## 1 Deriving \(r_{k}\)  
Observe that the point \(\left(x(s,t)=\frac{\cos t}{a_{\varepsilon}}(a_{\varepsilon}^{2}-b_{ \varepsilon}^{2}+s\cdot b_{\varepsilon}),y(s,t)=s\sin t\right)\) belongs to the coordinate axis \(OX\) for \(s=0\), while the point \((x^{\prime}=a_{\varepsilon}\cos t,y^{\prime}=b_{\varepsilon}\sin t)\) belongs to the frontier of the ellipse. Since \(x=\frac{\cos t}{a_{\varepsilon}^{2}-b_{\varepsilon}^{2}},\) then  
\(\cos t=\frac{a_{\varepsilon}x}{a_{\varepsilon}^{2}-b_{\varepsilon}^{2}},\)\(\sin t=\sqrt{1-\frac{a_{\varepsilon}^{2}x^{2}}{(a_{\varepsilon}^{2}-b_{ \varepsilon}^{2})^{2}}}\) for \(a_{\varepsilon}>b_{\varepsilon}\) and  
\(1-\frac{a_{\varepsilon}^{2}x^{2}}{(a_{\varepsilon}^{2}-b_{\varepsilon}^{2})^{2 }}\geq 0,\)\((a_{\varepsilon}^{2}-b_{\varepsilon}^{2})^{2}-a_{\varepsilon}^{2}x^{2}\geq 0,\)\(x^{2}\leq\frac{(a_{\varepsilon}^{2}-b_{\varepsilon}^{2})^{2}}{a_{\varepsilon}^ {2}},\)\(x\leq\frac{(a_{\varepsilon}^{2}-b_{\varepsilon}^{2})}{a_{\varepsilon}},\)\(x\leq a_{\varepsilon}-\frac{b_{\varepsilon}^{2}}{a_{\varepsilon}}.\)  
The tangent point \((x^{\prime},y^{\prime}),\)\(x^{\prime}=\frac{a_{\varepsilon}^{2}x}{a_{\varepsilon}^{2}-b_{ \varepsilon}^{2}},\)\(y^{\prime}=b\sqrt{1-\frac{a_{\varepsilon}^{2}x^{2}}{(a_{ \varepsilon}^{2}-b_{\varepsilon}^{2})^{2}}}\) belongs to the frontier of the ellipse. Figure 9: The optimized cover of the spheroid \(\mathbf{E}\) by \(M^{*}=21\) spheres. Since \(r^{2}=(x-x^{\prime})^{2}+(0-y^{\prime})^{2}\), then \(r^{2}=\left(x-\frac{a_{x}^{2}x}{a_{x}^{2}-b_{x}^{2}}\right)^{2}+y^{\prime 2}\) and thus \(r^{2}=\left(x-\frac{a_{x}^{2}x}{a_{x}^{2}-b_{x}^{2}}\right)^{2}+b_{x}^{2}\! \left(1-\frac{a_{x}^{2}x^{2}}{\left(a_{x}^{2}-b_{x}^{2}\right)^{2}}\right)=b_{x }^{2}\!\left(\!\frac{-(a_{x}^{2}-b_{x}^{2})^{2}+(a_{x}^{2}-b_{x}^{2})^{2}}{(a_{ x}^{2}-b_{x}^{2})^{2}}\!\right)=b_{x}^{2}\!\left(1-\frac{x^{2}}{a_{x}^{2}-b_{x} ^{2}}\right)\). Here we used \(\left(\frac{x(a_{x}^{2}-b_{x}^{2})}{a_{x}^{2}-b_{x}^{2}}-\frac{a_{x}^{2}x}{a_{ x}^{2}-b_{x}^{2}}\right)^{2}=\left(\frac{-b_{x}^{2}x}{a_{x}^{2}-b_{x}^{2}} \right)^{2}=\left(\frac{b_{x}^{2}x}{a_{x}^{2}-b_{x}^{2}}\right)^{2}=\frac{b_{ x}^{4}x^{2}}{(a_{x}^{2}-b_{x}^{2})^{2}}\). Therefore \(r_{k}=b_{x}\sqrt{1-\frac{x_{k}^{2}}{a_{x}^{2}-b_{x}^{2}}}\) at the \(k\)-th iteration. 2. **Deriving \({p^{\prime}}_{k}=({p^{\prime}}_{kx},{p^{\prime}}_{ky})\)**. The tangent point \({p^{\prime}}_{k}=({p^{\prime}}_{kx},{p^{\prime}}_{ky})\) is defined from the system \(\left\{ \begin{array}{l}\frac{p_{kx}^{2}}{a^{2}}+\frac{p_{xy}^{2}}{b^{2}}=1\\ \left(p_{kx}-x_{k}\right)^{2}+p_{ky}^{2}=r_{k}^{2}\end{array} \right.\) Therefore \(p_{ky}^{2}=r_{k}^{2}-(p_{kx}-x_{k})^{2}\), \({b^{2}}p_{kx}^{2}+a^{2}(r_{k}^{2}-(p_{kx}-x_{k})^{2})=a^{2}b^{2}\), \((b^{2}-a^{2})p_{kx}^{2}+2a^{2}x_{k}p_{kx}+a^{2}(r_{k}^{2}-x_{k}^{2}-b^{2})=0\) and thus \({p^{\prime}}_{kx}=\frac{-a^{2}x_{k}a}{(a^{2}x_{k}^{2}-(b^{2}-a^{2})(r_{k}^{2} -x_{k}^{2}-b^{2})}{(b^{2}-a^{2})},{p^{\prime}}_{ky}=\sqrt{r_{k}^{2}-(p_{kx}-x_ {k})^{2}}\). 3. **Deriving \(x_{k+1}\)**. From the system at Step2 we have \(\left\{ \begin{array}{l}\left(x_{k+1}-p_{kx}\right)^{2}+p_{ky}^{2}=r_{k}^{2} \\ b_{x}\sqrt{1-\frac{x_{k+1}^{2}}{a_{x}^{2}-b_{x}^{2}}}=r_{k}\end{array} \right..\) Therefore \((x_{k+1}-p_{kx})^{2}+p_{ky}^{2}=b_{x}^{2}\!\left(1-\frac{x_{k+1}^{2}}  
* _[PERSON] et al. (2016)_ [PERSON]; [PERSON] The raspberry model for protein-like particles: Ellipsoids and confinement in cylindrical pores. _Eur. Phys. J. Spec. Top._**2016**, _225_, 1643-1662. [CrossRef]
* _[PERSON] et al. (2019)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] Effect of Porosity on Mechanical Properties of 3D Printed Polymers: Experiments and Micromechanical Modeling Based on X-ray Computed Tomography Analysis. _Polymers_**2019**, _11_, 1154. [CrossRef]
* _[PERSON] et al. (2020)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] An experimental study of packing of ellipsoids under vibrations. _Powder Technol._**2020**, _361_, 45-51.