## 3 Union and Intersection of Thick Ellipsoid Enclosures
### Thick Ellipsoid Union of Two Ellipsoids with Different Midpoints  
#### 3.3.1 General Solution Procedure  
The thick ellipsoid union of two ellipsoids with different midpoints is a direct consequence of the procedure in Section 3.2.2. Firstly, the shape matrix \(\mathbf{Q}_{k}^{\mathrm{I}}\) of the thick ellipsoid  
\[\left(\mathcal{E}\right)_{\mathrm{U},k}=\left(\mathcal{E}\right)_{\mathrm{U},k }(\bar{\mu}_{k},\mathbf{\Gamma}_{\mathrm{U},k}\left[1\;;\;\eta_{k}\right])\quad \text{with}\quad\eta_{k}=\eta_{k}^{*}\geq 1\quad\text{and}\quad\mathbf{\Gamma}_{ \mathrm{U},k}=\left(\mathbf{Q}_{k}^{\mathrm{I}}\right)^{\frac{1}{2}} \tag{60}\]  
is determined as in Equation (47). The remaining degree of freedom \(\eta_{k}^{*}\) can be found according to the work of [PERSON] [50]. For that purpose, specific points on the ellipsoids to be merged were extracted in [24] with subsequently inflating the inner bound by the factor \(\eta_{k}^{*}\geq 1\) until all points are included in its interior. To avoid the extraction of individual points, which has to be performed with sufficient care so that the new outer bound is guaranteed to be conservative, Equation (50) with \(\zeta_{j,k}\) given in (44) is generalized to find the maximum scaling  
\[\eta_{k}^{*}=\max_{j\in\{1,2\}}\left\{\left(\min_{i\in\{1,\ldots,n\}}\left\{ \lambda_{i}\left(\left(\zeta_{j,k}^{2}\cdot\mathbf{Q}_{j,k}\right)^{-1}\cdot \mathbf{Q}_{k}^{\mathrm{I}}\right)\right\}\right)^{-\frac{1}{2}}\right\} \tag{61}\]  
so that all outer bounds of the thick ellipsoids in (46) are guaranteed to be contained in the union. Figure 4: Intersection of an ellipsoid \(\mathcal{E}_{1,k}\) with the degenerate ellipsoid \(\mathcal{E}_{2,k}\) (vertical strip) with identical and different midpoints. Inner bound of the resulting thick ellipsoid: \(\mathcal{E}_{\mathrm{D},k}^{\mathrm{I}}\); outer bound: \(\mathcal{E}_{\mathrm{D},k}^{\mathrm{O}}\) (\(*\) symbols denote the solution enclosure according to Equation (53)). (**a**) Case according to Section 3.2.1 with \(\mathbf{Q}_{2,k}\) according to (55). (**b**) Case according to Section 3.2.2 with the initial ellipsoid parameterization (56), (57), enclosure definition (49). (**c**) Case according to Section 3.2.2 with the initial ellipsoid parameterization (56), (57), enclosure definition (53). **Remark 10**. : _This inflation operation generalizes naturally to an arbitrary integer number \(j\) of ellipsoids to be merged after they have been converted into representations with a common midpoint._  
As an alternative parameterization (similar to the computation of the intersection of ellipsoids), both outer and inner bounds can be tuned by an alternative option. For that purpose, firstly define an approximation to the resulting shape matrix of the union according to the covariance prediction of a Kalman filter, where the state equation in the Kalman filter corresponds to computing the mean value of the data from each of the ellipsoids, with  
\[\mathbf{\tilde{Q}}_{k}=\frac{1}{4}\cdot\left(\mathbf{Q}_{1,k}+\mathbf{Q}_{2,k} \right)\enspace. \tag{62}\]  
Then, the scaling factors according to (49) and (53) yield the shape matrices  
\[\mathbf{Q}_{k}^{\mathrm{O},*}=\mathbf{\tilde{Q}}_{k}\cdot\left(\eta_{k}^{ \mathrm{O}}\right)^{2},\enspace\eta_{k}^{\mathrm{O}}=\max_{j\in\{1,2\}}\left\{ \left(\min_{i\in\{1,\ldots,n\}}\left\{\lambda_{i}\left(\left(\xi_{j,k}^{2} \cdot\mathbf{Q}_{jk}\right)^{-1}\cdot\mathbf{\tilde{Q}}_{k}\right)\right\} \right)^{-\frac{1}{2}}\right\} \tag{63}\]  
for the outer bound as well as  
\[\mathbf{Q}_{k}^{\mathrm{L}*}=\mathbf{\tilde{Q}}_{k}\cdot\left(\eta_{k}^{ \mathrm{I}}\right)^{2},\enspace\eta_{k}^{\mathrm{I}}=\left(\max_{i\in\{1, \ldots,n\}}\left\{\lambda_{i}\left(\left(\mathbf{Q}_{k}^{\mathrm{I}}\right)^{ -1}\cdot\mathbf{\tilde{Q}}_{k}\right)\right\}\right)^{-\frac{1}{2}} \tag{64}\]  
for the inner bound.