What role does the vertex play in the definition of the radius of curvature?

The vertex is crucial in defining the radius of curvature because it is the specific point on a curved surface (like a spherical mirror or lens) where the radius of curvature is measured. The radius of curvature is the distance from the vertex to the center of curvature, which is the center of the curved shape. By defining the radius in this way, we can establish consistent measurements that help in understanding how the curved surface interacts with light, such as how it reflects or refracts light rays. This concept is fundamental in optics as it affects the focal length and other important properties of lenses and mirrors.