## 4 Computational Results  
Computational experiments were run on an AMD Athlon 64 X2 5200+ computer. Local optimization was performed by the IPOPT solver [29], which is available at an open access noncommercial software depository ([[https://projects.coin-or.org/Ipopt](https://projects.coin-or.org/Ipopt)]([https://projects.coin-or.org/Ipopt](https://projects.coin-or.org/Ipopt))). To illustrate the algorithm performance, examples for ellipses and spheroids are considered. For each problem instance, the computational time required to construct the approximated cover is less than a second. Figure 4: Illustration of Stage 2. **Instance 1**. : _The ellipse E with semi-axes a \(=\)1.3, b =1 is given._  
1. _For_ \(\varepsilon=0.3\)_, the solution with one covering circle is presented in Figure_ 5_a:_ \[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]
2. _For_ \(\varepsilon=0.2\)_, the solution with an even number of covering circles is shown in Figure_ 5_b:_ \[\Lambda(\varepsilon^{\star},M^{\star})=\overset{2}{\underset{k=1}{\cup}}C_{k}( x_{k},0,0),\ \varepsilon^{\star}=0.087477,\ M^{\star}=2,\] \[\{x_{k},k=1,2\}=\{0.265385,-0.265385\},\text{and}\ \{r_{k},k=1,2\}=\{1.034615,\,1.034615\}.\]
3. _For_ \(\varepsilon=0.05\)_, the solution with an odd number of covering circles is given in Figure_ 5_c:_ \[\Lambda(\varepsilon^{\star},M^{\star})=\overset{3}{\underset{k=1}{\cup}}C_{k}( x_{k},0,0),\ \varepsilon^{\star}=0.040356,\ M^{\star}=3,\] \[\{x_{k},k=1,2,3\}=\{0.357061,\,-0.357061,\,0.000\},\ \{r_{k},k=1,2,3\}=\{0.942939,\,0.942939,\,1.040356\}.\]  
**Instance 2**. : _The ellipse E with semi-axes a \(=\)2.3, b =1 is given and \(\varepsilon=0.1\)._  
1. _The optimized cover with an even number of circles is presented in Figure_ 6_a:_ \[\Lambda(\varepsilon^{\star},M^{\star})=\overset{6}{\underset{k=1}{\cup}}C_{k}( x_{k},0,0),\ \varepsilon^{\star}=0.072085,\ M^{\star}=6,\] \[\{x_{k},k=1,\ldots,6\}=\{1.596217,\,-1.596217,\,1.008362,\,-1.008362,\,0.3447 54,\,-0.344754\},\] \[\{r_{k},k=1,\ldots,6\}=\{0.703783,\,0.703783,\,0.942524,\,0.942524\,1.057759, \,1.057759\}.\]
2. _The optimized cover with an odd number of circles is shown in Figure_ 6_b:_ \[\Lambda(\varepsilon^{\star},M^{\star})=\overset{7}{\underset{k=1}{\cup}}C_{k}( x_{k},0,0),\ \varepsilon^{\star}=0.05372,\ M^{\star}=7,\] \[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]0 \[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]1  
\[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]2 The optimized cover of the ellipse E: (a) by \(M^{\star}=1\) circle; (b) by \(M^{\star}=2\) circles; and (c) by \(M^{\star}=3\) circles. **Instance 3**. : _The spheroid \(\mathbf{E}\) with semi-axes \(a=\)1.9, \(b=c=\)1 is given and \(\varepsilon=0.1\)._  
1. _The optimized cover with an even number of spheres is presented in Figure 7a:_ \[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]3 \[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]4 \[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]5
2. _The optimized cover (for a =2) with an odd number of spheres is shown in Figure 7b:_ \[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]6 \[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]7 \[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]8  
\[\Lambda(\varepsilon^{\star},M^{\star})=C_{1}(x_{1}=0,0,0),\ \varepsilon^{\star}=0.3,\ M^{\star}=1.\]9 The optimized cover of the ellipse E: (**a**) by \(M^{*}=6\) circles; (**b**) by \(M^{*}=7\) circles. **Instance 4**.