Geometry definition

Geometry Model

Mulder geometries rely on a common model, inherited from Pumas [Nie22]. A Mulder geometry is defined as a set of propagation media, \(M = \{ m_i \,|\, i \in \left[\!\left[ 0, n - 1 \right]\!\right]\}\), where the media, \(m_i\), differ by their physical properties. The geometry media delimit volumes in the 3d-space, as described below.

Note

The spatial structure of a geometry instance is expected to be immutable. However, the physical properties of the constitutive media may be freely modified.

Space structure

The space structure of the geometry is defined by a Location function, \(L\), mapping a space position, \(\vec{r}\), to a medium index, \(i\), as

\[\begin{split}L: \left| \begin{array}{rcl} \mathbb{R}^3 & \longrightarrow & \left[\!\left[ 0, n \right]\!\right] \\ \vec{r} & \longmapsto & i \\ \end{array} \right. ,\end{split}\]

where the special index \(n = \left|M\right|\) indicates that \(\vec{r}\) lies outside of \(M\).

Simulating the geometry traversal of a muon is a ray-tracing problem. While the location function, \(L\), is sufficient for this purpose, it is not efficient. Therefore, the geometry model is complemented with a Tracing function, \(T\). Let us parametrise a ray as

\[R(\vec{r}, \vec{u}) = \{ \vec{r} + \lambda \vec{u} \,|\, \lambda \in \mathbb{R}_+ \} ,\]

where \(r\) is the ray origin and \(\vec{u} \in \mathbb{U}_3\) a unit vector of \(\mathbb{R}^3\). Then, the Tracing function is defined as

\[\begin{split}T: \left| \begin{array}{rcl} \mathbb{R}^3 \times \mathbb{U}_3 & \longrightarrow & \mathbb{R}_+ \times \left[\!\left[ 0, n \right]\!\right] \\ (\vec{r}, \vec{u}) & \longmapsto & (\lambda^*, j) \\ \end{array} \right. ,\end{split}\]

where the returned \(\lambda^*\) value is the distance to the first (closest from \(\vec{r}\)) intersected medium boundary, and \(j\) the index of the next medium, i.e. on the opposite side of the boundary w.r.t. the ray origin, \(\vec{r}\).

Physical properties

The physical properties of a medium are defined by,

• the name of a constitutive material, which maps to a Material or Composite object, specifying the medium atomic composition.

• a bulk density, that might differ from the material one, e.g., to account for a solid porosity or specific gas T,P conditions.