Cleaned up mathematical notation for physical units (#1995)

Co-authored-by: NikolaiFa <[email protected]>

docs/2_4_common_schema.adoc

@@ -340,21 +340,26 @@ A value latexmath:[v_{\mathit{unit}}] in `Unit` is converted to the base unit la
 
 [latexmath]
 ++++
-v_{\mathit{base}} = \texttt{factor} * v_{\mathit{unit}} + \texttt{if relativeQuantity then 0 else offset}
+v_{\mathit{base}} = \texttt{factor} \cdot v_{\mathit{unit}} + \left\{
+\begin{array}{ll}
+0 & \texttt{if relativeQuantity = true} \\
+\texttt{offset} & \texttt{if relativeQuantity = false}
+\end{array}
+\right.
 ++++
 where `factor` and `offset` are attributes of the `<BaseUnit>`, and  `relativeQuantity` an attribute of the `TypeDefinition` of a variable.
 
 _[For example, if_ latexmath:[{p_{\mathit{bar}}}] _is a pressure value in unit `bar`, and_ latexmath:[{p_{\mathit{Pa}}}] _is the pressure value in `<BaseUnit>`, then_
 
 [latexmath]
 ++++
-{p_{\mathit{Pa}} = 10^5 p_{\mathit{bar}}}
+{p_{\mathit{Pa}} = 10^5 \cdot p_{\mathit{bar}}}
 ++++
 
 _and therefore, `factor = 1.0e5` and `offset = 0.0`._
 
 _In the following table several unit examples are given._
-_Note that if in column `exponents` the definition_ latexmath:[\frac{kg \cdot m^2}{s^2}] _is present, then the attributes of `<BaseUnit>` are  `kg=1, m=2, s=-2`._
+_Note that if in column `exponents` the definition_ latexmath:[\mathrm{kg} \cdot \mathrm{m}^2 \cdot \mathrm{s}^{-2}] _is present, then the attributes of `<BaseUnit>` are  `kg=1, m=2, s=-2`._
 
 .Unit examples.
 [[table-unit-examples]]
@@ -370,75 +375,75 @@ h|offset
 
 |_Torque_
 |`N.m`
-|latexmath:[{kg \cdot m^2 / s^2}]
-|`1.0`
-|`0.0`
+|latexmath:[\mathrm{kg} \cdot \mathrm{m}^2 \cdot \mathrm{s}^{-2}]
+|latexmath:[1.0]
+|latexmath:[0.0]
 
 |_Energy_
 |`J`
-|latexmath:[{kg \cdot m^2 / s^2}]
-|`1.0`
-|`0.0`
+|latexmath:[\mathrm{kg} \cdot \mathrm{m}^2 \cdot \mathrm{s}^{-2}]
+|latexmath:[1.0]
+|latexmath:[0.0]
 
 |_Pressure_
 |`bar`
-|latexmath:[{\frac{kg}{m \cdot s^2}}]
-|`1.0e5`
-|`0.0`
+|latexmath:[\mathrm{kg} \cdot \mathrm{m}^{-1} \cdot \mathrm{s}^{-2}]
+|latexmath:[1.0 \cdot 10^5]
+|latexmath:[0.0]
 
 |_Angle_
 |`deg`
-|`rad`
-|`0.01745329251994330 (= pi/180)`
-|`0.0`
+|latexmath:[\mathrm{rad}]
+|latexmath:[0.01745329251994330 \ \left(= \frac{\pi}{180}\right)]
+|latexmath:[0.0]
 
 |_Angular velocity_
 |`rad/s`
-|`rad/s`
-|`1.0`
-|`0.0`
+|latexmath:[\mathrm{rad} \cdot \mathrm{s}^{-1}]
+|latexmath:[1.0]
+|latexmath:[0.0]
 
 |_Angular velocity_
 |`rpm`
-|`rad/s`
-|`0.1047197551196598 (= 2*pi/60)`
-|`0.0`
+|latexmath:[\mathrm{rad} \cdot \mathrm{s}^{-1}]
+|latexmath:[0.1047197551196598 \ \left(= 2 \cdot \frac{\pi}{60}\right)]
+|latexmath:[0.0]
 
 |_Frequency_
 |`Hz`
-|`rad/s`
-|`6.283185307179586 (= 2*pi)`
-|`0.0`
+|latexmath:[\mathrm{rad} \cdot \mathrm{s}^{-1}]
+|latexmath:[6.283185307179586 \ \left(= 2 \cdot \pi\right)]
+|latexmath:[0.0]
 
 |_Temperature_
 |`&#176;F`
-|`K`
-|`0.5555555555555556 (= 5/9)`
-|`255.3722222222222 (= 273.15-32*5/9)`
+|latexmath:[\mathrm{K}]
+|latexmath:[0.5555555555555556 \ \left(= \frac{5}{9}\right)]
+|latexmath:[255.3722222222222 \ \left(= 273.15 - 32 \cdot \frac{5}{9}\right)]
 
 |_Percent by length_
 |`%/m`
-|`1/m`
-|`0.01`
-|`0.0`
+|latexmath:[\mathrm{m}^{-1}]
+|latexmath:[0.01]
+|latexmath:[0.0]
 
 |_Parts per million_
 |`ppm`
-|`1`
-|`1.0e-6`
-|`0.0`
+|latexmath:[1]
+|latexmath:[1.0 \cdot 10^{-6}]
+|latexmath:[0.0]
 
 |_Length_
 |`km`
-|`m`
-|`1000`
-|`0.0`
+|latexmath:[\mathrm{m}]
+|latexmath:[1000]
+|latexmath:[0.0]
 
 |_Length_
 |`yd`
-|`m`
-|`0.9144`
-|`0.0`
+|latexmath:[\mathrm{m}]
+|latexmath:[0.9144]
+|latexmath:[0.0]
 |====
 
 _Note that `Hz` is typically used as `Unit.name` for a frequency quantity, but it can also be used as `<DisplayUnit>` for an angular velocity quantity (since `revolution/s`)._
@@ -452,10 +457,25 @@ _When only one of <<input>> `v2` and <<output>> `v1`, connected with equation `v
 _When two variables v1 and v2 are connected and for both of them `<BaseUnit>` elements are defined, then they must have identical exponents of their `<BaseUnit>`._
 _If `factor` and `offset` are also identical, again the connection equation `v2 = v1` holds._
 _If `factor` and `offset` are not identical, the tool may either trigger an error or, if supported, perform a conversion; in other words, use the connection equation (in this case the `relativeQuantity` of the `<TypeDefinition>`, see below, has to be taken into account in order to determine whether `offset` shall or shall not be utilized):_
-+
-`factor(v1) * v1 + (if relativeQuantity(v1) then 0 else offset(v1)) = factor(v2) * v2 + (if relativeQuantity(v2) then 0 else offset(v2))` +
+
+[latexmath]
+++++
+\texttt{factor(v1)} \cdot \texttt{v1} + \left\{
+\begin{array}{ll}
+0 & \texttt{if relativeQuantity(v1) = true} \\
+\texttt{offset(v1)} & \texttt{if relativeQuantity(v1) = false}
+\end{array}
+\right.
+=
+\texttt{factor(v2)} \cdot \texttt{v2} + \left\{
+\begin{array}{ll}
+0 & \texttt{if relativeQuantity(v2) = true} \\
+\texttt{offset(v2)} & \texttt{if relativeQuantity(v2) = false}
+\end{array}
+\right.
+++++
 _where_ `relativeQuantity(v1) = relativeQuantity(v2)` _is required_.
-+
+
 _As a result, wrong connections can be detected (for example, connecting a force with an angle-based variable would trigger an error) and conversions between, say, US and SI units can be either automatically performed or, if not supported, an error is triggered as well._
 +
 _This approach is not satisfactory for variables belonging to different quantities that have, however, the same `<BaseUnit>`, such as quantities `Energy` and `Torque`, or `AngularVelocity` and `Frequency`._
@@ -473,8 +493,8 @@ _Example:_
 [latexmath]
 ++++
 \begin{align*}
-J \cdot \alpha = \tau \rightarrow [kg.m^2]*[rad/s^2] = [kg.m^2/s^2] & \quad \text{// o.k. ("rad" is treated as "1")} \\
-J \cdot \alpha = f \rightarrow [kg.m^2]*[rad/s^2] = [kg.m/s^2] & \quad \text{// error, since dimensions do not agree}
+J \cdot \alpha = \tau \rightarrow [\mathrm{kg} \cdot \mathrm{m}^2] \cdot [\mathrm{rad} \cdot \mathrm{s}^{-2}] = [\mathrm{kg} \cdot \mathrm{m}^2 \cdot \mathrm{s}^{-2}] & \quad \text{// o.k. ("rad" is treated as "1")} \\
+J \cdot \alpha = f \rightarrow [\mathrm{kg} \cdot \mathrm{m}^2] \cdot [\mathrm{rad} \cdot \mathrm{s}^{-2}] = [\mathrm{kg} \cdot \mathrm{m} \cdot \mathrm{s}^{-2}] & \quad \text{// error, since dimensions do not agree}
 \end{align*}
 ++++
 
@@ -485,11 +505,11 @@ _If no unit computation is needed, `rad` is propagated._
 _If a unit computation is needed and one of the involved units has `rad` as a `<BaseUnit>`, then unit propagation is not possible._
 _Examples:_
 +
-- _a = b + c, and `Unit` of c is provided, but not `Unit` of a and b:_ +
+- _latexmath:[a = b + c], and `Unit` of c is provided, but not `Unit` of a and b:_ +
 _The Unit definition of `c` (in other words, `Unit.name`, `<BaseUnit>`, `<DisplayUnit>`) is also used for `a` and `b`._
 _For example, if BaseUnit(c) = `rad/s`, then BaseUnit(a) = BaseUnit(b) = `rad/s`._
 +
-- _a = b*c, and `Unit` of a and of c is provided, but not `Unit` of b:_ +
+- _latexmath:[a = b \cdot c], and `Unit` of a and of c is provided, but not `Unit` of b:_ +
 _If `rad` is either part of the `<BaseUnit>` of `a` and/or of `c`, then the `<BaseUnit>` of `b` cannot be deduced (otherwise it can be deduced)._
 _Example: If `BaseUnit(a) = kg.m/s2` and `BaseUnit(c) = m/s2`, then the `BaseUnit(b)` can be deduced to be `kg`._
 _In such a case `Unit.name` of b cannot be deduced from the `Unit.name` of `a` and `c`, and a tool would typically construct the `Unit.name` of `b` from the deduced `<BaseUnit>`.]_
@@ -512,8 +532,8 @@ A value latexmath:[v_{\mathit{unit}}] in `Unit` is converted to a value latexmat
 ++++
 v_{\mathit{display}} =
 \left\{\begin{array}{ll}
-\texttt{factor} *          v_{\mathit{unit}}  + \texttt{offset} &\text{if} \; \texttt{inverse = false} \\
-\texttt{factor} * \frac{1}{v_{\mathit{unit}}}  &\text{if} \; \texttt{inverse = true}
+\texttt{factor} \cdot          v_{\mathit{unit}}  + \texttt{offset} &\text{if} \; \texttt{inverse = false} \\
+\texttt{factor} \cdot \frac{1}{v_{\mathit{unit}}}  &\text{if} \; \texttt{inverse = true}
 \end{array}\right.
 ++++
 
@@ -523,10 +543,10 @@ _For example, if latexmath:[{T_K}] is the temperature value of `Unit.name` (in `
 
 [latexmath]
 ++++
-T_F = (9/5) * (T_K - 273.15) + 32
+T_F = \frac{9}{5} \cdot (T_K - 273.15) + 32
 ++++
 
-_and therefore, `factor = 1.8 (=9/5)` and `offset = -459.67 (= 32 - 273.15*9/5)`._
+_and therefore, latexmath:[\texttt{factor} = 1.8 \ \left(= \frac{9}{5}\right)] and latexmath:[\texttt{offset} = -459.67 \ \left(= 32 - 273.15 \cdot \frac{9}{5}\right)]._
 
 _Both the `DisplayUnit.name` definitions as well as the `Unit.name` definitions are used in the variable elements._
 
@@ -638,7 +658,7 @@ _[If, for example, an `<Enumeration>` is defined with `name1 = -4`, `name2 = 1`,
 If not defined and no other information about the nominal value is available, then `nominal = 1` is assumed.
 For array variables, this nominal value applies to all elements of the array. +
 _[The nominal value of a variable can be, for example, used to determine the absolute tolerance for this variable as needed by numerical algorithms:_ +
-`absoluteTolerance = nominal * tolerance * 0.01` +
+`absoluteTolerance = nominal \cdot tolerance \cdot 0.01` +
 _where `tolerance` is, for example, the relative tolerance defined in <<DefaultExperiment>>.]_
 
 |`unbounded`