13. 1.25 $\star \star$ Given the radius of the base of a cylinder is $r$, and the height is $h$, a section is made through a point on the top base, with an angle $\alpha$ to the base, where $0^{\circ}<\alpha<90^{\circ}$. Try to find the equation of the curve formed by the section when the side of the cube between the section and the base is unfolded into a plane.
First, assume the height of the cylinder is sufficiently large, so that the section forms a complete ellipse. Let the section intersect the axis of the cylinder at point \( S \). The lowest and highest points of the section are \( P \) and \( Q \), respectively.

Figure a
Figure b
Figure c

Cut the cylindrical surface along the generatrix passing through point \( P \) and unfold it into a plane. On this plane, take the generatrix \( l \) as the \( y \)-axis, and the point \( O \) on \( l \) at the same height as \( S \) as the origin, and establish a Cartesian coordinate system. Let \( A \) be any point on the section. Draw a plane \( M \) parallel to the base through \( S \), and draw a plane \( N \) perpendicular to \( M \) through \( A \). Plane \( N \) intersects plane \( M \) at \( A' B' \), and intersects plane \( QOS \) at \( BB' \) (as shown in Figure c). \( A' \) and \( B' \) are the projections of \( A \) and \( B \) on plane \( M \), respectively. It is easy to know that \( A'A = B'B = SB' \cdot \tan \alpha = SA' \cdot \cos \left(\pi - \angle OSA'\right) \cdot \tan \alpha = r \tan \alpha \cdot \sin \left(\frac{x}{r} - \frac{\pi}{2}\right) \). Here, \( x \) is the length of the arc \( \overparen{OA} \), which is the horizontal coordinate of \( A \) in the unfolded diagram, and \( A'A \) is the vertical coordinate of \( A \) in the unfolded diagram. Therefore, the equation of the section curve is
\[
y = r \tan \alpha \cdot \sin \left(\frac{x}{r} - \frac{\pi}{2}\right) \quad (0 < x < 2\pi r)
\]

The graph is a sine curve (one complete period). For the general case, draw the lines \( y - h_1 \) and \( y = -h_2 \), where \( h_1 \) and \( h_2 \) are the distances from \( S \) to the top and bottom bases, respectively \((h_1 + h_2 = h)\). The section curve is the part of (1) between these two lines.