum t t' t'' ) => Div (CGSDim lh'' mh'' t'') +> (CGSDim lh' mh' t') +> (CGSDim lh mh t) + +> instance ( N.Mul lh x lh' +> , N.Mul mh x mh' +> , N.Mul t x t' ) => Pow (CGSDim lh mh t) x +> (CGSDim lh' mh' t') + +> instance ( N.Div lh x lh' +> , N.Div mh x mh' +> , N.Div t x t' ) => Root (CGSDim lh mh t) x +> (CGSDim lh' mh' t') + + += Units = + +We define the base units of the system. By defining 'meter' with a +"scale" of 100 we get a scale of one for 'centi meter'. + +> meter :: Num a => Unit DLength a +> meter = Dimensional 100 +> gram :: Num a => Unit DMass a +> gram = Dimensional 1 +> second :: Num a => Unit DTime a +> second = Dimensional 1 + +We continue by defining the CGS equivalents of the other base SI +units. Actually we limit ourselves to 'ampere' since I am not sure +if or how the SI base dimensions other than current are expressed +in CGS. + +> ampere :: Floating a => Unit DElectricCurrent a +> ampere = prefix (recip 3.33564e-10) ((SI.centi meter ^ pos3) ^/ pos2 * gram ^/ pos2 * second ^ neg2) + +We also define the preferred CGS unit for charge. + +> franklin :: Floating a => Unit DCharge a -- Also known as "esu". +> franklin = gram ^/ pos2 * (SI.centi meter ^ pos3) ^/ pos2 / second + + += Conversion from SI = + +At some point we may wish to convert an SI quantity to a CGS quantity +or vice versa. + +In order to convert a 'Quantity' from the SI system to the CGS +system we use the strategy of dividing the quantity by the SI base +unit and multiplying the resulting number (sans dimension) by the +equivalent CGS unit. To realize this strategy we must be able to +obtain the SI base unit and the equivalent CGS unit for a given +quantity. We start with the SI unit since it is trivial. + +> unit_SI :: Num a => Quantity (Dim l m t i th n j) a -> Unit (Dim l m t i th n j) a +> unit_SI _ = Dimensional 1 + +(Perhaps the above function would be better defined in another +module.) + +Obtaining the CGS unit corresponding to the SI base unit of a +Quantity isn't quite as trivial. The function body itself is +straight-forward enough, the hairy part is the type signature. + +> unit_CGS :: forall a l m t i l2 m2 il it l' m' t'. +> ( Floating a +> , N.Mul Zero l Zero, N.Mul Pos2 l l2 +> , N.Mul Zero m Zero, N.Mul Pos2 m m2 +> , N.Mul Zero t Zero, N.Mul Pos1 t t +> , N.Sum l2 Zero l2 +> , N.Sum Zero m2 m2, N.Sum m2 Zero m2 +> , N.Sum Zero t t +> , N.Mul Pos3 i il +> , N.Mul Pos1 i i +> , N.Mul Neg2 i it +> , N.Sum l2 il l' +> , N.Sum m2 i m' +> , N.Sum t it t' +> ) => Quantity (Dim l m t i Zero Zero Zero) a -> Unit (CGSDim l' m' t') a +> unit_CGS _ = meter ^ (undefined :: l) +> * SI.kilo gram ^ (undefined :: m) +> * second ^ (undefined :: t) +> * ampere ^ (undefined :: i) + +Note that since the base dimensions of the CGS are a subset of those +of the SI the mapping of types from SI to CGS is unambiguous. + +Also note that complex as the type signature may be producing it is a +mostly mechanical process. + +With the above two functions we can define the function that converts +a unit from the SI. We omit the type signature since it is hairy +but can be readily inferred. + +> fromSI x = x /~ unit_SI x *~ unit_CGS x + + += Conversion to SI = + +We use the same strategy to convert from CGS to SI. However, when +converting from CGS to SI there may be several valid SI dimensionalities +for any given CGS dimensionality. We will handle this ambiguity by +requiring the user to specify the desired type (except when it is +inferable) of the resulting quantity. For example: + +] toSI (3.2 *~ centi meter) :: Length Double + +In order to do this we must employ lexically scoped type variables +and provide the hairy type signature for the 'toSI' function. + +> toSI :: forall a l m t i l2 m2 il it l' m' t'. +> ( Floating a +> , N.Mul Zero l Zero, N.Mul Pos2 l l2 +> , N.Mul Zero m Zero, N.Mul Pos2 m m2 +> , N.Mul Zero t Zero, N.Mul Pos1 t t +> , N.Sum l2 Zero l2 +> , N.Sum Zero m2 m2, N.Sum m2 Zero m2 +> , N.Sum Zero t t +> , N.Mul Pos3 i il +> , N.Mul Pos1 i i +> , N.Mul Neg2 i it +> , N.Sum l2 il l' +> , N.Sum m2 i m' +> , N.Sum t it t' +> ) => Quantity (CGSDim l' m' t') a -> Quantity (Dim l m t i Zero Zero Zero) a +> toSI x = x /~ unit_CGS (undefined :: Quantity (Dim l m t i Zero Zero Zero) a) +> *~ unit_SI (undefined :: Quantity (Dim l m t i Zero Zero Zero) a) + +Again, the type signature is complex but deriving it is a mechanical +process. + + += 'Show' instance = + +We round off by writing 'Show' instance for 'CGSDim' analogous to +that of 'Dim'. + +Out of laziness we use the notation "sqrt(cm)" to represent halves +of integral dimensions. Nothing is technically keeping us from doing +a better job here. + +> instance forall lh mh t. +> ( NumType lh +> , NumType mh +> , NumType t +> ) => Show (CGSDim lh mh t) where +> show _ = (Prelude.unwords Prelude.. catMaybes) +> [ dimUnit "sqrt(cm)" (undefined :: lh) +> , dimUnit "sqrt(g)" (undefined :: mh) +> , dimUnit "s" (undefined :: t) +> ] + + += Examples = + +Let us try the Coulomb attraction example from [2]. We start by +performing the calculation in the SI. + +> q_si = 1.6021773e-19 *~ SI.coulomb -- Elementary charge in SI. +> r_si = 0.1 *~ SI.nano SI.meter -- Distance in SI +> f_si = q_si ^ pos2 / (_4 * pi * e0 * r_si ^ pos2) +> where +> e0 = 8.8541878e-12 *~ (SI.ampere * SI.second / (SI.volt * SI.meter)) + +The same calculation in the CGS system. + +> q_cgs = fromSI q_si -- Elementary charge in CGS. +> r_cgs = fromSI r_si -- Distance in CGS +> f_cgs = q_cgs ^ pos2 / r_cgs ^ pos2 + +Inspecting the values in GHCi shows us that the results are consistent +(within reasonable accuracy) with [2]. + + *Numeric.Dimensional.CGS> f_si + 2.3070794737101255e-8 m kg s^-2 + *Numeric.Dimensional.CGS> f_cgs + 2.30708078598602e-3 sqrt(cm)^2 sqrt(g)^2 s^-2 + +To convert from CGS to SI we must specify the type of the SI 'Quantity'. + +> f_si' = toSI f_cgs :: SIQ.Force Double + + *Numeric.Dimensional.CGS> f_si' + 2.3070807859860202e-8 m kg s^-2 + +We follow up with another conversion example demonstrating the +ambiguity in the conversion from CGS to SI. + +> c = 1 *~ SI.farad -- A SI capacitance. +> c_cgs = fromSI c -- Capacitance has dimensionality L in CGS. +> c' = toSI c_cgs :: SIQ.Capacitance Double +> c'' = toSI c_cgs :: Length Double + + *Numeric.Dimensional.CGS> c + 1.0 m^-2 kg^-1 s^4 A^2 + *Numeric.Dimensional.CGS> c_cgs + 8.98755691740885e11 sqrt(cm)^2 + *Numeric.Dimensional.CGS> c' + 1.0 m^-2 kg^-1 s^4 A^2 + *Numeric.Dimensional.CGS> c'' + 8.98755691740885e9 m + + += Future work = + +This is a very rudimentary implementation. To make it more practical +a significant number of quantities and units, in particularly those +commonly used with the CGS, would need to be added. In the mean +time all units defined for the SI can be used with the CGS by +applying 'fromSI' to quantities defined from the SI units. + +If anyone is willing to add quantities/units (or other enhancements) +I will happily to accept patches. Personally I do not expect to use +this module and therefore do not intend to invest much more time +in it. If the module has other users I might reconsider. + +And of course, another direction of future work is to define +additional systems (e.g. natural, relativistic) using this module +as a template. I imagine this should be fairly straight forward. + + += References = + +[1] http://code.google.com/p/dimensional/wiki/ChuckBlake20070611 +[2] http://www.tf.uni-kiel.de/matwis/amat/mw1_ge/kap_2/basics/b2_1_14.html diff --git a/dimensional.cabal b/dimensional.cabal --- a/dimensional.cabal +++ b/dimensional.cabal @@ -1,8 +1,8 @@ Name: dimensional -Version: 0.7.1 +Version: 0.7.2 License: BSD3 License-File: LICENSE -Copyright: Bjorn Buckwalter 2006-2007 +Copyright: Bjorn Buckwalter 2006-2008 Author: Bjorn Buckwalter Maintainer: bjorn.buckwalter@gmail.com Stability: mostly stable @@ -26,7 +26,7 @@ Numeric.Units.Dimensional.Quantities, Numeric.Units.Dimensional.SIUnits, Numeric.Units.Dimensional.NonSI, - Numeric.Units.Dimensional.Extensible - --Numeric.Units.Dimensional.CGS + Numeric.Units.Dimensional.Extensible, + Numeric.Units.Dimensional.CGS ghc-options: -O