Arithmetic Rules for Difference vs Absolute Quantities (DRAFT)

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Exemplified with Temperature but applies more generally.

ArithmeticCharacteristic

Assuming ArithmeticCharacteristic(UNIFIED, ABSOLUTE, DIFFERENCE) being a property of Quantity.

public enum ArithmeticCharacteristic {

  /** No arithmetic distinction is made between absolute or difference on quantities 
   * having UNIFIED characteristic, because the physical interpretation is agnostic 
   * to such a distinction. Eg. any quantity of unit KELVIN
   */
  UNIFIED,
  
  /** Special arithmetic rules apply to quantities having ABSOLUTE characteristic, 
   * because the physical interpretation does require to distinguish between 
   * ABSOLUTE vs. DIFFERENCE. <p>
   * eg. an absolute quantity of unit CELSIUS, where 0°C is equivalent 273.15 K
   */
  ABSOLUTE,
  
  /** Special arithmetic rules apply to quantities having DIFFERENCE characteristic, 
   * because the physical interpretation does require to distinguish between 
   * ABSOLUTE vs. DIFFERENCE. <p>
   * eg. a difference quantity of unit CELSIUS, where 0°C is equivalent 0 K
   */
  DIFFERENCE
}

ArithmeticPolicy

Assuming ArithmeticPolicy(UNIFIED_ONLY, ABSOLUTE_OR_DIFFERENCE) being a property of Unit.

public enum ArithmeticPolicy{

  /** Quantities of Units with policy UNIFIED_ONLY can only have UNIFIED 
   * characteristic */
  UNIFIED_ONLY,

  /** Quantities of Units with policy ABSOLUTE_OR_DIFFERENCE can either have 
   * ABSOLUTE or DIFFERENCE characteristic */
  ABSOLUTE_OR_DIFFERENCE
}

Let

• scalar ... a number or dimensionless quantity
• A ... absolute Quantity type, exemplified with CELSIUS, unit °C_abs
• D ... difference Quantity type, exemplified with CELSIUS, unit °C_dif
• U ... Quantity type with 'unified' arithmetic characteristic exemplified with KELVIN

Quantity Conversion (pseudo-code)

• toUnified(A) -> U ... toUnified( 0°C_abs ) = 273.15 K
• toUnified(D) -> U ... toUnified( 1°C_dif ) = 1 K

Quantity Operations

Assuming commutativity of addition and multiplication, and let X - Y === X + (-1 * Y)

A ± A -> A ... 0°C_abs + 0°C_abs = 273.15°C_abs
A ± D -> A ... 2°C_abs + 3°C_dif = 5°C_abs
D ± D -> D ... 2°C_dif + 3°C_dif = 5°C_dif

U ± U -> U ... 1K + 2K = 3K
A ± U -> U ... 0°C_abs + 0K = 273.15K
     Note: breaks with the convention, that first operand gives the result's unit
D ± U -> U ... 0°C_dif + 0K = 0K
     Note: breaks with the convention, that first operand gives the result's unit

A * scalar -> A ... 0°C_abs * 2 = 273.15°C_abs
D * scalar -> D ... 5°C_dif * 2 = 10°C_dif
U * scalar -> U ... 5K * 2 = 10K

A * A -> yields unit K² ... by converting operands to Unified = Kelvin
D * D -> yields unit K² ... by converting operands to Unified = Kelvin
A * D -> yields unit K² ... by converting operands to Unified = Kelvin

A * U -> yields unit K² ... by converting operands to Unified = Kelvin
D * U -> yields unit K² ... by converting operands to Unified = Kelvin
U * U -> yields unit K² ... 5K * 2K = 10K²

D / D -> scalar ... 10°C_dif / 2°C_dif = 5
     Note: same result as if by converting operands to Unified
U / U -> scalar ... 10K / 2K = 5

A / A -> scalar ... by converting operands to Unified = Kelvin
A / D -> scalar ... by converting operands to Unified = Kelvin
D / A -> scalar ... by converting operands to Unified = Kelvin
A / U -> scalar ... by converting operands to Unified = Kelvin
D / U -> scalar ... by converting operands to Unified = Kelvin
U / D -> scalar ... by converting operands to Unified = Kelvin
U / A -> scalar ... by converting operands to Unified = Kelvin

1 / A -> yields unit 1/K ... by converting operand to Unified = Kelvin
     Note: is consistent with A/A, D/A and U/A
1 / D -> yields unit 1/K ... by converting operand to Unified = Kelvin
     Note: needs to be consistent with A/D, D/D and U/D (which it is, I guess)
1 / U -> yields unit 1/K

API Extension: Difference Operator

For convenience and usability a new difference operator (other than Quantity#subtract) is required to allow for the intended use-case of generating a DIFFERENCE quantity from 2 ABSOLUTE quantities:
A - A -> D ... 5°C_abs - 4°C_abs = 1°C_dif
     Note: new API !

Units and Characteristics

Class	Unit	System Unit	Arithmetic-Policy
Dimensionless	AbstractUnit.ONE	YES	UNIFIED_ONLY
ElectricCurrent	AMPERE	YES	UNIFIED_ONLY
LuminousIntensity	CANDELA	YES	???
Temperature	KELVIN	YES	UNIFIED_ONLY
Temperature	CELSIUS	NO	ABSOLUTE_OR_DIFFERENCE
Mass	KILOGRAM	YES	UNIFIED_ONLY
Length	METRE	YES	UNIFIED_ONLY
AmountOfSubstance	MOLE	YES	UNIFIED_ONLY
Time	SECOND	YES	UNIFIED_ONLY
Angle	RADIAN	YES	UNIFIED_ONLY
SolidAngle	STERADIAN	YES	UNIFIED_ONLY
Frequency	HERTZ	YES	UNIFIED_ONLY
Force	NEWTON	YES	UNIFIED_ONLY
Pressure	PASCAL	YES	UNIFIED_ONLY
Energy	JOULE	YES	UNIFIED_ONLY
Power	WATT	YES	UNIFIED_ONLY
ElectricCharge	COULOMB	YES	UNIFIED_ONLY
ElectricPotential	VOLT	YES	UNIFIED_ONLY
ElectricCapacitance	FARAD	YES	UNIFIED_ONLY
ElectricResistance	OHM	YES	UNIFIED_ONLY
ElectricConductance	SIEMENS	YES	UNIFIED_ONLY
MagneticFlux	WEBER	YES	UNIFIED_ONLY
MagneticFluxDensity	TESLA	YES	UNIFIED_ONLY
ElectricInductance	HENRY	YES	UNIFIED_ONLY
LuminousFlux	LUMEN	YES	???
Illuminance	LUX	YES	???
Radioactivity	BECQUEREL	YES	???
RadiationDoseAbsorbed	GRAY	YES	???
RadiationDoseEffective	SIEVERT	YES	???
CatalyticActivity	KATAL	YES	???
Speed	METRE_PER_SECOND	YES	UNIFIED_ONLY
Acceleration	METRE_PER_SQUARE_SECOND	YES	UNIFIED_ONLY
Area	SQUARE_METRE	YES	UNIFIED_ONLY
Volume	CUBIC_METRE	YES	UNIFIED_ONLY

API Extension: Quantity Creation 1

It would be desirable to extend the programming model, such that we can enforce this policy at compile-time, but this example does not:

1. For Unit with policy UNIFIED_ONLY use:
   Quantities.getQuantity(Number, Unit) ... a Quantity w/ characteristic UNIFIED

2. For Unit with policy ABSOLUTE_OR_DIFFERENCE use:
   Quantities.getQuantity(Number, Unit, ArithmeticCharacteristic) ... Quantity w/ either ABS. or DIF. characteristic

Examples utilizing checks at run-time

Quantities.getQuantity(5, KELVIN) ... OK
Quantities.getQuantity(5, KELVIN, ABSOLUTE) ... IllegalArgumentException or OK ???
Quantities.getQuantity(5, KELVIN, DIFFERENCE) ... IllegalArgumentException or OK ???

Quantities.getQuantity(5, CELSIUS) ... IllegalArgumentException
Quantities.getQuantity(5, CELSIUS, ABSOLUTE) ... OK
Quantities.getQuantity(5, CELSIUS, DIFFERENCE) ... OK

API Extension: Quantity Creation 2

Variant to 'Quantity Creation 1' w/o exposing ArithmeticCharacteristic to the public API

1. For Unit with policy UNIFIED_ONLY use:
   Quantities.getQuantity(Number, Unit) ... a Quantity w/ characteristic UNIFIED

2. For Unit with policy ABSOLUTE_OR_DIFFERENCE use:
   Quantities.getAbsoluteQuantity(Number, Unit) ... Quantity w/ ABSOLUTE characteristic Quantities.getDifferenceQuantity(Number, Unit) ... Quantity w/ DIFFERENCE characteristic

Examples utilizing checks at run-time

Quantities.getQuantity(5, KELVIN) ... OK
Quantities.getAbsoluteQuantity(5, KELVIN) ... IllegalArgumentException or OK ??? Quantities.getDifferenceQuantity(5, KELVIN) ... IllegalArgumentException or OK ???

Quantities.getQuantity(5, CELSIUS) ... IllegalArgumentException
Quantities.getAbsoluteQuantity(5, CELSIUS) ... OK
Quantities.getDifferenceQuantity(5, CELSIUS) ... OK