A cylinder is intersected by a plane inclined at an angle of $45^{\circ }$ to its axis. What is the shape of the resulting cross-section?

Let's consider the cross-sections formed by cutting a right circular cylinder with a plane:
1. If the cutting plane is perpendicular to the cylinder's central longitudinal axis ($0^{\circ }$ orientation change), the cross-section is a perfect circle.
2. If the cutting plane is parallel to the axis, the cross-section forms a rectangle or pair of parallel straight lines.
3. If the cutting plane passes completely through all sides of the cylinder at an oblique inclination angle (other than $90^{\circ }$ or $0^{\circ }$ relative to the base, such as $45^{\circ }$ to its axis) without intersecting the flat bases, the resulting bounded profile forms an elongated closed curve called an ellipse.

Therefore, a cross-section tilted at $45^{\circ }$ to the axis results in an ellipse.