Physlib.Units.ParametricUnits

Let $B$ be a basis. For any two unit scales $u_1, u_2$ mapping elements of $B$ to positive real numbers, and for any $b \in B$, the ratio of the magnitudes $\frac{u_1(b)}{u_2(b)}$ is non-zero.

Let $B$ be a finite set representing a basis of physical dimensions. Given two unit scales $u_1$ and $u_2$ (which assign a positive real magnitude to each base dimension $b \in B$), the dimension-scaling homomorphism $\text{dimScale}(u_1, u_2)$ is a monoid homomorphism from the group of dimensions $\text{Dimension } B$ to the non-negative real numbers $\mathbb{R}_{\ge 0}$. For any dimension $d$, the scaling factor is defined by the product: $$\text{dimScale}(u_1, u_2)(d) = \prod_{b \in B} \left( \frac{u_1(b)}{u_2(b)} \right)^{\text{exp}_b(d)}$$ where $\text{exp}_b(d)$ denotes the exponent of the base dimension $b$ in the dimension $d$. This map determines how a physical quantity of dimension $d$ rescales when the unit choice is changed from $u_1$ to $u_2$.

Let $B$ be a finite basis of physical dimensions. For any unit scale $u$ (which assigns a positive real magnitude to each base dimension $b \in B$) and any physical dimension $d \in \text{Dimension } B$, the dimension-scaling factor from $u$ to itself is equal to 1, i.e., $$\text{dimScale}(u, u)(d) = 1$$

Let $B$ be a finite set representing a basis of physical dimensions. For any two unit scales $u_1$ and $u_2$ over $B$, the dimension-scaling homomorphism $\text{dimScale}(u_1, u_2)$ maps the dimensionless identity $1 \in \text{Dimension } B$ to the multiplicative identity $1$ in $\mathbb{R}_{\ge 0}$. That is, $$\text{dimScale}(u_1, u_2)(1) = 1$$ This reflects the fact that dimensionless quantities do not change their numerical value when the system of units is rescaled.

Let $B$ be a finite set representing a basis of physical dimensions. For any three unit scales $u_1, u_2, u_3$ (which assign positive real magnitudes to each base dimension in $B$) and any physical dimension $d \in \text{Dimension } B$, the dimension-scaling factors satisfy the transitivity property: $$\text{dimScale}(u_1, u_2)(d) \cdot \text{dimScale}(u_2, u_3)(d) = \text{dimScale}(u_1, u_3)(d)$$ where $\text{dimScale}(u_i, u_j)(d)$ denotes the scaling factor applied to a quantity of dimension $d$ when changing the system of units from $u_i$ to $u_j$.

The function maps a set of unit choices $u$ (of type `LTMCTUnitChoices`) to a `UnitScale` over the basis of base dimensions $\{L, T, M, C, \Theta\}$. For each base dimension $b \in \{L, T, M, C, \Theta\}$, the resulting scale is defined as the positive real magnitude of the corresponding field in $u$ (length, time, mass, charge, or temperature).