Example 1. Suppose the moving point $P(x, y)$ is equidistant from the fixed line $x=$ - $\frac{p}{2}$ and the point $F\left(\frac{p}{2}, 0\right)$, find the parametric equation of the trajectory of point $P$.
Solution: Move the origin of coordinates to point $F$, and then rotate the axes by an angle $\theta$, so that the new horizontal axis passes through $P$ (Figure 1). Let the new coordinates of point $P$ be $\left(x^{\prime}, 0\right)$, then
$$
\left\{\begin{array}{l}
x=\frac{p}{2}+x^{\prime} \cos \theta, \\
y=x^{\prime} \sin \theta .
\end{array}\right.
$$

But $x^{\prime}=\frac{p}{2}+x$, so $x=\frac{p}{2}+\left(\frac{p}{2}+x\right) \cos \theta$, which gives $x=\frac{p}{2}+\frac{p}{2} \cos \theta+x \cos \theta$, thus
$$
x=\frac{p}{2} \cdot \frac{1+\cos \theta}{1-\cos \theta}=\frac{p}{2} \operatorname{ctg}^{2} \frac{\theta}{2} .
$$

Similarly, we get $y=p \operatorname{ctg} \frac{\theta}{2}$.