Example 5 On a piece of paper, there is a circle $\odot O$ with radius $R$ and a fixed point $A$ inside the circle, and $O A=a$. Fold the paper so that a point $A^{\prime}$ on the circumference coincides exactly with point $A$. Each such fold leaves a straight line crease. When $A^{\prime}$ takes all points on the circumference, find the set of points on all the crease lines.
(2003 National High School Mathematics Competition)
As shown in Figure 11-3, establish a rectangular coordinate system, then we have $A(a, 0)$. Connect $O A^{\prime}$ to intersect the crease $l$ at point $M$, and connect $M A$. Then $|M A|=\left|M A^{\prime}\right|,|O M|+|M A|=$ $\mid\left(M|+| M A^{\prime} \mid=R\right.$. By the definition of an ellipse, the locus of point $M$ is an ellipse with foci at $(0,0)$ and $A(a, 0)$, and the length of the major axis is $R$. Its equation is:
$$
\frac{\left(x-\frac{a}{2}\right)^{2}}{\left(\frac{R}{2}\right)^{2}}+\frac{y^{2}}{\left(\frac{R}{2}\right)^{2}-\left(\frac{a}{2}\right)^{2}}=1 .
$$

For any point $P$ on the crease $l$ other than $M$, we have
$$
|P O|+|P A|=|P O|+\left|P A^{\prime}\right|>O A^{\prime}=R,
$$

Thus, $P$ is outside the ellipse. Therefore, the set of all points on the lines of the creases is the set of points outside (including on) the ellipse.