What are some ways of finding the center of curvature of concave and convex mirrors (spherical)?
For example, I can double the distance of focal length on the principle axis to find the center of curvature of concave mirror.
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Yes, the radius of curvature is twice the focal length from the pole (A in upper diagram, P in lower diagram) : $R=2f$.
A concave mirror used for focusing light is parabolic, not spherical. There therefore isn't a "center of curvature".
If you know the object distance and image distance, you can solve for the focal length f using the mirror equation. Then you will also know the center of curvature since it is 2 times f.