6.2 Reflective Symmetries

Reflective symmetries are (beside the identity) the simplest types of symmetries.

Figure 6.4: Reflective symmetry

The set $A$ has a reflective symmetry indicated by the hyperplane with the normal vector $\bm{n}$ and the scalar $d$. It can be reconstructed by using the subset on the positive side $A^+$ and the reflection function $\operatorname{refl}_{\bm{n},d}$.

In other words, a set has a reflective symmetry, if there exists an affine coordinate system with an orthogonal matrix, where one coordinate is negated. The reflecting hyperplane always passes through the center of gravity of the set. For a reflective symmetric set $A$ with the reflecting hyperplane $\mathbb{H}_{n,d}$, the set $A$ can be reconstructed by using the subset on the positive side of the hyperplane $A^+ := \{ \bm{x} \in A \vert \bm{n} \cdot \bm{x} \geq d \}$:

$$A = A^+ \cup \operatorname{refl}_{\bm{n},d}(A^+)$$ (6.1)

In other words, the subset on one side of the reflecting hyperplane can be re-constructed by the subset on the other side using the reflection function $\operatorname{refl}_{\bm{n},d}$ as visualized in Figure 6.4. Additionally, every point on the reflecting plane is invariant, i.e., is a fixed point, under the reflection function: $\operatorname{refl}_{\bm{n},d} \vert_{A^0} = {\mathbb{I}}$ with $A^0 := \{ \bm{x} \in A \vert \bm{n} \cdot \bm{x} = d\}$.

Figure 6.5: Boundary patch partition of a set with reflective symmetry

The arrows on the right visualize the boundary patch relation. Each boundary patch partition element is only related to itself. The green line indicates the reflecting hyperplane.

A geometry ${\mathcal{G}}$ having a reflective symmetry can be represented by a templated geometry with just one template being ${\mathcal{G}}^+$ (or ${\mathcal{G}}^-$) and two instances with the transformation functions ${\mathbb{I}}$ and $\operatorname{refl}_{\bm{n},d}$. The same approach can also be applied to meshes having reflective symmetries. This templated structure has only one instance interface which is included in the reflection hyperplane.

The boundary patch partition of the sole template is fairly simple and consists of only two elements, being the boundary which is included in the reflecting hyperplane and the rest of the boundary. Both boundary patches are only related to itself in the boundary patch relation and the instance graph is regular. The boundary patch partition and boundary patch relation of a templated geometry with a reflective symmetry is visualized in Figure 6.5.

A multi-region geometry or a multi-region mesh is said to have a reflective symmetry with the reflecting hyperplane $\mathbb{H}_{\bm{n},d}$, if all regions have the same reflective symmetry. The corresponding multi-region templated structure has a template for each region and therefore additional instance interfaces are possible. However, no new transformation functions (besides ${\mathbb{I}}$ and $\operatorname{refl}_{\bm{n},d}$) are introduced for multi-region structures and the instance graph is still regular.

Using the property above, the following important Lemma for templated structures can be formulated:

Lemma 1 (Reflective symmetry conformity)   Let $\bm{n} \in {\mathbb{R}}^n$ with ${\left\lVert\bm{n}\right\rVert_2} = 1$, $d \in {\mathbb{R}}$, and ${\mathcal{G}}$ be a geometry with $\bm{n} \cdot \bm{x} \geq d$ for all $\bm{x} \in {\mathcal{G}}$. Then, for any region indicators $r_1, r_2 \in \{1,2\}$, ${\Lambda}= (({\mathcal{G}}, ({\mathbb{I}}, \operatorname{refl}_{\bm{n},d}), (r_1, r_2)))$ is a templated geometry.

Similarly, for a mesh ${\mathcal{M}}$ with $\bm{n} \cdot \bm{x} \geq d$ for all $\bm{x} \in {\operatorname{us}}({\mathcal{M}})$, the templated structure ${\Gamma}= (({\mathcal{M}}, ({\mathbb{I}}, \operatorname{refl}_{\bm{n},d}), (r_1, r_2)))$ is a templated mesh.

Proof. ${\Lambda}$ is a templated structure, because $\operatorname{int}^\star ({\mathbb{I}}({\mathcal{G}})) \cap \operatorname{int}^\star (\operatorname{refl}_{\bm{n},d}({\mathcal{G}})) = \emptyset$ and ${\operatorname{AT}}({\Lambda}) = {\mathbb{I}}({\mathcal{G}}) \cup \operatorname{refl}_{\bm{n},d}({\mathcal{G}})$ is a geometry (by definition of a geometry). Therefore, ${\Lambda}$ is a templated geometry.

${\Gamma}$ is a templated structure, because $\operatorname{int}^\star ({\operatorname{us}}({\mathbb{I}}({\mathcal{G}}))) \c... ...{\operatorname{us}}(\operatorname{refl}_{\bm{n},d}({\mathcal{G}}))) = \emptyset$. If ${\operatorname{AT}}({\Gamma}) = {\mathbb{I}}({\mathcal{M}}) \cup \operatorname{refl}_{\bm{n},d}({\mathcal{M}})$ is not conforming, then non-conformities can only occur on the instance interface which is included in the reflecting hyperplane. However, the reflecting hyperplane is invariant under the reflection function and therefore all elements are conforming. Thus, ${\Gamma}$ is a templated mesh. $\qedsymbol$

Lemma 6.1 can also be formulated and proven for multi-region geometries and multi-region meshes. This implies that any mesh ${\mathcal{M}}^+$ which geometry-conforms to a geometry ${\mathcal{G}}^+$ can be used as a mesh template and it automatically results in a conforming templated mesh. This drastically simplifies the process of generating templated meshes based on geometries with reflective symmetry. Theoretically, any algorithm presented in Section 5.1 can be used for templated mesh generation. However, due to Lemma 6.1, Algorithm 6.1 presents a much simpler alternative: At first, the positive multi-region sub geometry ${({\mathcal{G}}, {\widetilde{\xi}})}^+$ is extracted (Line 3). Then, a conventional mesh generation algorithm is used to create a multi-region mesh ${({\mathcal{M}}, {\xi})}^+$ which geometry-conforms ${({\mathcal{G}}, {\widetilde{\xi}})}^+$ (Line 5). Finally, the resulting templated mesh is obtained by using every region of ${({\mathcal{M}}, {\xi})}^+$ as a mesh template with each two instances indicated by the transformation functions ${\mathbb{I}}$ and $\operatorname{refl}_{\bm{n},d}$ and the corresponding region indicator (Line 6).

Even though reflective symmetries are easy to detect automatically (cf. Section 3.3.1) and the mesh generation process has no conformity and dependence issues, there is a theoretical upper bound on the improvements in memory and mesh generation runtime of a factor of two. Benchmark results for the improvements are given in Section 7.2. The approaches presented in this section can also by applied iteratively, if the geometry or mesh template itself again has a reflective symmetry.