## 3 Solution Algorithm
### Solution Algorithm for Stage 1  
Let the value of \(\varepsilon>0\) be given.  
At Stage 1, the objective is to find the minimum number of circles \(C_{k}(x_{k},0)\), \(k=0,1,\ldots,2m\) covering the ellipse \(E\) centered at \((0,0)\) for the given value of \(\varepsilon\).  
**Step 1**. Set \(r_{0}=b+\varepsilon\), \(x_{0}=0\), \(k=0\), \(a_{\varepsilon}=a+\varepsilon\), \(b_{\varepsilon}=b+\varepsilon\), where \(a_{\varepsilon}\), \(b_{\varepsilon}\) are the sizes of the ellipse \(E^{\varepsilon}\).  
**Step 2**. Get the intersection point \({p^{\prime}}_{k}=({p^{\prime}}_{kx},{p^{\prime}}_{ky})\) of the ellipse \(E\) and the circle \(C_{k}\) solving the following system:  
\[\left\{\begin{array}{l}\left(x_{k}-p_{kx}\right)^{2}+p_{ky}^{2}=r_{k}^{2},\\ \frac{p_{kx}^{2}}{a^{2}}+\frac{p_{ky}^{2}}{b^{2}}=1,p_{kx}\geq 0,p_{ky}\geq 0 \end{array}\right.,\]  
where \(r_{k}\) is the radius of the circle \(C_{k}\) centered at the point \((x_{k},0)\).  
Using the parametric description of the ellipse [28], the point \({p^{\prime}}_{k}=({p^{\prime}}_{kx},{p^{\prime}}_{ky})\) is presented analytically by the following formula (see the Appendix A for details):  
\[{p^{\prime}}_{kx}=\frac{a^{2}x_{k}+a\sqrt{a^{2}{x_{k}}^{2}-(b^{2}-a^{2})(r_{k }^{2}-{x_{k}}^{2}-b^{2})}}{a^{2}-b^{2}},\;{p^{\prime}}_{ky}=\sqrt{r_{k}^{2}- \left(x_{k}-{p^{\prime}}_{kx}\right)^{2}}.\]  
**Step 3**. Set \(k=k+1\).  
**Step 4**. Solve the following nonlinear programming problem:  
\[t_{k}^{\star}=\min_{(r_{k},r_{k})\in V}t_{k}\]  
and get its optimal solution \((r_{k}^{\star},t_{k}^{\star})\). Here, the feasible set \(V\) is defined by the following system of inequalities:  
\[r_{k}^{2}-(\frac{a_{\varepsilon}}{\cos{t_{k}}}\cdot({a_{\varepsilon}}^{2}-{b_ {\varepsilon}}^{2})-{p_{(k-1)x}}^{2})^{2}-{p_{(k-1)y}^{2}}\geq 0, \tag{5}\]  
\[\frac{b_{\varepsilon}}{a_{\varepsilon}}\cdot\sqrt{{b_{\varepsilon}}^{2}\cos^ {2}{t_{k}}+{a_{\varepsilon}}^{2}\sin^{2}{t_{k}}}-r_{k}\geq 0, \tag{6}\]  
\[t_{k}\leq t_{k-1}^{\star}, \tag{7}\]  
\[r_{k}>0, \tag{8}\]  
where inequality (5) assures that the point \(p_{k-1}\) is inside the circle \(C_{k}\) of radius \(r_{k}\) centered at \((x_{k},0)\); inequality (6) guarantees that the circle \(C_{k}\) of radius \(r_{k}\) centered at \((x_{k},0)\) is inside the ellipse \(E^{\varepsilon}\) with semi-axes \(a_{\varepsilon}=a+\varepsilon\), \(b_{\varepsilon}=b+\varepsilon\); inequality (7) reflects the monotonous decrease of the corresponding angle \(t_{k}\); and the last inequality describes the natural constraint for the radius of the circle \(C_{k}\). Find  
Figure 1: The center of the first approximating circle: (**a**) for odd circular covering and (**b**) for even circular covering.  
\(x_{k}=\frac{a_{\varepsilon}}{\cos r_{k}^{2}}\cdot({a_{\varepsilon}}^{2}-{b_{ \varepsilon}}^{2})\). Using the parametric description of the ellipse [28], the parameters \(x_{k}\) and \(r_{k}\) are presented analytically in the following form (see the Appendix A for details):  
\[x_{k}=\frac{(a_{\varepsilon}^{2}-b_{\varepsilon}^{2})p_{xx}+ \sqrt{(a_{\varepsilon}^{2}-b_{\varepsilon}^{2})(a_{\varepsilon}^{2}(b_{ \varepsilon}^{2}-p_{xy}^{2})-p_{xx}^{2}b_{\varepsilon}^{2})}}{a_{\varepsilon }^{2}},\] \[\text{if }x_{k}\leq\frac{a_{\varepsilon}^{2}-b_{\varepsilon}^{2}}{a _{\varepsilon}}\text{ then }r_{k}=b_{\varepsilon}\sqrt{1-\frac{x_{k}^{2}}{a_{ \varepsilon}^{2}-b_{\varepsilon}^{2}}},\text{ otherwise }r_{k}=a_{\varepsilon}-x_{k}.\]  
**Step 5.** If the point \((a,0)\) is inside the circle \(C_{k}\) of radius \(r_{k}\) centered at \((x_{k},0)\), then the solution is obtained and the algorithm stops. Otherwise, go to Step 2.  
Figures 2 and 3 illustrate an iterative procedure for the algorithm described above.